The Geometry of Myller Configurations. Applications to Theory of Surfaces and Nonholonomic Manifolds
aa r X i v : . [ m a t h . DG ] F e b RADU MIRON
THE GEOMETRY OF MYLLERCONFIGURATIONS.Applications to Theory of Surfaces andNonholonomic Manifolds
Editura Academiei Romˆane
Bucure¸sti, 2010
Dedicated to academicians
Alexandru Myller and
Octav Mayer ,two great Romanian mathematicians
MOTTO
It is possible to bring for the great disappearedscientists varied homages. Sometimes, the strongwind of progress erases the trace of their steps. Tonot forget them means to continue their works,connecting them to the living present.
Octav Mayer reface
In 2010, the Mathematical Seminar of the “Alexandru Ioan Cuza” Uni-versity of Ia¸si comes to its 100th anniversary of prodigious existence.The establishing of the Mathematical Seminar by Alexandru Mylleralso marked the beginning of the School of Geometry in Ia¸si, developed intime by prestigious mathematicians of international fame, Octav Mayer,Gheorghe Vrˆanceanu, Grigore Moisil, Mendel Haimovici, Ilie Popa, Dim-itrie Mangeron, Gheorghe Gheorghiev and many others.Among the first paper works, published by Al. Myller and O. Mayer,those concerning the generalization of the Levi-Civita parallelism must bespecified, because of the relevance for the international recognition of theacademic School of Ia¸si. Later on, through the effort of two generationsof Ia¸si based mathematicians, these led to a body of theories in the areaof differential geometry of Euclidian, affine and projective spaces.At the half-centenary of the Mathematical Seminary, in 1960, the au-thor of the present opuscule synthesized the field ′ s results and laid it outin the form of a “whole, superb theory”, as mentioned by Al. Myller. Inthe same time period, the book The geometry of the Myller configura-tions was published by the Technical Publishing House. It represents, asmentioned by Octav Mayer in the Foreword, “the most precious tributeever offered to Alexandru Myller”. Nowadays, at the 100th anniversaryof the Mathematical Seminary “Alexandru Myller” and 150 years afterthe enactment, made by Alexandru Ioan Cuza, to set up the Universitythat carries his name, we are going to pay homage to those two historicalacts in the Romanian culture, through the publishing, in English, of the volume
The Geometry of the Myller Configurations , completed with anample chapter, containing new results of the Romanian geometers, re-garding applications in the studying of non-holonomic manifolds in theEuclidean space. On this occasion, one can better notice the undeniablevalue of the achievements in the field made by some great Romanianmathematicians, such as Al. Myller, O. Mayer, Gh. Vr˘anceanu, Gr.Moisil, M. Haimovici, I. Popa, I. Creang˘a and Gh. Gheorghiev.The initiative for the re-publishing of the book belongs to the leader-ship of the “Alexandru Ioan Cuza” University of Ia¸si, to the local branchof the Romanian Academy, as well as NGO Formare Studia Ia¸si. A com-petent assistance was offered to me, in order to complete this work, by myformer students and present collaborators, Professors Mihai Anastasieiand Ioan Buc˘ataru, to whom I express my utmost gratitude.Ia¸si, 2010 Acad. Radu Miron reface of the book
Geometriaconfigurat¸iilor Myl ler written in1966 by Octav Mayer, formermember of the RomanianAcademy (Translated from Romanian)
One can bring to the great scientists, who passed away, various homages.Some times, the strong wind of the progress wipes out the trace of theirsteps. To not forget them means to continue their work, connectingthem to the living present. In this sense, this scientific work is the mostprecious homage which can be dedicated to Alexandru Myller.Initiator of a modern education of Mathematics at the Universityof Iassy, founder of the Geometry School, which is still flourishing to-day, in the third generation, Alexandru Myller was also a hardworkingresearcher, well known inside the country and also abroad due to hispapers concerning Integral Equations and Differential Geometry.Our Academy elected him as a member and published his scientificwork. The “A. Humboldt” University from Berlin awarded him the titleof “doctor Honoris Causa” for “special efforts in creating an independentRomanian Mathematical School”.Some of Alexandru Myller ′ s discoveries have penetrated fruitfully theimpetuous torrent of ideas, which have changed the Science of Geometry during the first decades of the century. Among others, it is the case of“Myller configurations”. The reader would be perhaps interested to findout some more details about how he discovered these configurations.Let us imagine a surface S , on which there is drawn a curve C and,along it, let us circumscribe to S a developing surface Σ; then we ap-ply (develop) the surface Σ on a plane π . The points of the curve C ,connected to the respective tangent planes, which after are developedoverlap each other on the plane π , are going to represent a curve C ′ intothis plane. In the same way, a series ( d ) of tangent directions to thesurface S at the points of the curve C becomes developing, on the plane π , a series ( d ′ ) of directions getting out from the points of the curve C ′ .The directions ( d ) are parallel on the surface S if the directions ( d ′ ) areparallel in the common sense.Going from this definition of T. Levi-Civita parallelism (valid in theEuclidian space), Alexandru Myller arrived to a more general conceptin a sensible process of abstraction. Of course, it was not possible toleave the curve C aside, neither the directions ( d ) whose parallelism wasgoing to be defined in a more general sense. What was left aside was thesurface S .For the surface Σ was considered, in a natural way, the enveloping ofthe family of planes constructed from the points of the curve C , planesin which are given the directions ( d ). Keeping unchanged the remainderof the definition one gets to what Alexandru Myller called “parallelisminto a family of planes”.A curve C , together with a family of planes on its points and a familyof given directions (in an arbitrary way) in these planes constitutes whatthe author called a “Myller configuration”.It was considered that this new introduced notion had a central placeinto the classical theory of surfaces, giving the possibility to interpret andlink among them many particular facts. It was obvious that this notioncan be successfully applied in the Geometry of other spaces different fromthe Euclidian one. Therefore the foreworded study worthed all the effortsmade by the author, a valuable mathematician from the third generation of the Ia¸si Geometry School.By recommending this work, we believe that the reader (who needsonly basic Differential Geometry) will be attracted by the clear lectureof Radu Miron and also by the beauty of the subject, which can still bedeveloped further on. Octav Mayer, 1966 short biography ofAl. Myller ALEXANDRU MYLLER was born in Bucharest in 1879, and diedin Ia¸si on the 4th of July 1965. Romanian mathematician. HonoraryMember (27 May 1938) and Honorific Member (12 August 1948) of theRomanian Academy. High School and University studies (Faculty ofScience) in Bucharest, finished with a bachelor degree in mathematics.He was a Professor of Mathematics at the Pedagogical Seminar; hesustained his PhD thesis
Gewohnliche Differential Gleichungen HohererOrdnung , in G¨ottingen, under the scientific supervision of David Hilbert.He worked as a Professor at the Pedagogical Seminar and the School of Post and Telegraphy (1907 - 1908), then as a lecturer at the Univer-sity of Bucharest (1908 - 1910). He is appointed Professor of AnalyticalGeometry at the University of Ia¸si (1910 - 1947); from 1947 onward - con-sultant Professor. In 1910, he sets up the Mathematical Seminar at the“Alexandru Ioan Cuza“ University of Ia¸si, which he endows with a libraryfull of didactic works and specialty journals, and which, nowadays, bearshis name. Creator of the mathematical school of Ia¸si, through whichmany reputable Romanian mathematicians have passed, he conveyed tous research works in fields such as integral equations, differential geom-etry and history of mathematics. He started his scientific activity withpapers on the integral equations theory, including extensions of Hilbert ′ sresults, and then he studied integral equations and self- adjoint of evenand odd order linear differential equations, being the first mathematicianto introduce integral equations with skew symmetric kernel.He was the first to apply integral equations to solve problems forpartial differential equations of hyperbolic type. He was also interestedin the differential geometry, discovering a generalization of the notion ofparallelism in the Levi-Civita sense, and introducing the notion todayknown as concurrence in the Myller sense. All these led to “the differ-ential geometry of Myller configurations”(R. Miron, 1960). Al. Myller,Gh. T¸ it¸eica and O. Mayer have created “the differential centro-affinegeometry“, which the history of mathematics refers to as “purely Ro-manian creation!“ He was also concerned with problems from the geom-etry of curves and surfaces in Euclidian spaces. His research outcomescan be found in the numerous memoirs, papers and studies, publishedin Ia¸si and abroad: Development of an arbitrary function after Bessel ′ sfunctions (1909); Parallelism in the Levi-Civita sense in a plane system(1924); The differential centro-affine geometry of the plane curves (1933)etc. - within the volume “Mathematical Writings“ (1959), Academy Pub-lishing House. Alexandru Myller has been an Honorary Member of theRomanian Academy since 1938. Also, he was a Doctor Honoris Causa ofthe Humboldt University of Berlin.Excerpt from the volume “Members of the Romanian Academy 1866 - 1999. Dictionary.“ - Dorina N. Rusu, page 360. ntroduction In the differential geometry of curves in the Euclidean space E oneintroduces, along a curve C , some versor fields, as tangent, principalnormal or binormal, as well as some plane fields as osculating, normal orrectifying planes.More generally, we can consider a versor field ( C, ξ ) or a plane field(
C, π ). A pair { ( C, ξ ) , ( C, π ) } for which ξ ∈ π , has been called in 1960 [23]by the present author, a Myller configuration in the space E , denoted by M ( C, ξ, π ). When the planes π are tangent to C then we have a tangentMyller configuration M t ( C, ξ, π ).Academician Alexandru Myller studied in 1922 the notion of paral-lelism of (
C, ξ ) in the plane field (
C, π ) obtaining an interesting general-ization of the famous parallelism of Levi-Civita on the curved surfaces.These investigations have been continued by Octav Mayer which intro-duced new fundamental invariants for M ( C, ξ, π ). The importance ofthese studies was underlined by Levi Civita in Addendum to his book
Lezioni di calcolo differentiale assoluto , 1925.Now, we try to make a systematic presentation of the geometry ofMyller configurations M ( C, ξ, π ) and M t ( C, ξ, π ) with applications to thedifferential geometry of surfaces and to the geometry of nonholonomicmanifolds in the Euclidean space E .Indeed, if C is a curve on the surface S ⊂ E , s is the natural param-eter of curve C and ξ ( s ) is a tangent versor field to S along C and π ( s ) istangent planes field to S along C , we have a tangent Myller configuration M t ( C, ξ, π ) intrinsic associated to the geometric objects
S, C, ξ . Conse- quently, the geometry of the field ( C, ξ ) on surface S is the geometry ofthe associated Myller configurations M t ( C, ξ, π ). It is remarkable thatthe geometric theory of M t is a particular case of that of general Myllerconfiguration M ( C, ξ, π ).For a Myller configuration M ( C, ξ, π ) we determine a Darboux frame,the fundamental equations and a complete system of invariants
G, K, T called, geodesic curvature, normal curvature and geodesic torsion, re-spectively, of the versor field (
C, ξ ) in Myller configuration M ( C, ξ, π ).A fundamental theorem, when the functions ( G ( s ) , K ( s ) , T ( s )) are given,can be proven.The invariant G ( s ) was discovered by Al. Myller (and named by himthe deviation of parallelism). G ( s ) = 0 on curve C characterizes theparallelism of versor field ( C, ξ ) in M , [23], [24], [31], [32], [33]. Thesecond invariant K ( s ) was introduced by O. Mayer (it was called thecurvature of parallelism). Third invariant T ( s ) was found by E. Bortolotti[3]. In the particular case, when M is a tangent Myller configuration M t ( C, ξ, π ) associated to a tangent versor field (
C, ξ ) on a surface S , the G ( s ) is an intrinsic invariant and G ( s ) = 0 along C leads to the LeviCivita parallelism.In configurations M t ( C, ξ, π ) there exists a natural versor field (
C, α ),where α ( s ) are the tangent versors to curve C . The versor field ( C, α )has a Darboux frame R = ( P ( s ); α, µ ∗ , ν ) in M t where ν ( s ) is normal toplane π ( s ) and µ ∗ ( s ) = ν ( s ) × α ( s ) . The moving equations of R are: drds = α ( s ) , s ∈ ( s , s ) dαds = κ g ( s ) µ ∗ + κ n ( s ) ν,dµ ∗ ds = − κ g ( s ) α + τ g ( s ) ν,dνds = − κ n ( s ) α − τ g ( s ) µ ∗ . The functions κ g ( s ) , κ n ( s ) and τ g ( s ) form a complete system of invariants of the curve in M t .A theorem of existence and uniqueness for the versor fields ( C, α ) in M t ( C, α, π ), when the invariant κ g ( s ), k n and τ g ( s ) are given, is proved.The function κ g ( s ) is called the geodesic curvature of the curve C in M t ; κ n ( s ) is the normal curvature and τ g ( s ) is the geodesic torsion of C in M t .The condition κ g ( s ) = 0, ∀ s ∈ ( s , s ) characterizes the geodesic(autoparallel lines) in M t ; κ n ( s ) = 0 , ∀ s ∈ ( s , s ) give us the asymptoticlines and τ g ( s ) = 0 , ∀ s ∈ ( s , s ) characterizes the curvature lines C in M t One can remark that in the case when M t is the associated Myllerconfiguration to a curve C on a surface S we obtain the classical theoryof curves on surface S . It is important to remark that the Mark Krien ′ sformula (2.10.3) leads to the integral formula (2.10.9) of Gauss-Bonnetfor surface S , studied by R. Miron in the book [62].Also, if C is a curve of a nonholonomic manifold E in E , we uniquelydetermine a Myller configuration M t ( C, α, π ) in which (
C, α ) is the tan-gent versor field to C and ( C, π ) is the tangent plane field to E along C , [54], [64], [65].In this case the geometry of M t is the geometry of curves C in the non-holonomic manifolds E . Some new notions can be introduced as: con-currence in Myller sense of versor fields ( C, ξ ) on E , extremal geodesictorsion, the mean torsion and total torsion of E at a point, a remark-able formula for geodesic torsion and an indicatrix of Bonnet for geodesictorsion, which is not reducible to a pair of equilateral hyperbolas, as inthe case of surfaces.The nonholonomic planes, nonholonomic spheres of Gr. Moisil, canbe studied by means of techniques from the geometry of Myller configu-rations M t .We finish the present introduction pointing out some important de-velopments of the geometry of Myller configurations:The author extended the notion of Myller configuration in Rieman-nian Geometry [24], [25], [26], [27]. Izu Vaisman [76] has studied the Myller configurations in the symplectic geometry.Mircea Craioveanu realized a nice theory of Myller configurations ininfinit dimensional Riemannian manifolds. Gheorghe Gheorghiev devel-oped the configuration M , [11], [55] in the geometry of versor fields inthe Euclidean space and applied it in hydromechanics.N.N. Mihalieanu studied the Myller configurations in Minkowski spaces[61]. For Myller configurations in a Finsler, Lagrange or Hamilton spaceswe refer to the paper [58] and to the books of R. Miron and M. Anastasiei[68], R. Miron, D. Hrimiuc, H. Shimada and S. Sab˘au [70].All these investigations underline the usefulness of geometry of Myllerconfigurations in differential geometry and its applications. hapter 1Versor fields. Plane fields in E First of all we investigate the geometry of a versor field (
C, ξ ) in theEuclidean space E , introducing an invariant frame of Frenet type, themoving equations of this frame, invariants and proving a fundamentaltheorem. The invariants are called K -curvature and K -torsion of ( C, ξ ).Geometric interpretations for K and K are pointed out. The parallelismof ( C, ξ ), concurrence of (
C, ξ ) and the enveloping of versor field (
C, ξ )are studied, too.A similar study is made for the plane fields (
C, π ) taking into accountthe normal versor field (
C, ν ), with ν ( s ) a normal versor to the plane π ( s ). ( C, ξ ) In the Euclidian space a versor field (
C, ξ ) can be analytical representedin an orthonormal frame R = (0; i , i , i ), by r = r ( s ) , ξ = ξ ( s ) , s ∈ ( s , s )(1.1.1) E where s is the arc length on curve C , r ( s ) = OP ( s ) = x ( s ) i + y ( s ) i + z ( s ) i and ξ ( s ) = ξ ( r ( s )) = −→ P Q = ξ ( s ) i + ξ ( s ) i + ξ ( s ) i , k ξ ( s ) k = h ξ ( s ) , ξ ( s ) i = 1 . All geometric objects considered in this book are assumed to be ofclass C k , k ≥
3, and sometimes of class C ∞ . The pair ( C, ξ ) has ageometrical meaning. It follows that the pair (cid:18)
C, dξds (cid:19) has a geometricmeaning, too. Therefore, the norm: K ( s ) = (cid:13)(cid:13)(cid:13)(cid:13) dξ ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) (1.1.2)is an invariant of the field ( C, ξ ).We denote ξ ( s ) = ξ ( s )(1.1.3)and let ξ ( s ) be the versor of vector dξ ds . Thus we can write dξ ( s ) ds = K ( s ) ξ ( s ) . Evidently, ξ ( s ) is orthogonal to ξ ( s ).It follows that the frame R F = ( P ( s ); ξ , ξ , ξ ) , ξ ( s ) = ξ × ξ (1.1.4)is orthonormal, positively oriented and has a geometrical meaning. R F is called the Frenet frame of the versor field ( C, ξ ).We have:
Theorem 1.1.1
The moving equations of the Frenet frame R F are: drds = a ( s ) ξ + a ( s ) ξ + a ( s ) ξ a ( s ) + a ( s ) + a ( s ) = 1(1.1.5) .1. Versor fields ( C, ξ ) 21 and dξ ds = K ( s ) ξ ,dξ ds = − K ( s ) ξ ( s ) + K ( s ) ξ , (1.1.6) dξ ds = − K ( s ) ξ ( s ) , where K ( s ) > . The functions K ( s ) , K ( s ) , a ( s ) , a ( s ) , a ( s ) , s ∈ ( s , s ) are invariants of the versor field ( C, ξ ) . The proof does not present difficulties.The invariant K ( s ) is called the curvature of ( C, ξ ) and has the samegeometric interpretation as the curvature of a curve in E . K ( s ) is calledthe torsion and has the same geometrical interpretation as the torsion ofa curve in E .The equations (1.1.5), (1.1.6) will be called the fundamental or Frenetequations of the versor field ( C, ξ ).In the case a ( s ) = 1 , a ( s ) = 0 , a ( s ) = 0 the tangent versor drds isdenoted by(1 . . ′ ) α ( s ) = drds ( s )The equations (1.1.5), (1.1.6) are then the Frenet equations of a curve inthe Euclidian space E .For the versor field ( C, ξ ) we can formulate a fundamental theorem:
Theorem 1.1.2
If the functions K ( s ) > , K ( s ) , a ( s ) , a ( s ) , a ( s ) , ( a + a + a = 1) of class C ∞ are apriori given, s ∈ [ a, b ] there existsa curve C : [ a, b ] → E parametrized by arclengths and a versor field ξ ( s ) , s ∈ [ a, b ] , whose the curvature, torsion and the functions a i ( s ) are K ( s ) , K ( s ) and a i ( s ) . Any two such versor fields ( C, ξ ) differ by aproper Euclidean motion. E For the proof one applies the same technique like in the proof ofTheorem 11, p. 45 from [75], [76].
Remark 1.1.1
1. If K ( s ) = 0 , s ∈ ( s , s ) the versors ξ ( s ) are parallelin E along the curve C.
2. The versor field (
C, ξ ) determines a ruled surface S ( C, ξ ).3. The surface S ( C, ξ ) is a cylinder iff the invariant K ( s ) vanishes.4. The surface S ( C, ξ ) is with director plane iff K ( s ) = 0.5. The surface S ( C, ξ ) is developing iff the invariant a ( s ) vanishes.If the surfaces S ( C, ξ ) is a cone we say that the versors field (
C, ξ ) is concurrent . Theorem 1.1.3
A necessary and sufficient condition for the versor field ( C, ξ ) , ( K ( s ) = 0) , to be concurrent is the following dds (cid:18) a ( s ) K ( s ) (cid:19) − a ( s ) = 0 , a ( s ) = 0 , ∀ s ∈ ( s , s ) . ( C, ξ ) Consider the sphere Σ with center a fix point O ∈ E and radius 1. Definition 1.2.1
The spherical image of the versor field ( C, ξ ) is thecurve C ∗ on sphere Σ given by ξ ∗ ( s ) = −−→ OP ∗ ( s ) = ξ ( s ) , ∀ s ∈ ( s , s ) . From this definition we have: dξ ∗ ( s ) = ξ ( s ) K ( s ) ds. (1.2.1)Some immediate properties:1. It follows ds ∗ = K ( s ) ds. (1.2.2)Therefore: .3. Plane fields ( C, π ) 23
The arc length of the curve C ∗ is s ∗ = s ∗ + Z ss K ( σ ) dσ (1.2.3)where s ∗ is a constant and [ s , s ] ⊂ ( s , s ).2. The curvature K ( s ) of ( C, ξ ) at a point P ∗ ( s ) is expressed by K ( s ) = ds ∗ ds (1.2.4)3. C ∗ is reducible to a point iff ( C, ξ ) is a parallel versor field in E .4. The tangent line at point P ∗ ∈ C ∗ is parallel with principal normalline of ( C, ξ ).5. Since ξ ( s ) × ξ ( s ) = ξ ( s ) it follows that the direction of binormalversor field ξ is the direction tangent to Σ orthogonal to tangent line of C ∗ at point P ∗ .6. The geodesic curvature κ g of the curve C ∗ at a point P ∗ verifiesthe equation κ g ds ∗ = K ds. (1.2.5)7. The versor field ( C, ξ ) is of null torsion (i.e. K ( s ) = 0) iff κ g = 0 . In this case C ∗ is an arc of a great circle on Σ. ( C, π ) A plane field (
C, π ) is defined by the versor field (
C, ν ( s )) where ν ( s ) isnormal to π ( s ) in every point P ( s ) ∈ C. We assume the π ( s ) is oriented.Consequently, ν ( s ) is well determined.Let R π = ( P ( s ) , ν ( s ) , ν ( s ) , ν ( s )) be the Frenet frame, ν ( s ) = ν ( s ),of the versor field ( C, ν ).It follows that the fundamental equations of the plane field (
C, π ) are: drds ( s ) = b ( s ) ν + b ( s ) ν + b ( s ) ν , b + b + b = 1 . (1.3.1) E dν ds = χ ( s ) ν ,dν ds = − χ ν + χ ( s ) ν (1.3.2) dν ds = − χ ( s ) ν . The invariant χ ( s ) = (cid:13)(cid:13)(cid:13)(cid:13) dν dr (cid:13)(cid:13)(cid:13)(cid:13) is called the curvature and χ ( s ) is thetorsion of the plane field ( C, π ( s )).The following properties hold:1. The characteristic straight lines of the field of planes ( C, π ) crossthrough the corresponding point P ( s ) ∈ C iff the invariant b ( s ) = 0, ∀ s ∈ ( s , s ).2. The planes π ( s ) are parallel along the curve C iff the invariant χ ( s ) vanishes, ∀ s ∈ ( s , s ).3. The characteristic lines of the plane field ( C, π ) are parallel iff theinvariant χ ( s ) = 0, ∀ s ∈ ( s , s ).4. The versor field ( C, ν ) determines the directions of the character-istic line of ( C, π ).5. The versor field (
C, ν ), with χ ( s ) = 0, is concurrent iff: b ( s ) = 0 , b ( s ) + ddt (cid:18) b ( s ) χ ( s ) (cid:19) = 0 .
6. The curve C is on orthogonal trajectory of the generatrices of theruled surface R ( C, ν ) if b ( s ) = 0 , ∀ s ∈ ( s , s ).7. By means of equations (1.3.1), (1.3.2) we can prove a fundamentaltheorem for the plane field ( C, π ) . hapter 2Myller configurations M ( C, ξ , π ) The notions of versor field (
C, ξ ) and the plane field (
C, π ) along to thesame curve C lead to a more general concept named Myller configuration M ( C, ξ, π ), in which every versor ξ ( s ) belongs to the corresponding plane π ( s ) at point P ( s ) ∈ C. The geometry of M = M ( C, ξ, π ) is much morerich as the geometries of (
C, ξ ) and (
C, π ) separately taken. For M onecan define its geometric invariants, a Darboux frame and introduce a newidea of parallelism or concurrence of versor ( C, ξ ) in M . The geometryof M is totally based on the fundamental equations of M . The basicidea of this construction belongs to Al. Myller [31], [32], [33], [34] andit was considerable developed by O. Mayer [20], [21], R. Miron [23], [24],[62] (who proposed the name of Myller Configuration and studied itscomplete system of invariants). Definition 2.1.1
A Myller configuration M = M ( C, ξ, π ) in the Eu-clidean space E is a pair ( C, ξ ) , ( C, π ) of versor field and plane field, M ( C, ξ, π ) having the property: every ξ ( s ) belongs to the plane π ( s ) . Let ν ( s ) be the normal versor to plane π ( s ). Evidently ν ( s ) is uniquelydetermined if π ( s ) is an oriented plane for ∀ s ∈ ( s , s ).By means of versors ξ ( s ) , ν ( s ) we can determine the Darboux frame of M : R D = ( P ( s ); ξ ( s ) , µ ( s ) , ν ( s )) , (2.1.1)where µ ( s ) = ν ( s ) × ξ ( s ) . (2.1.2) R D is geometrically associated to M . It is orthonormal and positivelyoriented.Since the versors ξ ( s ) , µ ( s ) , ν ( s ) have a geometric meaning, the sameproperties have the vectors dξds , dµds and dνds .Therefore, we can prove, without difficulties: Theorem 2.1.1
The moving equations of the Darboux frame of M areas follows: drds = α ( s ) = c ( s ) ξ + c ( s ) µ + c ( s ) ν ; c + c + c = 1(2.1.3) and dξds = G ( s ) µ + K ( s ) ν,dµds = − G ( s ) ξ + T ( s ) ν, (2.1.4) dνds = − K ( s ) ξ − T ( s ) µ and c ( s ) , c ( s ) , c ( s ); G ( s ) , K ( s ) and T ( s ) are uniquely determined andare invariants. The previous equations are called the fundamental equations of theMyller configurations M .Terms: .1. Fundamental equations of Myller configuration 27 G ( s ) – is the geodesic curvature, K ( s ) – is the normal curvature, T ( s )– is the geodesic torsion of the versor field ( C, ξ ) in Myller configuration M . For M a fundamental theorem can be stated: Theorem 2.1.2
Let be a priori given C ∞ functions c ( s ) , c ( s ) , c ( s ) , [ c + c + c = 1] , G ( s ) , K ( s ) , T ( s ) , s ∈ [ a, b ] . Then there is a Myllerconfiguration M ( C, ξ, π ) for which s is the arclength of curve C and thegiven functions are its invariants. Two such configuration differ by aproper Euclidean motion.Proof. By means of given functions c ( s ) , . . . , G ( s ) , . . . we can write thesystem of differential equations (2.1.3), (2.1.4). Let the initial conditions r = −→ OP , ( ξ , µ , ν ), an orthonormal, positively oriented frame in E .From (2.1.4) we find an unique solution ( ξ ( s ) , µ ( s ) , ν ( s )), s ∈ [ a, b ]with the property ξ ( s ) = ξ , µ ( s ) = µ , ν ( s ) = ν , with s ∈ [ a, b ] and ( ξ ( s ) , µ ( s ) , ν ( s )) being an orthonormal, positivelyoriented frame.Then consider the following solution of (2.1.3) r ( s ) = r + Z ss [ c ( σ ) ξ ( σ ) + c ( σ ) µ ( σ ) + c ( σ ) ν ( σ )] dσ, which has the property r ( s ) = r , and (cid:13)(cid:13)(cid:13)(cid:13) drds (cid:13)(cid:13)(cid:13)(cid:13) = 1 . Thus s is the arc length on the curve r = r ( s ).Now, consider the configuration M ( C, ξ, π ( s )), π ( s ) being the planeorthogonal to versor ν ( s ) at point P ( s ).We can prove that M ( C, ξ, π ) has as invariants just c , c , c , G, K, T. The fact that two configurations M and M ′ , obtained by changingthe initial conditions ( r ; ξ , µ , ν ) to ( r ′ , ξ ′ , µ ′ , ν ′ ) differ by a proper M ( C, ξ, π ) Euclidian motions follows from the property that the exists an uniqueEuclidean motion which apply ( r ; ξ , µ , ν ) to ( r ′ , ξ ′ , µ ′ , ν ′ ). The invariants c ( s ) , c ( s ) , c ( s ) have simple geometric interpretations: c ( s ) = cos ∢ ( α, ξ ) , c ( s ) = cos ∢ ( α, µ ) , c = cos ∢ ( α, ν ) . We can find some interpretation of the invariants G ( s ) , K ( s ) and T ( s )considering a variation of Darboux frame R D ( P ( s ); ξ ( s ) , µ ( s ) , ν ( s )) → R ′ D ( P ′ ( s + ∆ s ) , ξ ( s + ∆ s ) , µ ( s + ∆ s ) , ν ( s + ∆ s )) , obtained by the Taylor expansion r ( s + ∆ s ) = r ( s ) + ∆ s drds + (∆ s ) d rds + . . . ++ (∆ s ) n n ! (cid:18) d n rds n + ω ( s, ∆ s ) (cid:19) ,ξ ( s + ∆ s ) = ξ ( s ) + ∆ s dξds + (∆ s ) d ξds + . . . ++ (∆ s ) n n ! (cid:18) d n ξds n + ω ( s, ∆ s ) (cid:19) ,µ ( s + ∆ s ) = µ ( s ) + ∆ s dµds + (∆ s ) d µds + . . . ++ (∆ s ) n n ! (cid:18) d n µds n + ω ( s, ∆ s ) (cid:19) ,ν ( s + ∆ s ) = ν ( s ) + ∆ s dνds + (∆ s ) d νds + . . . +(∆ s ) n n ! (cid:18) d n νds n + ω ( s, ∆ s ) (cid:19) , (2.2.1) .2. Geometric interpretations of invariants 29 where lim ∆ s → ω i ( s, ∆ s ) = 0 , ( i = 0 , , , . (2.2.2)Using the fundamental formulas (2.1.3), (2.1.4) we can write, for n =1 : r ( s + ∆ s ) = r ( s ) + ∆ s ( c ξ + c µ + c ν + ω ( s, ∆ s )) ,ξ ( s + ∆ s ) = ξ ( s ) + ∆ s ( G ( s ) µ + K ( s ) ν + ω ( s, ∆ s )) , (2.2.3) µ ( s + ∆ s ) = µ ( s ) + ∆ s ( − G ( s ) ξ + T ( s ) ν + ω ( s, ∆ s )) ν ( s ) + ∆ s = ν ( s ) + ∆ s ( − K ( s ) ξ − T ( s ) µ + ω ( s, ∆ s )) . and (2.2.2) being verified.Let ξ ∗ ( s + ∆ s ) be the orthogonal projection of the versor ξ ( s + ∆ s )on the plane π ( s ) at point P ( s ) and let ∆ ψ be the oriented angle of theversors ξ ( s ) , ξ ∗ ( s + ∆ s ). Thus, we have Theorem 2.2.1
The invariant G ( s ) of versor field ( C, ξ ) in Myller con-figuration M ( C, ξ, π ) is given by G ( s ) = lim ∆ s → ∆ ψ ∆ s . By means of second formula (2.2.3), this Theorem can be provedwithout difficulties.Therefore the name of geodesic curvature of (
C, ξ ) in M is justified.Consider the plane ( P ( s ); ξ ( s ) , ν ( s ))-called the normal plan of M which contains the versor ξ ( s ).Let be the vector ξ ∗∗ ( s + ∆ s ) the orthogonal projection of versor ξ ( s + ∆ s ) on the normal plan ( P ( s ); ξ ( s ) , ν ( s )). The angle ∆ ψ = ∢ ( ξ ( s ) , ξ ∗∗ ( s + ∆ s )) is given by the formasin ∆ ψ = h ξ ( s ) , ξ ( s + ∆ s ) , µ ( s ) ik ξ ∗∗ ( s + ∆ s ) k . M ( C, ξ, π ) By (2.2.3) we obtainsin ∆ ψ = K ( s ) + h ξ ( s ) , ω ( s, ∆ s ) , µ ( s ) ik ξ ∗∗ ( s + ∆ s ) k ∆ s. Consequently, we have:
Theorem 2.2.2
The invariant K ( s ) has the following geometric inter-pretation K ( s ) = lim ∆ s → ∆ ψ ∆ s . Based on the previous result we can call K ( s ) the normal curvature of ( C, ξ ) in M .A similar interpretation can be done for the invariant T ( s ). Theorem 2.2.3
The function T ( s ) has the interpretation: T ( s ) = lim ∆ s → ∆ ψ ∆ s , where ∆ ψ is the oriented angle between µ ( s ) and µ ∗ ( s + ∆ s ) -which isthe orthogonal projection of µ ( s + ∆ s ) on the normal plane ( P ( s ); µ ( s ) ,ν ( s )) . This geometric interpretation allows to give the name geodesic torsion for the invariant T ( s ). G, K, T
The fundamental formulae (2.1.3), (2.1.4) allow to calculate the expres-sions of second derivatives of the versors of Darboux frame R D . Wehave: d rds = (cid:18) dc ds − Gc − Kc (cid:19) ξ + (cid:18) dc ds + Gc − T c (cid:19) µ ++ (cid:18) dc ds + Kc + T c (cid:19) ν (2.3.1) .4. Relations between the invariants of the field ( C, ξ ) and the invariants of (
C, ξ ) in M ( C, ξ, π ) 31 and d ξds = − ( G + T ) ξ + (cid:18) dGds − KT (cid:19) µ + (cid:18) dKds + GT (cid:19) ν,d µds = − (cid:18) dGds + KT (cid:19) ξ − ( G + T ) µ + (cid:18) dTds − GT (cid:19) ν,d νds = (cid:18) − dKds + GT (cid:19) ξ − (cid:18) dTds + GT (cid:19) µ − ( K + T ) ν. (2.3.2)These formulae will be useful in the next part of the book.From the fundamental equations (2.1.3), (2.1.4) we get Theorem 2.3.1
The following formulae for invariants G ( s ) , K ( s ) and T ( s ) hold: G ( s ) = (cid:28) ξ, dξds , ν (cid:29) , (2.3.3) K ( s ) = (cid:28) dξds , ν (cid:29) = − (cid:28) ξ, dνds (cid:29) , T ( s ) = (cid:28) ξ, ν, dνds (cid:29) . (2.3.4)Evidently, these formulae hold in the case when s is the arclength of thecurve C . ( C, ξ ) and the invariants of ( C, ξ ) in M ( C, ξ, π ) The versors field (
C, ξ ) in E has a Frenet frame R F = ( P ( s ) , ξ , ξ , ξ )and a complete system of invariants ( a , a , a ; K , K ) verifying the equa-tions (1.1.5) and (1.1.6).The same field ( C, ξ ) in Myller configuration M ( C, ξ, π ) has a Dar-boux frame R D = ( P ( s ) , ξ, µ, ν ) and a complete system of invariants( c , c , c ; G, K, T ). If we relate R F to R D we obtain ξ ( s ) = ξ ( s ) , M ( C, ξ, π ) ξ ( s ) = µ ( s ) sin ϕ + ν ( s ) cos ϕ,ξ ( s ) = − µ ( s ) cosϕ + ν ( s ) sin ϕ, with ϕ = ∢ ( ξ , ν ) and h ξ , ξ , ξ i = h ξ, µ, ν i = 1 . Then, from (1.1.5) and (2.1.3) we can determine the relations betweenthe two systems of invariants.From drds = α ( s ) = a ξ + a ξ + a ξ = c ξ + c µ + c ν it follows c ( s ) = a ( s ) c ( s ) = a ( s ) sin ϕ − a ( s ) cos ϕ (2.4.1) c ( s ) = a ( s ) cos ϕ + a ( s ) sin ϕ. And, (1.1.6), (2.1.4) we obtain G ( s ) = K ( s ) sin ϕK ( s ) = K ( s ) cos ϕ (2.4.2) T ( s ) = K ( s ) + dϕds . These formulae, allow to investigate some important properties of(
C, ξ ) in M when some invariants G, K, T vanish. ( C, ν ) and invariants G, K, T
The plane field (
C, π ) is characterized by the normal versor field (
C, ν ),which has as Frenet frame R F = ( P ( s ); ν , ν , ν ) with ν = ν and has( b , b , b , χ , χ ) as a complete system of invariants. They satisfy theformulae (1.3.1), (1.3.2). But the frame R F is related to Darboux frame .6. Meusnier ′ s theorem. Versor fields ( C, ξ ) conjugated with tangent versor (
C, α ) 33 R D of ( C, ξ ) in M by the formulae ν = ν ( s ) − ν = sin σξ + cos σµ (2.5.1) ν = − cos σξ + sin σµ where σ = ∢ ( ξ ( s ) , ν ( s )).Proceeding as in the previous section we deduce Theorem 2.5.1
The following relations hold: c = − b sin σ + b cos σ − c = b cos σ + b sin σ (2.5.2) c = b and K = χ sin σT = χ cos σ (2.5.3) G = χ + dσds . A first consequence of previous formulae is given by
Theorem 2.5.2
The invariant K + T depends only on the plane field ( C, π ) . We have K + T = χ . (2.5.4)The proof is immediate, from (2.5.3). ′ s theorem. Versor fields ( C, ξ ) conjugated with tangent versor ( C, α ) Consider the vector field ξ ∗∗ ( s + ∆ s ), ( | ∆ s | < ε, ε > ξ ( s + ∆ s ) on the normal plane ( P ( s ); ξ ( s ) , ν ( s )). M ( C, ξ, π ) Since, up to terms of second order in ∆ s , we have ξ ∗∗ ( s + ∆ s ) = ξ ( s ) + ∆ s K ( s ) ν ( s ) + θ ( s, ∆ s ) (∆ s ) , for ∆ s → dξ ∗∗ ds = K ( s ) ν ( s ) . (2.6.1)Assuming K ( s ) = 0 we consider the point P ∗∗ c -called the center ofcurvature of the vector field ( C, ξ ∗∗ ), given by −→ P P ∗∗ c = 1 K ( s ) ν ( s ) . On the other hand the field of versors (
C, ξ ) have a center of curvature P c given by −→ P P c = 1 K ( s ) ξ .The formula (2.4.2), i.e., K ( s ) = K ( s ) cos ϕ, shows that the orthog-onal projection of vector −→ P P ∗∗ c on the (osculating) plane ( P ( s ); ξ , ξ ) isthe vector −→ P P c .Indeed, we have cos ϕK = 1 K (2.6.2)As a consequence we obtain a theorem of Meusnier type: Theorem 2.6.1
The curvature center of the field ( C, ξ ) in M is theorthogonal projection on the osculating plane ( M ; ξ , ξ ) of the curvaturecenter P ∗∗ c . Definition 2.6.1
The versor field ( C, ξ ) is called conjugated with tan-gent versor field ( C, α ) in the Myller configuration M ( C, ξ, π ) if the in-variant K ( s ) vanishes. Some immediate consequences:1. (
C, ξ ) is conjugated with (
C, α ) in M iff the line ( P ; ξ ) is parallelin E to the characteristic line of the planes π ( s ), s ∈ ( s , s ).2. ( C, ξ ) is conjugated with (
C, α ) in M iff | T ( s ) | = χ ( s ). .7. Versor field ( C, ξ ) with null geodesic torsion 35
3. (
C, ξ ) is conjugated with (
C, α ) in M iff the osculating planes( P ; ξ , ξ ) coincide to the planes π ( s ) of M .4. ( C, ξ ) is conjugated with (
C, ξ ) iff the asimptotic planes of theruled surface R ( C, ξ ) coincide with the planes π ( s ) of M . ( C, ξ ) with null geodesic tor-sion A new relation of conjugation of versor field (
C, ξ ) with the tangent versorfield (
C, α ) is obtained in the case T ( s ) = 0 . Definition 2.7.1
The versor field ( C, ξ ) is called orthogonal conjugatedwith the tangent versor field ( C, α ) in M if its geodesic torsion T ( s ) = 0 , ∀ s ∈ ( s , s ) .Some properties1. ( C, ξ ) is orthogonal conjugated with ( C, α ) in M iff µ ( s ) are parallelwith the characteristic line of planes π ( s ) along the curve C .2. ( C, ξ ) is orthogonal conjugated with ( C, α ) in M if | K ( s ) | = χ ( s ) ,along C . Theorem 2.7.1
Assuming that the versor field ( C, ξ ) in the configura-tion M ( C, ξ, π ) has two of the following three properties then it has thethird one, too:a. The osculating planes ( P ; ξ , ξ ) are parallel in E along C .b. The osculating planes ( P ; ξ , ξ ) have constant angle with the plans π ( s ) on C .c. The geodesic torsion T ( s ) vanishes on C . The proof is based on the formula T ( s ) = K ( s ) + dϕds , ϕ = ∢ ( ξ , ν ). M ( C, ξ, π ) Consider two Myller configurations M ( C, ξ, π ) and M ′ ( C, ξ, π ′ ) whichhave in common the versor field ( C, ξ ). Denote by ϕ = ∢ ( ξ , ν ), ϕ ′ = ∢ ( ξ , ν ′ ). Then the geodesic torsions of ( C, ξ ) in M and M ′ are follows: T ( s ) = K ( s ) + dϕds , T ′ ( s ) = K ( s ) + dϕ ′ ds . Evidently, we have ϕ − ϕ ′ = ∢ ( ν, ν ′ ).By means of these relations we can prove, without difficulties: Theorem 2.7.2
If the Myller configurations M ( C, ξ, π ) and M ′ ( C, ξ, π ′ ) have two of the following properties:a) ( C, ξ ) has the null geodesic torsion, T ( s ) = 0 in M .b) ( C, ξ ) has the null geodesic torsion T ′ ( s ) in M ′ .c) The angle ∢ ( ν, ν ′ ) is constant along C , then M and M ′ have thethird property. Remark 2.7.1
The versor field (
C, ν ) is orthogonally conjugated withthe tangent versors ( C, α ) in the configuration M ( C, ξ, π ). M Consider (
C, V ) a vector field, along the curve C . We denote V ( s ) = V ( r ( s )) and say that ( C, V ) is a vector field in the configuration M = M ( C, ξ, π ) if the vector V ( s ) belongs to plane π ( s ) , ∀ s ∈ ( s , s ). Definition 2.8.1
The vectors field ( C, V ) in M ( C, ξ, π ) is parallel inMyller sense if the vector field dVds is normal to M , i.e. dVds = λ ( s ) ν ( s ) , ∀ s ∈ ( s , s ) . .8. The vector field parallel in Myller sense in configurations M The parallelism in Myller sense is a direct generalization of Levi-Civita parallelism of tangent vector fields along a curve C of a surface S . It is not difficult to prove that the vector field V ( s ) is parallel inMyller sense if the vector field V ′ ( s + ∆ s ) = pr π ( s ) V ( s + ∆ s )is parallel in ordinary sens in E -up to terms of second order in ∆ s .In Darboux frame, V ( s ) can be represented by its coordinate as fol-lows: V ( s ) = V ( s ) ξ ( s ) + V ( s ) µ ( s ) . (2.8.1)By virtue of fundamental equations (2.1.4) we find: dVds = (cid:18) dV ds − GV (cid:19) ξ + (cid:18) dV ds + GV (cid:19) µ + ( KV + T V ) ν. (2.8.2)Taking into account the Definition 2.8.1, one proves: Theorem 2.8.1
The vector field V ( s ) , (2 . . is parallel in Myller sensein configuration M ( C, ξ, π ) iff coordinates V ( s ) , V ( s ) are solutions ofthe system of differential equations: dV ds − GV = 0 , dV ds + GV = 0 . (2.8.3)In particular, for V ( s ) = ξ ( s ), we obtain Theorem 2.8.2
The versor field ξ ( s ) is parallel in Myller sense in M ( C, ξ, π ) iff the geodesic curvature G ( s ) of ( C, ξ ) in M vanishes. This is a reason that Al. Myller says that G is the deviation ofparallelism [31]. Later we will see that G ( s ) is an intrinsec invariant inthe geometry of surfaces in E .By means of (2.8.3) we have M ( C, ξ, π ) Theorem 2.8.3
There exists an unique vector field V ( s ) , s ∈ ( s ′ , s ′ ) ⊂ ( s , s ) parallel in Myller sense in the configuration M ( C, ξ, π ) whichsatisfy the initial condition V ( s ) = V , s ∈ ( s ′ , s ′ ) and h V , ν ( s ) i = 0 . Evidently, theorem of existence and uniqueness of solutions of system(2.8.3), is applied in this case.In particular, if G ( s ) = constant, then the general solutions of (2.8.3)can be obtained by algebric operations.An important property of parallelism in Myller sense is expressed inthe next theorem. Theorem 2.8.4
The Myller parallelism of vectors in M preserves thelengths and angles of vectors.Proof. If dVds = λ ( s ) ν , then dds h V , V i = 0 . Also, dV ′ ds = λ ( s ) ν , dUds = λ ′ ( s ) ν , then dds h V ( s ) , U ′ ( s ) i = 0 The notions of adjoint point, adjoint curve and concurrence in Myllersense in a configuration M have been introduced and studied by O. Mayer[20] and Gh. Gheorghiev [11], [55]. They applied these notions, to thetheory of surfaces, nonholomorphic manifolds and in the geometry ofversor fields in Euclidean space E .In the present book we introduce these notions in a different way.Consider the vector field ξ ∗ ( s + ∆ s ) = pr π ( s ) ξ ( s + ∆ s ) . Taking into account the formula (2.2.1) ′ we can write up to terms ofsecond order in ∆ s : ξ ∗ ( s + ∆ s ) = ξ ( s ) + ∆ s ( Gµ ( s ) + ω ∗ ( s, ∆ s ))(2.9.1) .9. Adjoint point, adjoint curve and concurrence in Myller sense 39 with ω ∗ ( s, ∆ s ) → , (∆ s → . Let C ′ be the orthogonal projection of the curve C on the plane π ( s ).A neighbor point P ′ ( s + ∆ s ) is projected on plane π ( s ) in the point P ∗ ( s + ∆ s ) given by r ∗ ( s + ∆ s ) = r ( s ) + ∆ s ( c ξ + c µ + ω ∗ ( s, ∆ s )) ,ω ∗ ( s, ∆ s ) → , (∆ s → . (2.9.2) Definition 2.9.1
The adjoint point of the point P ( s ) with respect to ξ ( s ) in M is the characteristic point P a on the line ( P ; ξ ) of the plane ruledsurface R ( C ∗ , ξ ∗ ) . One proves that the position vector R ( s ) of adjoint point P a for G = 0,is as follows: R ( s ) = r ( s ) − c G ξ ( s ) . (2.9.3)The vector field ( C ∗ , ξ ∗ ) from (2.9.3) is called geodesic field . A resultestablished by O. Mayer [20] holds: Theorem 2.9.1
If the versor field ( C, ξ ) is enveloping in space E , thenthe adjoint point P a of the point P ( s ) in M is the contact point of theline ( P, ξ ) with the cuspidale line. Definition 2.9.2
The geometric locus of the adjoint points correspond-ing to the versor field ( C, ξ ) in M is the adjoint curve C a of the curve C in M . The adjoint curve C a has the vector equations (2.9.3) for ∀ s ∈ ( s , s ).Now, we can introduce Definition 2.9.3
The versor field ( C, ξ ) is concurrent in Myller sensein M ( C, ξ, π ) if, at every point P ( s ) ∈ C the geodesic vector field ( C ∗ , ξ ∗ ) is concurrent. M ( C, ξ, π ) For G ( s ) = 0, we have Theorem 2.9.2
The versor field ( C, ξ ) is concurrent in Myller sense in M iff the following equation hold dds (cid:16) c G (cid:17) = c . (2.9.4)For the proof see Section 1.1, Chapter 1, Theorem 1.1.3. M In the Section 2, Chapter 1 we defined the spherical image of a versorfield (
C, ξ ). Applying this idea to the normal vectors field (
C, ν ) to aMyller configuration M ( C, ξ, π ) we define the notion of spherical image C ∗ of M as being ν ∗ ( s ) = OP ∗ ( s ) = ν ( s )(2.10.1)Thus, the relations between the curvature χ and torsion χ of ( C, ν ) andgeodesic curvature κ ∗ g of C ∗ at a point P ∗ ∈ C ∗ and arclength s ∗ are asfollows χ = ds ∗ ds , χ ds = κ ∗ g ds ∗ . The properties enumerated in Section 2, ch 1, can be obtained for thespheric image C ∗ of configuration M .Consider the versor field −−−→ P ∗ P ∗ = ξ ∗ ( s ) = ξ ( s ) and the angle θ = ∢ ( ν , ξ ). It follows K = − ds ∗ ds cos θ, ( f orχ = 0) . (2.10.2)Thus, for θ = ± π K = 0 , which leads to a new interpretationof the fact that ξ ( s ) is conjugated to α ( s ) in Myller configurations.Analogous one can obtain T = − ds ∗ ds cos e θ, e θ = ∢ ( ν , µ ∗ ) (with µ ∗ ( s ) = µ ( s ), applied at the point P ∗ ( s ) ∈ C ∗ ). For e θ = ± π .10. Spherical image of a configuration M lows that ξ ( s ) are orthogonally conjugated with α ( s ) . The problem is to see if the Gauss-Bomet formula can be extendedto Myller configurations M ( C, ξ, π ). In the case ( α ( s ) ∈ π ( s ), (i.e. α ( s ) ⊥ ν ( s )) such a problem was suggested by Thomson [18] and it hasbeen solved by Mark Krein in 1926, [18].Here, we study this problem in the general case of Myller configura-tion, when h α ( s ) , ν ( s ) i 6 = 0 . First of all we prove
Lemma 2.10.1
Assume that we have:1. M ( C, ξ, π ) a Myller configuration of class C , in which C is aclosed curve, having s as arclength.2. The spherical image C ∗ of M determines on the support sphere Σ a simply connected domain of area ω .In this hypothesis we have the formula ω = 2 π − Z C (cid:18) ν, dνds , d νds (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13) dνds (cid:13)(cid:13)(cid:13)(cid:13) ds. (2.10.3) Proof.
Let Σ be the unitary sphere of center O ∈ E and a simply con-nected domain D , delimited by C ∗ on Σ . Assume that D remains to leftwith respect to an observer looking in the sense of versor ν ( s ), when heis going along C ∗ in the positive sense.Thus, we can take the following representation of Σ: x = cos ϕ sin θx = sin ϕ sin θ (2.10.4) x = cos ϕ, ϕ ∈ [0 , π ) , θ ∈ (cid:16) − π , π (cid:17) . The curve C ∗ can be given by ϕ = ϕ ( s ) , θ = θ ( s ) , s ∈ [0 , s ](2.10.5)with ϕ ( s ) , θ ( s ) of class C and C ∗ being closed: ϕ (0) = ϕ ( s ), θ (0) = M ( C, ξ, π ) θ ( s ).The area ω of the domain D is ω = R s R θ ( s )0 sin θdθ = R s (1 − cos θ ) dϕds ds == 2 π − R s cos θ dϕds ds. (2.10.6)Noticing that the versor ν ∗ = ν ( s ) has the coordinate (2.10.4) a straight-forward calculus leads to (cid:28) ν, dνds , d νds (cid:29) / (cid:13)(cid:13)(cid:13)(cid:13) dνds (cid:13)(cid:13)(cid:13)(cid:13) = cos θ dϕds + dds arctg sin θ dϕdsdθds . Denoting by ψ the angle between the meridian ϕ = ϕ and curve C ∗ ,oriented with respect to the versor ν we havetg ψ = (cid:18) sin θ dϕds (cid:19) / dθds . (2.10.7)The previous formulae lead to (cid:28) ν, dνds , d νds (cid:29) / (cid:13)(cid:13)(cid:13)(cid:13) dνds (cid:13)(cid:13)(cid:13)(cid:13) = cos θ dϕds + dψds . (2.10.8)But in our conditions of regularity Z C dψds ds = 0 . Thus (2.10.7), (2.10.8)implies the formula (2.10.3)It is not difficult to see that the formula (2.10.3) can be generalizedin the case when the curve C of the configuration M has a finite numberof angular points. The second member of the formula (2.10.3) will beadditive modified with the total of variations of angle ψ at the angularpoints corresponding to the curve C .Now, one can prove the generalization of Mark Krein formula. Theorem 2.10.1
Assume that we have1. M ( C, ξ, π ) a Myller configuration of class C ( i.e. C is of class .10. Spherical image of a configuration M C and ξ ( s ) , ν ( s ) are the class C ) in which C is a closed curve, having s as natural parameter.2. The spherical image C ∗ of M determine on the support sphere Σ a simply connected domain of area ω. σ the oriented angle between the versors ν ( s ) and ξ ( s ) . In theseconditions the following formula hold: ω = 2 π − Z C G ( s ) ds + Z C dσ. (2.10.9) Proof.
The first two conditions allow to apply the Lemma 2.10.1. Thefundamental equations (1.3.1), (1.3.2) of (
C, ν ) give us for ν = ν : dν ds = χ ν , d ν ds = dχ ds ν + χ ( − χ ν + χ ν ) . So, (cid:28) ν , dν ds , d ν ds (cid:29) = χ χ . Thus, the formula (2.10.3), leads to the following formula ω = 2 π − Z C χ ( s ) ds. But, we have G ( s ) = χ ( s ) + dσds , G ( s ) being the geodesic curvature of( C, ξ ) in M . Then the last formula is exactly (2.10.9).If G = 0 for M , then we have ω = 2 π. Indeed G ( s ) = 0 along the curve C imply ω = 2 π + 2 kπ , k ∈ N . Butwe have 0 ≤ ω < π, so k = 0 . A particular case of Theorem 2.10.1 is the famous result of Jacobi:
The area ω of the domain D determined on the sphere Σ by the closedcurve C ∗ -spherical image of the principal normals of a closed curve C in E , assuming D a simply connected domain, is a half of area of sphere Σ . In this case we consider the Myller configuration M ( C, α, π ), π ( s ) M ( C, ξ, π ) being the rectifying planes of C . hapter 3Tangent Myller configurations M t The theory of Myller configurations M ( C, ξ, π ) presented in the Chapter2 has an important particular case when the tangent versor fields α ( s ), ∀ s ∈ ( s , s ) belong to the corresponding planes π ( s ). These Myllerconfigurations will be denoted by M t = M t ( C, ξ, π ) and named tangent
Myller configuration .The geometry of M t is much more rich that the geometrical theory of M because in M t the tangent field has some special properties. So, ( C, α )in M t has only three invariants κ g , κ n and τ g called geodesic curvature , normal curvature and geodesic torsion , respectively, of the curve C in M t .The curves C with κ g = 0 are geodesic lines of M t ; the curves C with the property κ n = 0 are the asymptotic lines of M t and the curve C for which τ g = 0 are the curvature lines for M t . The mentionedinvariants have some geometric interpretations as the geodesic curvature,normal curvature and geodesic torsion of a curve C on a surfaces S in theEuclidean space E . M t M t Consider a tangent Myller configuration M t = ( C, ξ, π ). Thus we have h α ( s ) , ν ( s ) i = 0 , ∀ s ∈ ( s , s ) . (3.1.1)The Darboux frame R D is R D = ( P ( s ); ξ ( s ) , µ ( s ) , ν ( s )) with µ ( s ) = ν ( s ) × ξ ( s ).The fundamental equations of M t are obtained by the fundamentalequations (2.1.3), (2.1.4), Chapter 2 of a general Myller configuration M for which the invariant c ( s ) vanishes. Theorem 3.1.1
The fundamental equations of the tangent Myller con-figuration M t ( C, ξ, π ) are given by the following system of differentialequations: drds = c ( s ) ξ ( s ) + c ( s ) µ ( s ) , ( c + c = 1) , (3.1.2) dξds = G ( s ) µ ( s ) + K ( s ) ν ( s ) dµds = − G ( s ) ξ ( s ) + T ( s ) ν ( s )(3.1.3) dνds = − K ( s ) ξ ( s ) − T ( s ) µ ( s ) . The invariants c , c , G, K, T have the same geometric interpretationsand the same denomination as in Chapter 2. So, G ( s ) is the geodesiccurvature of the field ( C, ξ ) in M t , K ( s ) is the normal curvature and T ( s ) is the geodesic torsion of ( C, ξ ) in M t .The cases when some invariants G, K, T vanish can be investigateexactly as in the Chapter 2.In this respect, denoting ϕ = ∢ ( ξ ( s ) , ν ( s )) and using the Frenetformulae of the versor field ( C, ξ ) we obtain the formulae G = K sin ϕ, K = K cos ϕ, T = K + dϕds . (3.1.4) .2. The invariants of the curve C in M t In §
5, Chapter 2 we get the relations between the invariants of (
C, ξ ) in M t and the invariants of normal versor field ( C, ν ), (Theorem 2.5.1, Ch2.) For σ = ∢ ( ξ ( s ) , ν ( s )) we have K = χ sin σ, T = χ cos σ, G = χ + dσds . (3.1.5)Others results concerning M t can be deduced from those of M .For instance Theorem 3.1.2 (Mark Krein) Assuming that we have: . M t ( C, ξ, π ) a tangent Myller configuration of class C in which C is a closed curve, having s as natural parameter. . The spherical image C ∗ of M t determines on the support sphere Σ a simply connected domain of area ω . . σ = ∢ ( ξ, ν ) .In these conditions the following Mark-Krein ′ s formula holds: ω = 2 π − Z C G ( s ) ds + Z C dσ. (3.1.6) C in M t A smooth curve C having s as arclength determines the tangent versorfield ( C, α ) with α ( s ) = drds . Consequently we can consider a particulartangent Myller configuration M t ( C, α, π ) defined only by curve C andtangent planes π ( s ).In this case the geometry of Myller configurations M t ( C, α, π ) is calledthe geometry of curve C in M t . The Darboux frame of curve C in M t is R D = ( P ( s ); α ( s ) , µ ∗ ( s ) , ν ( s )), µ ∗ ( s ) = ν ( s ) × α ( s ). It will be called theDarboux frame of the curve C in M t . Theorem 3.2.1
The fundamental equations of the curve C in the Myllerconfiguration M t ( C, α, π ) are given by the following system of differential M t equations: drds = α ( s ) , (3.2.1) dαds = κ g ( s ) µ ∗ ( s ) + κ n ( s ) ν ( s ) ,dµ ∗ ds = − κ g ( s ) α ( s ) + τ g ( s ) ν ( s ) , (3.2.2) dνds = − κ n ( s ) α ( s ) − τ g ( s ) µ ∗ ( s ) . Of course (3 . . , (3 . .
2) are the moving equations of the Darboux frame R D of the curve C .The invariants κ g , κ n and τ g are called: the geodesic curvature , normalcurvature and geodesic torsion of the curve C in M t . Of course, we can prove a fundamental theorem for the geometry ofcurves C in M t : Theorem 3.2.2
A priori given C ∞ -functions κ g ( s ) , κ n ( s ) , τ g ( s ) , s ∈ [ a, b ] , there exists a Myller configuration M t ( C, α, π ) for which s is ar-clength on the curve C and given functions are its invariants. Two suchconfigurations differ by a proper Euclidean motion. The proof is the same as proof of Theorem 2.1.2, Chapter 2.
Remark 3.2.1
Let C be a smooth curve immersed in a C ∞ surface S in E . Then the tangent plans π to S along C uniquely determines a tan-gent Myller configuration M t ( C, α, π ). Its Darboux frame R D and theinvariants k g , κ n , τ g of C in M t are just the Darboux frame and geodesiccurvature, normal curvature and geodesic torsion of curve C on the sur-face S .Let R F = ( P ( s ); α ( s ) , α ( s ) , α ( s )), be the Frenet frame of curve C with α ( s ) = α ( s ).The Frenet formulae hold: drds = α ( s )(3.2.3) .3. Geodesic, asymptotic and curvature lines in M t ( C, α, π ) 49 dα ds = κ ( s ) α ( s ); dα ds = − κ ( s ) α ( s ) + τ ( s ) α ( s ) dα ds = − τ ( s ) α ( s )(3.2.4)where κ ( s ) is the curvature of C and τ ( s ) is the torsion of C .The relations between the invariants κ g , κ n , τ g and κ, τ can be ob-tained like in Section 4, Chapter 2. Theorem 3.2.3
The following formulae hold good κ g ( s ) = κ sin ϕ ∗ , κ n ( s ) = κ cos ϕ ∗ , τ g ( s ) = τ + dϕ ∗ ds , (3.2.5) with ϕ ∗ = ∢ ( α ( s ) , ν ( s )) . In the case when we consider the relations between the invariants κ g , κ n , τ g and the invariants χ , χ of the normal versor field ( C, ν ) wehave from the formulae (3.1.5): κ n = χ sin σ, τ g = χ cos σ, κ g = χ + dσds , (3.2.6)where σ = ∢ ( α ( s ) , ν ( s )).It is clear that the second formula (3.2.5) gives us a theorem ofMeusnier type, and for ϕ ∗ = 0 or ϕ ∗ = ± π we have κ g = 0 , κ n = ± κ , τ g = τ. For σ = 0 , or σ = ± π, from (3.2.6) we obtain κ n = 0 , τ g = ± χ , κ g = χ . M t ( C, α, π ) The notion of parallelism of a versor field (
C, α ) in M t along the curve C , investigated in the Section 8, Chapter 2 for the general case, can be M t applied now for the particular case of tangent versor field ( C, α ). It isdefined by the condition κ g ( s ) = 0 , ∀ s ∈ ( s , s ). The curve C with thisproperty is called geodesic line for M t (or autoparallel curve ).The following properties hold:1. The curve C is a geodesic in the configuration M t iff at every point P ( s ) of C the osculating plane of C is normal to M t . C is geodesic in M t iff the equality | κ n | = κ holds along C . If C is a straight line in M t then C is a geodesic of M t . Asymptotics
The curve C is called asymptotic in M t if κ n = 0 , ∀ s ∈ ( s , s ). Anasymptotic C is called an asymptotic line , too. The following propertiescan be proved without difficulties:1. C is asymptotic line in M t iff at every point P ( s ) ∈ C the oscu-lating plane of C coincides to the plane π ( s ) . If C is a straight line then C is asymptotic in M t . C is asymptotic in M t iff along C , | κ g | = κ. If C is asymptotic in M t , then τ g = τ along C . If α ( s ) is conjugate to α ( s ) then C is asymptotic line in M t . Therefore we may say that the asymptotic line in M t are the auto-conjugated lines. Curvature lines
The curve C is called the curvature line in the configurations M t ( C, ξ, π ) if the ruled surface R ( C, ν ) is a developing surface.One knows that R ( C, ν ) is a developing surface iff the following equa-tion holds: h α ( s ) , ν ( s ) , dνds ( s ) i = 0 , ∀ s ∈ ( s , s ) . Taking into account the fundamental equations of M t one gets:1. C is a curvature line in M t iff its geodesic torsion τ ( s ) = 0 , ∀ s . C is a curvature line in M t iff the versors field ( C, µ ∗ ) are conju-gated to the tangent α ( s ) in M t . .4. Mark Krein ′ s formula 51 Theorem 3.3.1
If the curve C of the tangent Myller configuration M t ( C,ξ, π ) satisfies two from the following three conditionsa) C is a plane curve.b) C is a curvature line in M t .c) The angle ϕ ∗ ( s ) = ∢ ( α ( s ) , ν ( s )) is constant,then the curve C verifies the third condition, too. For proof, one can apply the Theorem 3.2.3, Chapter 3.More general, let us consider two tangent Myller configurations M t ( C,α, π ) and M ∗ t ( C, α, π ∗ ) and ϕ ∗∗ ( s ) = ∢ ( ν ( s ) , ν ∗ ( s )).We can prove without difficulties: Theorem 3.3.2
Assuming satisfied two from the following three condi-tions1. C is curvature line in M t .2. C is curvature line in M ∗ t .3. The angle ϕ ∗∗ ( s ) is constant for s ∈ ( s , s ) . Then, the third con-dition is also verified. ′ s formula In the Section 10, Chapter 2, we have defined the spherical image ofa general Myller configuration M . Of course the definition applies fortangent configurations M t , too.But now will appear some new special properties.If in the formulae from Section 10, Chapter 2 we take ξ ( s ) = α ( s ), ∀ s , the invariants G, K, T reduce to geodesic curvature, normal curvatureand geodesic torsion, respectively, of the curve C in M t .Such that we obtain the formulae: κ n = − ds ∗ ds cos θ , θ = ∢ ( ν , α )(3.4.1) M t τ g = − ds ∗ ds cos θ , θ = ∢ ( ν , µ ∗ )(3.4.2)which have as consequences:1. C is asymptotic in M t iff θ = ± π . C is curvature line in M t iff θ = ± π . The following Mark Krein ′ s theorem holds: Theorem 3.4.1
Assume that we have . A Myller configuration M t ( C, ξ, π ) of class C k , ( k ≥ where s isthe natural parameter on the curve C and C is a closed curve. . The spherical image C ∗ of M t determine on the support sphere Σ a simply connected domain of area ω . . σ = ∢ ( ν , α ) .In these conditions Mark Krein ′ s formula holds: ω = 2 π − Z C κ g ( s ) ds + Z C dσ. (3.4.3)Indeed the formula (2.10.9) from Theorem 2.10.1, Chapter 2 is equiv-alent to (3.4.3).The remarks from the end of Section 10, Chapter 2 are valid, too. G, K, T of the versor field ( C, ξ ) in M t ( C, ξ, π ) and the invariants of tangent versor field ( C, α ) in M t Let M t ( C, ξ, π ) be a tangent Myller configuration, R D = ( P ( s ) , ξ ( s ) ,µ ( s ) ,ν ( s )) its Darboux frame and the tangent Myller configuration M ′ t ( C, a, π )determined by the plane field (
C, π ( s )), with α = drds -tangent versors field .5. Relations between the invariants G, K, T of the versor field (
C, ξ ) in M t ( C, ξ, π ) andthe invariants of tangent versor field (
C, α ) in M t to the oriented curve C . The Darboux frame R ′ D = ( P ( s ); α ( s ) , µ ∗ ( s ) ,ν ) and oriented angle λ = ∢ ( α, ξ ) allow to determine R D by means offormulas: ξ = α cos λ + µ ∗ sin λµ = − α sin λ + µ ∗ cos λ (3.5.1) ν = ν. The moving equations (3.1.2), (3.1.3) and (3.2.1), (3.2.3) of R D and R ′ D lead to the following relations between invariants κ g , κ n and τ g of thecurve C in M ′ t and the invariants G, K, T of versor field (
C, ξ ) in M t : K = κ n cos λ + τ g sin λT = − κ n sin λ + τ g cos λ (3.5.2) G = κ g + dλds . The two first formulae (3.5.2) imply K + T = κ n + τ g = (cid:18) ds ∗ ds (cid:19) . (3.5.3)The formula (3.5.2) has some important consequences: Theorem 3.5.1
1. The curve C has κ g ( s ) as geodesic curvature on thedeveloping surface E generated by planes π ( s ) , s ∈ ( s , s ) .2. G is an intrinsec invariant of the developing surface E .Proof.
1. Since the planes π ( s ) pass through tangent line ( P ( s ) , α ( s ))to curve C , the developing surface E , enveloping by planes π ( s ) passesthrough curve C . So, κ g is geodesic curvature of C ⊂ E at point P ( s ).2. The invariants κ g ( s ) and λ ( s ) are the intrinsic invariants of surface E . It follows that the invariant G has the same property.Now, we investigate an extension of Bonnet result. M t Theorem 3.5.2
Supposing that the versor field ( C, ξ ) in tangent Myllerconfiguration M t ( C, ξ, π ) has two from the following three properties:1. ξ ( s ) is a parallel versor field in M t ,2. The angle λ = ∢ ( α, ξ ) is constant,3. The curve C is geodesic in M t , then it has the third property, too. The proof is based on the last formula (3.5.2). Also, it is not difficultto prove:
Theorem 3.5.3
The normal curvature and geodesic torsion of the ver-sor field ( C, µ ∗ ) in tangent Myller configuration M t ( C, ξ, π ) , respectivelyare the geodesic torsion and normal curvature with opposite sign of thecurve C in M t . The same formulae (3.5.2) for K ( s ) = 0 , s ∈ ( s , s ) and τ g ( s ) = 0imply: tg λ = − κ n ( s ) τ g ( s ) . (3.5.4)In the theory of surfaces immersed in E this formula was indepen-dently established by E. Bortolotti [3] and Al. Myller [31-34]. Definition 3.5.1
A curve C is called a geodesic helix in tangent config-uration M t ( C, ξ, π ) if the angle λ ( s ) = ∢ ( α, µ ) is constant ( = 0 , π ) . The following two results can be proved without difficulties.
Theorem 3.5.4 If C is a geodesic helix in M t ( C, ξ, π ) , then κ n /τ g = ± κτ . Another consequence of the previous theory is given by: .5. Relations between the invariants
G, K, T of the versor field (
C, ξ ) in M t ( C, ξ, π ) andthe invariants of tangent versor field (
C, α ) in M t Theorem 3.5.5 If C is geodesic in M t ( C, ξ, π ) , it is a geodesic helix for M t iff C is a cylinder helix in the space E . We can make analogous consideration, taking τ = 0 in the formula(3.5.2).We stop here the theory of Myller configurations M ( C, ξ, π ) in Eu-clidean space E . Of course, it can be extended for Myller configurations M ( C, ξ, π ) with π a k -plane in the space E n , n ≥ k < n or in moregeneral spaces, as Riemann spaces, [26], [27]. hapter 4Applications of theory ofMyller configuration M t inthe geometry of surfaces in E A first application of the theory of Myller configurations M ( C, ξ, π ) inthe Euclidean space E can be realized to the geometry of surfaces S embedded in E . We obtain a more clear study of curves C ⊂ S , anatural definition of Levi-Civita parallelism of vector field V , tangentto S along C , as well as the notion of concurrence in Myller sense ofvector field, tangent to S along C . Some new concepts, as those of meantorsion of S , the total torsion of S , the Bonnet indicatrix of geodesictorsion and its relation with the Dupin indicatrix of normal curvaturesare introduced. A new property of Bacaloglu-Sophie-Germain curvatureis proved, too. Namely, it is expressed in terms of the total torsion of thesurface S . M t E S be a smooth surface embedded in the Euclidean space E . Sincewe use the classical theory of surfaces in E [see, for instance: MikeSpivak, Diff. Geom. vol. I, Bearkley, 1979], consider the analyticalrepresentation of S , of class C k , ( k ≥ , or k = ∞ ): r = r ( u, v ) , ( u, v ) ∈ D. (4.1.1) D being a simply connected domain in plane of variables ( u, v ), and thefollowing condition being verified: r u × r v = 0 for ∀ ( u, v ) ∈ D. (4.1.2)Of course we adopt the vectorial notations r u = ∂r∂u , r v = ∂r∂v . Denote by C a smooth curve on S , given by the parametric equations u = u ( t ) , v = v ( t ) , t ∈ ( t , t ) . (4.1.3)In this chapter all geometric objects or mappings are considered of C ∞ -class, up to contrary hypothesis.In space E , the curve C is represented by r = r ( u ( t ) , v ( t )) , t ∈ ( t , t ) . (4.1.4)Thus, the following vector field dr = r u du + r v dv (4.1.5)is tangent to C at the points P ( t ) = P ( r ( u ( t )) , v ( t )) ∈ C. The vectors r u .1. The fundamental forms of surfaces in E and r v are tangent to the parametric lines and dr is tangent vector to S at point P ( t ) . From (4.1.2) it follows that the scalar function∆ = k r u × r v k (4.1.6)is different from zero on D .The unit vector ν ν = r u × r v √ ∆(4.1.7)is normal to surface S at every point P ( t ) . The tangent plane π at P to S , has the equation h R − r ( u, v ) , ν i = 0 . (4.1.8)Assuming that S is orientable, it follows that ν is uniquely determinedand the tangent plane π is oriented (by means of versor ν ), too.The first fundamental form φ of surface S is defined by φ = h dr, dr i , ∀ ( u, v ) ∈ D (4.1.9)or by φ ( du, dv ) = h dr ( u, v ) , dr ( u, v ) i , ∀ ( u, v ) ∈ D. Taking into account the equality (4.1.5) it follows that φ ( du, dv ) is aquadratic form: φ ( du, dv ) = Edu + 2 F dudv + Gdv . (4.1.10)The coefficients of φ are the functions, defined on D : E ( u, v ) = h r u , r u i , F ( u, v ) = h r u , r v i , G ( u, v ) = h r v , r v i . (4.1.11)But we have the discriminant ∆ of φ :∆ = EG − F . (4.1.12) M t E > , G > , ∆ > . (4.1.13)Consequence: the first fundamental form φ of S is positively defined.Since the vector dr does not depend on parametrization of S , it followsthat φ has a geometrical meaning with respect to a change of coordinatesin E and with respect to a change of parameters ( u, v ) on S .Thus ds, given by: ds = φ ( du, dv ) = Edu + 2 F dudv + Gdv (4.1.14)is called the element of arclength of the surface S .The arclength of a curve C an S is expressed by s = Z tt ( E (cid:18) dudσ (cid:19) + 2 F dudσ dvdσ + G (cid:18) dvdσ (cid:19) ) / dσ. (4.1.15)The function s = s ( t ) , t ∈ [ a, b ] ⊂ ( t , t ), t ∈ [ a, b ] is a diffeomor-phism from the interval [ a, b ] → [0 , s ( t )]. The inverse function t = t ( s ),determines a new parametrization of curve C : r = r ( u ( s ) , v ( s )). Thetangent vector drds is a versor: α = drds = r u duds + r v dvds . (4.1.16)For two tangent versors α = drds , α = δrδs at point P ∈ S the angle ∢ ( α, α ) is expressed bycos ∢ ( α, α )= Eduδu + F ( duδv + δudv )+ Gdvδv √ Edu +2 F dudv + Gdv ·√ Eδu +2 F δuδv + Gδv . (4.1.17) The second fundamental form ψ of S at point P ∈ S is defined by ψ = h ν, d r i = −h dr, dν i ∀ ( u, v ) ∈ D. (4.1.18)Also, we adopt the notation ψ ( du, dv ) = −h dr ( u, v ) , dν ( u, v ) i . ψ is a .2. Gauss and Weingarten formulae 61 quadratic form: ψ ( du, dv ) = Ldu + 2 M dudv + N dv , (4.1.19)having the coefficients functions of ( u, v ) ∈ D : L ( u, v ) = 1 √ ∆ h r u , r u , r uu i , M ( u, v ) = 1 √ ∆ h r u , r v , r uv i N ( u, v ) = 1 √ ∆ < r u , r v , r vv > . (4.1.20)Of course from ψ = −h dr, dν i it follows that ψ has geometric meaning.The two fundamental forms φ and ψ are enough to determine a surface S in Euclidean space E under proper Euclidean motions. This propertyresults by the integration of the Gauss-Weingarten formulae and by theirdifferential consequences given by Gauss-Codazzi equations. Consider the moving frame R = ( P ( r ); r u , r v , ν )on the smooth surface S (i.e. S is of class C ∞ ).The moving equations of R are given by the Gauss and Weingartenformulae.The Gauss formulae are as follows: r uu = ( ) r u + ( ) r v + Lν,r uv = ( ) r u + ( ) r v + M νr vv = ( ) r u + ( ) r v + N ν. (4.2.1) M t The coefficients ( ijk ) , ( i, j, k = 1 , u = u , v = u ) are called theChristoffel symbols of the first fundamental form φ : (cid:26) (cid:27) = − √ D h ν, r v , r uu i , (cid:26) (cid:27) = (cid:26) (cid:27) = − √ ∆ h ν, r v , r uv i , (4.2.2) (cid:26) (cid:27) = − √ ∆ h ν, r v , r vv i , (cid:26) (cid:27) = − √ ∆ h ν, r u , r uu i , (cid:26) (cid:27) = (cid:26) (cid:27) = − √ ∆ h ν, r u , r uv i , (cid:26) (cid:27) = − √ ∆ h ν, r u , r vv i . The Weingarten formulae are given by: ∂ν∂u = 1 √ ∆ { ( F M − GL ) r u + ( F L − EM ) r v } (4.2.3) ∂ν∂v = 1 √ ∆ { ( F N − GM ) r u + ( F M − EN ) r v } . Of course, the equations (4.2.1) and (4.2.3) express the variation of themoving frame R on surface S .Using the relations r uuv = r uvu , r vuv = r vvu and ∂ ν∂uv = ∂ ν∂vu appliedto (4.2.1), (4.2.3) one deduces the so called fundamental equations ofsurface S , known as the Gauss-Codazzi equations.A fundamental theorem can be proved, when the first and secondfundamental from φ and ψ are given and Gauss-Codazzi equations areverified (Spivak [75]). .3. The tangent Myller configuration M t ( C, ξ, π ) associated to a tangent versor field(
C, ξ ) on a surface S M t ( C,ξ, π ) associated to a tangent versor field ( C, ξ ) on a surface S Assuming that C is a curve on surface S in E and ( C, ξ ) is a versor fieldtangent to S along C , there is an uniquely determined tangent Myllerconfiguration M t = M t ( C, ξ, π ) for which (
C, π ) is tangent field planesto S along C .The invariants ( c , c , G, K, T ) of ( C, ξ ) in M t ( C, ξ, π ) will be calledthe invariants of tangent versor field (
C, ξ ) on surface S .Evidently, these invariants have the same values on every smoothsurface S ′ which contains the curve C and is tangent to S .Using the theory of M t from Chapter 3, for ( C, ξ ) tangent to S along C we determine:1. The Darboux frame R D = ( P ( s ); ξ ( s ) , µ ( s ) , ν ( s )) , (4.3.1)where s is natural parameter on C and ν = 1 √ ∆ r u × r v , µ = ν × ξ. (4.3.2)Of course R D is orthonormal and positively oriented.The invariants ( c , c , G, K, M ) of ( C, ξ ) on S are given by (3.1.2),(3.1.3) Chapter 3. So we have the moving equations of R. Theorem 4.3.1
The moving equations of the Darboux frame of tangentversor field ( C, ξ ) to S are given by drds = α ( s ) = c ( s ) ξ + c ( s ) µ, ( c + c = 1)(4.3.3) and M t dξds = G ( s ) µ ( s ) + K ( s ) ν ( s ) ,dµds = − G ( s ) ξ ( s ) + T ( s ) ν ( s ) , (4.3.4) dνds = − K ( s ) ξ − T ( s ) µ ( s ) , where c ( s ) , c ( s ) , G ( s ) , K ( s ) , T ( s ) are invariants with respect to the changesof coordinates on E , with respect of transformation of local coordinateson S , ( u, v ) → ( e u, e v ) and with respect to transformations of natural pa-rameter s → s + s . A fundamental theorem can be proved exactly as in Chapter 2.
Theorem 4.3.2
Let be c ( s ) , c ( s ) , [ c + c = 1] , G ( s ) , K ( s ) , T ( s ) , s ∈ [ a, b ] , a priori given smooth functions. Then there exists a tangent Myllerconfiguration M t ( C, ξ, π ) for which s is the arclength of curve C and thegiven functions are its invariants. M t is determined up to a proper Eu-clidean motion. The given functions are invariants of the versor field ( C, ξ ) for any smooth surface S , which contains the curve C and is tan-gent planes π ( s ) , s ∈ [ a, b ] . The geometric interpretations of the invariants G ( s ) , K ( s ) and T ( s )are those mentioned in the Section 2, Chapter 3.So, we get G ( s ) = lim ∆ s → ∆ ψ ∆ s , with ∆ ψ = ∢ ( ξ ( s ) , pr π ( s ) ξ ( s + ∆ s )) ,K ( s ) = lim ∆ s → ∆ ψ ∆ s , with ∆ ψ = ∢ ( µ, pr ( P ; ξ,ν ) ξ ( s + ∆ s )) ,T ( s ) = lim ∆ s → ∆ ψ ∆ s , with ∆ ψ = ∢ ( µ, pr ( P ; µ,ν ) ξ ( s + ∆ s )) . For this reason G ( s ) is called the geodesic curvature of ( C, ξ ) on S ; K ( s ) is called the normal curvature of ( C, ξ ) on S and T ( s ) is named the geodesic torsion of the tangent versor field ( C, ξ ) on S . .4. The calculus of invariants G, K, T on a surface 65
The calculus of invariants
G, K, T is exactly the same as it has beendone in Chapter 2, (2.3.3), (2.3.4): G ( s ) = (cid:28) ξ, dξds , ν (cid:29) , K ( s ) = (cid:28) dξds , ν (cid:29) = − (cid:28) ξ, dνds (cid:29) ,T ( s ) = (cid:28) ξ, ν, dνds (cid:29) . (4.3.5)The analytical expressions of invariants G, K, T on S are given in nextsection. G, K, T on asurface
The tangent versor α = drds to the curve C in S is α ( s ) = drds = r u duds + r v dvds . (4.4.1) α ( s ) is the versor of the tangent vector dr , from (4,4) dr = r u du + r v dv. (4.4.2)The coordinate of a vector field ( C, V ) tangent to surface S along thecurve C ⊂ S , with respect to the moving frame R = ( P ; r u , r v , ν ) are the C ∞ functions V ( s ) , V ( s ): V ( s ) = V ( s ) r u + V ( s ) r v . The square of length of vector V ( s ) is: h V ( s ) , V ( s ) i = E ( V ) + 2 F V V + G ( V ) and the scalar product of two tangent vectors V ( s ) and U ( s ) = U ( s ) r u + M t U ( s ) r v is as follows h U , V i = EU V + F ( U V + V U ) + GU V . Let C and C ′ two smooth curves on S having P ( s ) as common point.Thus the tangent vectors dr, δr at a point P to C , respectively to C ′ are dr = r u du + r v dv,δr = r u δu + r v δv. They correspond to tangent directions ( du, dv ), ( δu, δv ) on surface S .Evidently, α ( s ) = drds and ξ ( s ) = δrδs are the tangent versors to curves C and C ′ , respectively at point P ( s ), and ( C, α ), (
C, ξ ) are the tangentversor fields along the curve C .The Darboux frame R D = ( P ( s ); ξ ( s ) , µ ( s ) , ν ( s )) has the versors ξ ( s ) , µ ( s ) , ν ( s ) given by ξ ( s ) = r u δuδs + r v δvδs ,µ ( s ) = 1 √ ∆ (cid:20) ( Er v − F r u ) δuδs + ( F r v − Gr u ) δvδs (cid:21) , (4.4.3) ν ( s ) = 1∆ ( r u × r v ) . Taking into account (4.3.3) and (4.4.3) it follows: c = h α, ξ i = Eduδu + F ( duδv + dvδu ) + Gdvδv p φ ( du, dv ) p φ ( δu, δv )(4.4.4) c = h α, µ i = √ ∆( δudv − δvdu ) p φ ( du, dv ) p φ ( δu, δv )where φ ( du, dv ) = h dr, dr i , φ ( δu, δv ) = h δr, δr i . In order to calculate the invariants
G, K, T we need to determine the .4. The calculus of invariants
G, K, T on a surface 67 vectors dξds , dνds .By means of (4.4.3) we obtain dξds = dds (cid:18) δrδs (cid:19) = r uu duds δuδs + r uv (cid:18) duds δvδs + δuδs dvds (cid:19) + r vv dvds δvδs + r u dds (cid:18) δuδs (cid:19) + r v dds (cid:18) δνδs (cid:19) . (4.4.5)The scalar mixt h r u , r v , ν ) i is equal to √ ∆. It results: (cid:28) ξ, dξds , ν (cid:29) = √ ∆ (cid:20) dds (cid:18) δvδs (cid:19) duds − dds (cid:18) δuδs (cid:19) dvds (cid:21) + h r u , r uu , ν i duds (cid:18) δuδs (cid:19) ++ h r u , r uv , ν i (cid:18) duds δvδs + dvds δuδs (cid:19) δuδs +(4.4.6) + h r u , r vv , ν i δuδs dvds δvδs ++ h r v , r uu , ν i duds δuδs δvδs + h r v , r uv , ν i (cid:18) duds δvδs + dvds δuδs (cid:19) δvδs + h r v , r vv , ν i dvds (cid:18) δvδs (cid:19) . Taking into account (4.2.1), (4.2.2), (4.3.5) and (4.4.5) we have
Proposition 4.4.1
The geodesic curvature of the tangent versor field ( C, ξ ) on S , is expressed as follows: G ( δ, d ) = √ ∆ { dds (cid:18) δvδs (cid:19) δuδs − dds (cid:18) δuδs (cid:19) δvδs + (cid:18)(cid:26) (cid:27) duds + (cid:26) (cid:27) dvds (cid:19) (cid:18) δuδs (cid:19) (4.4.7) (cid:18)(cid:18)(cid:26) (cid:27) − (cid:26) (cid:27)(cid:19) duds − (cid:18)(cid:26) (cid:27) − (cid:26) (cid:27)(cid:19) dvds (cid:19) δuδs δvδs − (cid:18)(cid:26) (cid:27) duds + (cid:26) (cid:27) dvds (cid:19) (cid:18) δvδs (cid:19) } . M t Remark that the Christoffel symbols are expressed only by means ofthe coefficients of the first fundamental form φ of surface S and theirderivatives. It follows a very important result obtained by Al Myller[34]: Theorem 4.4.1
The geodesic curvature G of a tangent versor field ( C, ξ ) on a surface S is an intrisic invariant of S . The invariant G was named by Al. Myller the deviation of parallelism of the tangent field ( C, ξ ) on surface S .The expression (4.4.7) of G can be simplified by introducing the fol-lowing notations Dds (cid:18) δu i δs (cid:19) = dds (cid:18) δu i δs (cid:19) + X j,k =1 (cid:26) ijk (cid:27) δu j δs du k ds ( u = u ; v = u ; i = 1 , . (4.4.8)The operator (4.4.8) is the classical operator of covariant derivative withrespect to Levi-Civita connection.Denoting by G ij ( δ, d ) = √ ∆ (cid:26) Dds (cid:18) δu i δs (cid:19) δu j δs − Dds (cid:18) δu j δs (cid:19) δu i δs (cid:27) , ( i, j = 1 , G ij ( δ, d ) = − G ji ( δ, d ) we have G = G = 0. Proposition 4.4.2
The following formula holds: G ( δ, d ) = G ( δ, d ) . (4.4.10)Indeed, the previous formula is a consequence of (4.4.7), (4.4.8) and(4.4.9). Corollary 4.4.1
The parallelism of tangent versor field ( C, ξ ) to S alongthe curve C is characterized by the differential equation G ij ( δ, d ) = 0 ( i, j = 1 , . (4.4.11) .4. The calculus of invariants G, K, T on a surface 69
In the following we introduce the notations G ( s ) = G ( δ, d ) , K ( s ) = K ( δ, d ) , T ( s ) = T ( δ, d ) , (4.4.12)since ξ ( s ) = r u δuδs + r v δvδs , α ( s ) = r u duds + r v dvds .The second formula (4.3.5), by means of (4.4.5) gives us the expressionof normal curvature K ( δ, d ) in the form K ( δ, d ) = Lduδu + M ( duδv + dvδu ) + N dvδv √ Edu + 2 F dudv + Gdv √ Eδu + 2 F δuδv + Gδv . (4.4.13)It follows K ( δ, d ) = K ( d, δ )(4.4.14) Remark 4.4.1
The invariant K ( δ, d ) can be written as follows K ( δ, d ) = ψ ( δ, d ) p φ ( d, d ) p φ ( δ, δ )where ψ ( δ, d ) is the polar form of the second fundamental form ψ ( d, d )of surface S . K ( δ, d ) = 0 gives us the property of conjugation of ( C, ξ ) with (
C, α ).The calculus of invariant T ( δ, d ) can be made by means of Weingartenformulae (4.2.3).Since we have dνds = 1∆ (cid:26) [( F M − GL ) r u + ( F L − EM ) r v ] duds + [( F N − GM ) r u + ( F M − EN ) r v ] dvds (cid:27) , (4.4.15)we deduce T ( δ, d ) = 1 √ ∆ (cid:26)(cid:20) ( F M − GL ) duds + ( F N − GM ) dvds (cid:21) δvδs − (cid:20) ( F L − EM ) duds + ( F M − EN ) dvds (cid:21) δuδs (cid:27) . (4.4.16) M t For simplicity we write T ( δ, d ) from previous formula in the followingform T ( δ, d ) = 1 √ ∆ p φ ( d, d ) p φ ( δ, δ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Eδu + F δv F δu + GδvLdu + M dv M du + N dv (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (4.4.17)Remember the expression of mean curvature H and total curvature K t of surface S : H = EN − F M + GN EG − F ) , K t = LN − M EG − F . (4.4.18)It is not difficult to prove, by means of (4.4.16), the following formula T ( δ, d ) − T ( d, δ ) = 2 √ ∆ H (cid:18) δuδs dvds − δvδs duds (cid:19) . (4.4.19)It has the following nice consequence: Theorem 4.4.2
The geodesic torsion T ( δ, d ) is symmetric with respectof directions δ and d , if and only if S is a minimal surface. Finally, (
C, ξ ) is orthogonally conjugated with (
C, α ) if and only if T ( δ, d ) = 0. Remark 4.4.2
The invariant K ( δ, d ) was discovered by O. Mayer [20]and the invariant T ( δ, d ) was found by E. Bertoltti [3]. S The notion of Myller parallelism of vector field (
C, V ) tangent to S alongcurve C , in the associated Myller configuration M t ( C, ξ, π ) is exactly theLevi-Civita parallelism of tangent vector field (
C, V ) to S along the curve C . Indeed, taking into account that the tangent versor field ( C, ξ ) has ξ ( s ) = ξ ( s ) r u + ξ ( s ) r v (4.5.1) .5. The Levi-Civita parallelism of vectors tangent to S and expression of the operator Dds is Dξ i ds = dξ i ds + X j,k =1 ξ j ( ijk ) du k ds , ( i = 1 , G ij = Dξ i ds ξ j − Dξ j ds ξ i , ( i = 1 , . Thus the parallelism of versor (
C, ξ ) along C on S is expressed by G ij = 0 . But these equations are equivalent to D ( λ ( s ) ξ i ( s )) ds = 0 ( i = 1 , , ( λ ( s ) = 0) . (4.5.3)This is the definition of Levi-Civita parallelism of vectors V ( s ) = λ ( s ) ξ ( s ),( λ ( s ) = k V k ) tangent to S along C . Writing V = V ( s ) r u + V ( s ) r v andputting DV i ds = dV i ds + X j,k =1 V j (cid:26) ijk (cid:27) du k ds (4.5.4)the Levi-Civita parallelism is expressed by DV i ds = 0 . (4.5.5)In the case V ( s ) = ξ ( s ) is a versor field, parallelism of ( C, ξ ) in theassociated Myller configurations M t ( C, ξ, π ) is called the parallelism ofdirections tangent to S along C . We can express the condition of paral-lelism by means of invariants of the field ( C, ξ ), as follows:1. (
C, ξ ) is parallel in the Levi-Civita sense on S along C iff G ( δ, d ) =0 .
2. (
C, ξ ) is parallel in the Levi-Civita sense on S along C iff theversor ξ ( s ′ ) , {| s ′ − s | < ε, ε > , s ′ ∈ ( s , s ) } is parallel in the space E with the normal plan ( P ( s ); ξ ( s ) , ν ( s )) . A necessary and sufficient condition for the versor field ( C, ξ ) to be M t parallel on S along C in Levi-Civita sense is that developing on a planethe ruled surface E generated by tangent planes field ( C, π ) along C to E - the directions ( C, ξ ) after developing to be parallel in Euclidean sense. The directions ( C, ξ ) are parallel in the Levi-Civita sense on S along C iff the versor field ( C, ξ ) is normal to S .
5. (
C, ξ ) is parallel on S along C , iff | K | = K . If ( C, ξ ) is parallel on S along C , then | K | = K , T = K . If the ruled surface R ( C, ξ ) is not developable, then ( C, ξ ) is parallelin the Levi-Civita sense iff C is the striction line of surface R ( C, ξ ) . Taking into account the system of differential equations (4.5.5) in thegiven initial conditions, we have assured the existence and uniqueness ofthe Levi-Civita parallel versor fields on S along C .Other results presented in Section 9, Chapter 2 can be particularizedhere without difficulties.A first application. The Tchebishev nets on S are defined as a net of S for which the tangent lines to a family of curves of net are parallel on S along to every curve of another family of curves of net and conversely. A Bianchi resultTheorem 4.5.1
In order that the net parameter ( u = u , v = v ) on S to be a Tchebishev is necessary and sufficient to have the followingconditions: (cid:26) (cid:27) = (cid:26) (cid:27) = 0 . (4.5.6)Indeed, we have G ( δ, d ) = G ( d, δ ) = 0 if (4.5.6) holds.But (4.5.6) is equivalent to the equations ∂E∂v = ∂G∂u = 0 . So, withrespect to a Tchebishev parametrization of S its arclength element ds = φ ( d, d ) is given by ds = E ( u ) du + 2 F ( u, v ) dudv + G ( v ) dv . We finish this paragraph remarking that the parallelism of vectors(
C, V ) on S or the concurrence of vectors ( C, V ) on S can be studied .6. The geometry of curves on a surface 73 using the corresponding notions in configurations M t described in theChapter 3. The geometric theory of curves C embedded in a surface S can be derivedfrom the geometry of tangent Myller configuration M t ( C, α, π ) in which α ( s ) is tangent versor to C at point P ( s ) ∈ C and π ( s ) is tangent planeto S at point P for any s ∈ ( s , s ).Since M t ( c, α, π ) is geometrically associated to the curve C on S wecan define its Darboux frame R D , determine the moving equations of R D and its invariants, as belonging to curve C on surface S .Applying the results established in Chapter 3, first of all we have Theorem 4.6.1
For a smooth curve C embedded in a surface S , thereexists a Darboux frame R D = ( P ( s ); α ( s ) , µ ∗ ( s ) , ν ( s )) and a system ofinvariants κ g ( s ) , κ n ( s ) and τ g ( s ) , satisfying the following moving equa-tions drds = α ( s ) , ∀ s ∈ ( s , s )(4.6.1) and dαds = κ g ( s ) µ ∗ + κ n ( s ) ν,dµ ∗ ds = − κ g ( s ) α + τ g ( s ) ν, (4.6.2) dνds = − κ n ( s ) α − τ g ( s ) µ ∗ , ∀ s ∈ ( s , s ) . The functions κ g ( s ) , κ n ( s ) , τ g ( s ) are called the geodesic curvature , the normal curvature and geodesic torsion of curve C at point P ( s ) on surface S, respectively.Exactly as in Chapter 3, a fundamental theorem can be enounced andproved.The invariants κ g , κ n and τ g have the same values along C on any M t smooth surface S ′ which passes throught curve C and is tangent to surface S along C .The geometric interpretations of these invariants and the cases ofcurves C for which κ g ( s ) = 0 , or κ n ( s ) = 0 or τ g ( s ) = 0 can be studiedas in Chapter 3.The curves C on S for which κ g ( s ) = 0 are called (as in Chapter 3)geodesics (or autoparallel curves ) of S . If C has the property κ n ( s ) = 0 , ∀ s ∈ ( s , s ), it is asymptotic curve of S and the curve C for which τ g ( s ) = 0 , ∀ s ∈ ( s , s ) is the curvature line of S .The expressions of these invariants can be obtained from those of theinvariants G ( δ, d ), K ( δ, d ) and T ( δ, d ) for ξ ( s ) = α ( s ) given in Chapter4: κ g = G ( d, d ) , κ n ( s ) = K ( d, d ) , τ g ( s ) = T ( d, d ) . (4.6.3)So, we have κ g = √ ∆ (cid:26) Dds (cid:18) du i ds (cid:19) du j ds − Dds (cid:18) du j ds (cid:19) du i ds (cid:27) , ( i, j = 1 , κ n = Ldu + 2 M dudv + N dv Edu + 2 F dudv + Gdv ,τ g = 1 √ ∆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ldu + M dv M du + N dvEdu + F dv F du + Gdv (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
Edu + 2 F dudv + Gdv . (4.6.4)Evidently, κ g is an intrinsic invariant of surface S and κ n is the ratio offundamental forms ψ and Φ.The Mark Krein formula (3.4.3), Chapter 3 gives now the Gauss-Bonnet formula for a surface S . .7. The formulae of O. Mayer and E. Bortolotti 75 The geometers O. Mayer and E. Bortolotti gave some new forms of theinvariants K ( δ, d ) and T ( δ, d ) which generalize the Euler or Bonnet for-mulae from the geometry of surfaces in Euclidean space E .Let S be a smooth surface in E having the parametrization given bycurvature lines. Thus the coefficients F and M of the first fundamentaland the second fundamental form vanish.We denote by θ = ∢ (cid:18) α, r u √ E (cid:19) and obtaincos θ = √ Edu √ Edu + Gdv , sin θ = √ Gdv √ Edu + Gdv . (4.7.1)The principal curvature are expressed by1 R = LE , R = NG . (4.7.2)The mean curvature and total curvature are H = 12 (cid:18) R + 1 R (cid:19) = 12 (cid:18) LE + NG (cid:19) K t = 1 R R = LNEG . (4.7.3)Consider a tangent versor field (
C, ξ ), ξ = r u δuδs + r v δvδs and let be σ = ∢ (cid:18) ξ, r u √ E (cid:19) . One getscos σ = √ Eδu √ Eδu + Gδv , sin σ = √ Gδv √ Eδu + Gδv . (4.7.4)Consequently, the expressions (4.4.13) and (4.4.17) of the normal M t curvature and geodesic torsion of ( C, ξ ) on S are as follows K ( δ, d ) = cos σ cos θR + sin σ sin θR (4.7.5) T ( δ, d ) = cos σ sin θR − sin σ cos θR . (4.7.6)The first formula was established by O. Mayer [20] and second formulawas given by E. Bortolotti [3].In the case ξ ( s ) = α ( s ) these formulae reduce to the known Euler andBonnet formulas, respectively: κ n = cos θR + sin θR τ g = 12 (cid:18) R − R (cid:19) sin 2 θ. (4.7.7)For θ = 0 or θ = π , κ n is equal to 1 R and 1 R respectively. For θ = ± π τ g takes the extremal values.1 T = 12 (cid:18) R − R (cid:19) , T = − (cid:18) R − R (cid:19) . (4.7.8)Thus T m = 12 (cid:18) T + 1 T (cid:19) , T t = 1 T T (4.7.9)are the mean torsion of S at point P ∈ S and the total torsion of S atpoint P ∈ S. For surfaces S , T m and T t have the following properties: T m = 0 , T t = − (cid:18) R − R (cid:19) . (4.7.10) Remark 4.7.1
1. As we will see in the next chapter the nonholomorphicmanifolds in E have a nonvanishing mean torsion T m .2. T t from (4.7.10) gives us the Bacaloglu curvature of surfaces [35]. .7. The formulae of O. Mayer and E. Bortolotti 77 Consider in plane π ( s ) the cartesian orthonormal frame ( P ( s ); i , i ), i = r u √ E , i = r v √ G and the point Q ∈ π ( s ) with the coordinates ( x, y ),given by −→ P Q = xi + yi , −→ P Q = | κ n | − α. But α = cos θi + sin θi . So we have the coordinates ( x, y ) of point Q : x = | κ n | − cos θ ; y = | κ n | − sin θ. (4.7.11)The locus of the points Q , when θ is variable in interval (0 , π ) isobtained eliminating the variable θ between the formulae (4.7.7) and(4.7.11). One obtains a pair of conics: x R + y R = ± Dupin indicatrix of normal curvatures. This indicatrix is im-portant in the local study of surfaces S in Euclidean space.Analogously we can introduce the Bonnet indicatrix. Consider in theplane π ( s ) tangent to S at point P ( s ) the frame ( P ( s ) , i , i ) and thepoint Q ′ given by −→ P Q ′ = | τ g | − α = xi + yi . Then, the locus of the points Q ′ , when θ verifies (4.7.7) and x = | τ g | − cos θ , y = | τ g | − sin θ , defines the Bonnet indicatrix of geodesic torsions: (cid:18) R − R (cid:19) xy = ± K ( δ, d ) and T ( δ, d ).So, consider the angles θ = ∢ ( α, i ), σ = ∢ ( ξ, i ) and U ∈ π ( s ). The M t point U has the coordinates x, y with respect to frame ( P ( s ); i , i ) givenby x = | K ( δ, d ) | − cos σ, y = | K ( δ, d ) | − sin σ. (4.7.14)The locus of points U is obtained from (4.7.5) and (4.7.14): x cos θR + y sin θR = ± . (4.7.15)Therefore (4.7.15) is the indicatrix of the normal curvature K ( δ, d ) ofversor field ( C, ξ ) . It is a pair of parallel straight lines.Similarly, for x = | T ( δ, d ) | − cos σ, y = | T ( δ, d ) | − sin σ and the formula (4.7.6) we determine the indicatrix of geodesic torsion T g ( δ, d ) of the versor field ( C, ξ ) : x sin θR − y cos θR = ± . It is a pair of parallel straight lines.Finally, we can prove without difficulties the following formula: κ n ( θ ) κ n ( σ )+ τ g ( θ ) τ g ( σ ) = 2 HK ( σ, θ ) cos( σ − θ ) − K t cos 2( σ − θ ) , (4.7.16)where κ n ( θ ) = κ n ( d, d ) , κ n ( σ ) = κ n ( δ, δ ), K ( σ, θ ) = K ( d, δ ).For σ = θ, (i.e. ξ = α ) the previous formulas leads to a knownBaltrami-Enneper formula: κ n + τ g − Hκ n + K t = 0 , (4.7.17)for every point P ( s ) ∈ S. Along the asymptotic curves we have κ n = 0 , and the previous equa- .7. The formulae of O. Mayer and E. Bortolotti 79 tions give us the Enneper formula τ g + K t = 0 , ( κ n = 0) . Along to the curvature lines, τ g = 0 and one obtains from (4.7.17) κ n − Hκ n + K t = 0 , ( τ g = 0) . The considerations made in Chapter 4 of the present book allow toaffirm that the applications of the theory of Myller configurations thegeometry of surfaces in Euclidean space E are interesting. Of course,the notion of Myller configuration can be extended to the geometry ofnonholomonic manifolds in E which will be studied in next chapter. Itcan be applied to the theory of versor fields in E which has numer-ous applications to Mechanics, Hydrodynamics (see the papers by Gh.Gheorghiev and collaborators).Moreover, the Myller configurations can be defined and investigatedin Riemannian spaces and applied to the geometry of submanifolds ofthese spaces, [24], [25]. They can be studied in the Finsler, Lagrange orHamilton spaces [58], [68], [70]. M t hapter 5Applications of theory ofMyller configurations in thegeometry of nonholonomicmanifolds from E The second efficient applications of the theory of Myller configurationscan be done in the geometry of nonholonomic manifolds E in E . Someimportant results, obtained in the geometry of manifolds E by Issalyl ′ Abee, D. Sintzov [40], Gh. Vr˘anceanu [44], [45], Gr. Moisil [72], M.Haimovici [14], [15] Gh. Gheorghiev [10], [11], I. Popa [12], G. Th. Ghe-orghiu [13], R. Miron [30], [64], [65], I. Creang˘a [4], [5], [6], A. Dobrescu[7], [8] and I. Vaisman [41], [42], can be unitary presented by means of as-sociated Myller configuration to a curve embedded in E . Now, a numberof new concepts appears, the mean torsion, the total torsion, concurrenceof tangent vector field. The new formulae, as Mayer, Bortolotti-formulas,Dupin and Bonnet indicatrices etc. will be studied, too. E E R = ( P ; I , I , I ) be an orthonormed positively oriented frame in E . The application P ∈ E → R = ( P ; I , I , I ) of class C k , k ≥ C k in E . If r = −→ OP = xe + ye + ze isthe vector of position of point P , and P is the application point of theversors I , I , I , thus the moving equation of R can be expressed in theform (see Spivak, vol. I [76] and Biujguhens [2]): dr = ω I + ω I + ω I , (5.1.1)where ω i ( i = 1 , ,
3) are independent 1 forms of class C k − on E , and dI i = X j =1 ω ij I j , ( i = 1 , , , (5.1.2)with ω ij , ( i, j = 1 , ,
3) are the rotation Ricci coefficients of the frame R .They are 1-forms of class C k − , satisfying the skewsymmetric conditions ω ij + ω ji = 0 , ( i, j = 1 , , . (5.1.3)In the following, it is convenient to write the equations (5.1.2) in theform(5 . . ′ dI = rI − qI dI = pI − rI dI = qI − pI . Thus, thus structure equations of the moving frame R can be obtainedby exterior differentiating the equations (5.1.1) and (5.1.2) ′ modulo thesame system of equations (5.1.1), (5.1.2) ′ .One obtains, without difficulties Theorem 5.1.1
The structure equations of the moving frame R are dω = r ∧ ω , − q ∧ ω , dp = r ∧ q, .1. Moving frame in Euclidean spaces E dω = p ∧ ω − r ∧ ω , dq = p ∧ r, (5.1.4) dω = q ∧ ω − p ∧ ω , dr = q ∧ p. In the vol. II of the book of Spivak [76], it is proved the theorem ofexistence and uniqueness of the moving frames:
Theorem 5.1.2
Let ( p, q, r ) be 1-forms of class C k − , ( k ≥ on E which satisfy the second group of structure equation (5 . . , thus: ◦ . In a neighborhood of point O ∈ E there is a triple of vector fields ( I , I , I ) , solutions of equations (5.1.2) ′ , which satisfy initial conditions ( I (0) , I (0) , I (0)) -positively oriented, orthonormed triple in E . ◦ . In a neighborhood of point O ∈ E there exists a moving frame R = ( P ; I , I , I ) orthonormed positively oriented for which r = −→ OP isgiven by (5.1.1), where ω i ( i = 1 , , satisfy the first group of equations(5.1.4). ( I , I , I ) are given in ◦ and the initial conditions are verified: ( P = −→ OO ; I (0) , I (0) , I (0)) -the orthonormed frame at point O ∈ E . Remarking that the 1-forms ω , ω , ω are independent we can express1-forms p, q, r with respect to ω , ω , ω in the following form: p = p ω + p ω + p ω , q = q ω + q ω + q ω , (5.1.5) r = r ω + r ω + r ω . The coefficients p i , q i , r i are function of class C k − on E .Of course we can write the structure equations (5.1.4) in terms ofthese coefficients. Also, we can state the Theorem 5.1.2 by means ofcoefficients of 1-forms p, q, r .The 1-forms p, q, r determine the rotation vector Ω of the movingframe: Ω = pI + qI + rI . (5.1.6)Its orthogonal projection on the plane ( P ; I , I ) is θ = pI + qI . (5.1.7) E Let C be a curve of class C k , k ≥ E , given by r = r ( s ) , s ∈ ( s , s ) , (5.1.8)where s is natural parameter. If the origin P of moving frame describesthe curve C we have R = ( P ; I ( s ) , I ( s ) , I ( s )), where P ( s ) = P ( r ( s )), I j ( s ) = I j ( r ( s )), ( j = 1 , , drds to curve C is: drds = ω ( s ) ds I + ω ( s ) ds I + ω ( s ) ds I (5.1.9)and ds = h dr, dr i is: ds = ( ω ( s )) + ( ω ( s )) + ( ω ( s )) , (5.1.10)where ω i ( s ) are 1-forms ω i restricted to C .The restrictions p ( s ) , q ( s ) , r ( s ) of the 1-forms p, q, r to C give us: p ( s ) ds = p ( s ) ω ( s ) ds + p ( s ) ω ( s ) ds + p ( s ) ω ( s ) ds ,q ( s ) ds = q ( s ) ω ( s ) ds + q ( s ) ω ( s ) ds + q ( s ) ω ( s ) ds ,r ( s ) ds = r ( s ) ω ( s ) ds + r ( s ) ω ( s ) ds + r ( s ) ω ( s ) ds . (5.1.11) E Definition 5.2.1
A nonholonomic manifold E on a domain D ⊂ E isa nonintegrable distribution D of dimension 2, of class C k − , k ≥ . One can consider D as a plane field π ( P ), P ∈ D, the application P → π ( P ) being of class C k − .Also D can be presented as the plane field π ( P ) orthogonal to a versorfield ν ( P ) normal to the plane π ( P ), ∀ P ∈ D. .2. Nonholonomic manifolds E Assuming that ν ( P ) is the versor of vector V ( P ) = X ( x, y, z ) e + Y ( x, y, z ) e + Z ( x, y, z ) e and −→ OP = xe + ye + ze thus the nonholonomic manifold E is givenby the Pfaff equation: ω = X ( x, y, z ) dx + Y ( x, y, z ) dy + Z ( x, y, z ) dz = 0(5.2.1)which is nonintegrable, i.e. ω ∧ dω = 0 . (5.2.2)Consider a moving frame R = ( P ; I , I , I ) in the space E and thenonholonomic manifold E , on a domain D orthogonal to the versorsfield I . It is given by the Pfaff equations ω = h I , dr i = 0 , on D. (5.2.3)By means of (5.1.4) we obtain ω ∧ dω = − ( p + q ) ω ∧ ω ∧ ω . (5.2.4)So, the Pfaff equation ω = 0 is not integrable if and only if we have p + q = 0 , on D. (5.2.5)It is very known that:In the case p + q = 0 on the domain D , there are two functions h ( x, y, z ) = 0 and f ( x, y, z ) on D with the property hω = df. (5.2.6) E Thus the equation hω = 0 , have a general solution f ( x, y, z ) = c, ( c = const. ) , P ( x, y, z ) ∈ D. (5.2.7)The manifold E , in this case is formed by a family of surfaces, given by(5.2.7).For simplicity we assume that the class of the manifold E is C ∞ andthe same property have the geometric object fields or mappings whichwill be used in the following parts of this chapter.A smooth curve C embedded in the nonholonomic manifolds E , hasthe tangent vector α = drds given by (5.1.9) and ω = 0 . This is α ( s ) = drds = ω ( s ) ds I + ω ( s ) ds I , ∀ P ( r ( s )) ∈ D. (5.2.8)The square of arclength element, ds is ds = ( ω ( s )) + ( ω ( s )) . (5.2.9)And the arclength of curve C on the interval [ a, b ] ⊂ ( s , s ) is s = Z ba p ( ω ( σ )) + ( ω ( σ )) dσ. (5.2.10)A tangent versor field ξ ( s ) at the same point P ( s ) to another curve C ′ ,P ∈ C ′ and P ∈ C, can be given in the same form (5.2.8): ξ ( s ) = δrδs = e ω ( s ) δs I + e ω ( s ) δs I , (5.2.11)with δs = ( e ω ( s )) + ( e ω ( s )) .Thus, ξ ( s ) is a versor field tangent to the nonholonomic manifold E along to the curve C .To any tangent vector field ( C, ξ ) to E along the curve C belonging to E we uniquely associated the tangent Myller configuration M t ( C, ξ, I ). .3. The invariants G, R, T of a tangent versor field (
C, ξ ) in E We say that: the geometry of the associated tangent Myller configuration M t ( C, ξ, I ) is the geometry of versor field ( C, ξ ) on E . In particularthe geometry of configuration M t ( C, α, I ) is the geometry of curve C inthe nonholonomic manifold E .The Darboux frame of ( C, ξ ) on E is R D = ( P ; ξ, µ, I ), with µ = I × ξ , i.e.: µ = e ω ( s ) δs I − e ω ( s ) δs I . (5.2.12)In Darboux frame R D , the tangent versor α = drds to C can be expressedin the following form: α ( s ) = cos λ ( s ) ξ ( s ) + sin λ ( s ) µ ( s ) . (5.2.13)Taking into account the relations (5.2.8), (5.2.11) and (5.2.13), one gets: ω ds = cos λ e ω δs − sin λ e ω δs ,ω ds = sin λ e ω δs + cos λ e ω δs . (5.2.14)For λ ( s ) = 0, we have α ( s ) = ξ ( s ). G, R, T of a tangent ver-sor field ( C, ξ ) in E G, K, T of (
C, ξ ) satisfy the fundamental equations dξds = Gµ + KI , dµds = − Gξ + T I , dI ds = − Kξ − T µ. (5.3.1)Consequently, G ( δ, d ) = (cid:28) ξ, dξds , I (cid:29) E K ( δ, d ) = (cid:28) dξds , I (cid:29) = − (cid:28) ξ, dI ds (cid:29) (5.3.2) T ( δ, d ) = (cid:28) ξ, I , dI ds (cid:29) . The proofs and the geometrical meanings are similar to those fromChapter 3.By means of expression (5.2.11) of versors ξ ( s ) we deduce: dξds = (cid:20) dds (cid:18) e ω δs (cid:19) − rds e ω δs (cid:21) I + (cid:20) dds (cid:18) e ω δs (cid:19) + r e ω δs (cid:21) I ++ (cid:20) pds e ω δs − qds e ω δs (cid:21) I , (5.3.3)and dI ds = (cid:16) q ω ds + q ω ds (cid:17) I − (cid:16) p ω ds + p ω ds (cid:17) I . (5.3.4)The formulae (5.3.2) lead to the following expressions for geodesiccurvature G ( δ, d ), normal curvature K ( δ, d ) and geodesic torsion T ( δ, d )of the tangent versor field ( C, ξ ) on the nonholonomic manifold E : G ( δ, d ) = e ω δs (cid:20) dds (cid:18) e ω δs (cid:19) + rds e ω δs (cid:21) − e ω δs (cid:20) dds (cid:18) e ω δs − rds e ω δs (cid:19)(cid:21) , (5.3.5) K ( δ, d ) = p e ω − q e ω δs · ds (5.3.6) T ( δ, d ) = p e ω + q e ω δsds . (5.3.7)All these formulae take very simple forms if we consider the angles α = ∢ ( α, I ), β = ∢ ( ξ, I ) since we have ω ds = cos α, ω ds = sin α, e ω δs = cos β, e ω δs = sin β. (5.3.8) .3. The invariants G, R, T of a tangent versor field (
C, ξ ) in E With notations G ( δ, d ) = G ( β, α ); K ( δ, d ) = K ( β, α ) , T ( δ, d ) = T ( β, α )(5.3.9)the formulae (5.3.2) can be written: G ( β, α ) = dβds + r cos α + r sin α, (5.3.10) K ( β, α ) = ( p cos α + p sin α ) sin β − ( q cos α + q sin α ) cos β, (5.3.11) T ( β, α ) = ( p cos α + p sin α ) cos β + ( q cos α + q sin α ) sin β. (5.3.12)Immediate consequences:Setting T m = p + q , (5.3.13)(called the mean torsion of E at point P ∈ E ), H = p − q (5.3.14)(called the mean curvature of E at point P ∈ E ), from (5.3.11) and(5.3.12) we have: K ( β, α ) − K ( α, β ) = T m cos( α − β ) ,T ( β, α ) − T ( α, β ) = H cos( α − β ) . (5.3.15) Theorem 5.3.1
The following properties hold:1. Assuming β − α = ± π , the normal curvature K ( β, α ) is symmetricwith respect to the versor field ( C, ξ ) , ( C, α ) at every point P ∈ E , if andonly if the mean torsion T m of E vanishes ( E is integrable ) .2. For β − α = ± π , the geodesic torsion T ( β, α ) is symmetric withrespect to the versor fields ( C, ξ ) , ( C, α ) , if and only if the nonholonomicmanifold E has null mean curvature ( E is minimal ) . E The notion of conjugation of versor field (
C, ξ ) with tangent field (
C, α )on the nonholonomic manifold E is that studied for the associated Myllerconfiguration M t ,So that ( C, ξ ) are conjugated with (
C, α ) on E if and only if K ( δ, d ) =0 i.e. ( p ω + p ω ) e ω − ( q ω + q ω ) e ω = 0(5.4.1)or, by means of (5.3.11):( p cos α + p sin α ) sin β − ( q cos α + q sin α ) cos β = 0 . (5.4.2)All propositions established in section 3.3, Chapter 3, can be applied.The notion of orthogonal conjugation of ( C, ξ ) with (
C, α ) is given by T ( δ, d ) = 0 or by( p ω + p ω ) e ω + ( q ω + q ω ) e ω = 0(5.4.3)or, by means of (5.3.11), it is characterized by( p cos α + p sin α ) cos β + ( q cos α + q sin α ) sin β = 0 . (5.4.4)The relation of conjugation is symmetric iff E is integrable ( T m = 0)and that of orthogonal conjugation is symmetric iff E is minimal (i.e. H = 0).Now we study the case of tangent versor field ( C, ξ ) parallel along C ,in E . Applying the theory of parallelism in M t ( C, ξ, π ), from Chapter3 we obtain:
Theorem 5.4.1
The versors ( C, ξ ) , tangent to the manifold E alongthe curve C is parallel in the Levi-Civita sense if and only if the following .4. Parallelism, conjugation and orthonormal conjugation 91 system of equations holds e ω δs (cid:20) dds (cid:18) e ω δs (cid:19) + rds e ω δs (cid:21) − e ω δs (cid:20) dds (cid:18) e ω δs (cid:19) − rds e ω δs (cid:21) = 0 . (5.4.5)But, the previous system is equivalent to the system: dds (cid:18) e ω δs (cid:19) − rds e ω δs = h ( s ) e ω δsdds (cid:18) e ω δs (cid:19) + rds e ω δs = h ( s ) e ω δs , ∀ h ( s ) = 0 . (5.4.6)Since, the tangent vector field h ( s ) ξ ( s ) has the same direction with theversor ξ ( s ), from (5.2.11) we obtain G ( e δ, d ) = h G ( δ, d ), e δ being thedirection of tangent vector h ( s ) ξ . Therefore, the equations G ( δ, d ) =0 is invariant with respect to the applications ξ ( s ) → h ( s ) ξ ( s ). Thusthe equations (5.4.3) or (5.4.4) characterize the parallelism of directions( C, h ( s ) ξ ( s )) tangent to E along C .All properties of the parallelism of versors ( C, ξ ) studied for the con-figuration M t in sections 2.8, Chapter 2 are applied here.Theorem of existence and uniqueness of parallel of tangent versors( C, ξ ) on E can be formulated exactly as for this notion in M t .But a such kind of theorem can be obtained by means of the followingequation, given by (5.3.10): G ( β, α ) ≡ dβds + r ( s ) cos α + r ( s ) sin α = 0 . (5.4.7)The parallelism of vector field ( C, V ) tangents to E along the curve C can be studied using the form V ( s ) = k V ( s ) k ξ ( s )where ξ is the versor of V ( s ).Also, we can start from the definition of Levi-Civita parallelism of E tangent vector field ( C, V ), expressed by the property dVds = h ( s ) I . (5.4.8)Setting V ( s ) = V ( s ) I + V ( s ) I (5.4.9)and using (5.4.8) we obtain the system of differential equations dV ds − rds V = 0 , dV ds + rds V = 0 . (5.4.10)All properties of parallelism in Levi-Civita sense of tangent vectors( C, V ) studied in section 2.8, Chapter 2 for Myller configuration are valid.For instance
Theorem 5.4.2
The Levi-Civita parallelism of tangent vectors ( C, V ) in the manifold E preserves the lengths and angle of vectors. The concurrence in Myller sense of tangent vector fields (
C, ξ ) is char-acterized by Theorem 2.9.2 Chapter 2, by the following equations dds (cid:16) c G (cid:17) = c , (5.4.11)where c = h α, ξ i , c = h α, µ i , G = G ( δ, d ) = 0 . Consequence: the concurrence of tangent versor field (
C, ξ ) in E for G = 0 is characterized by dds (cid:20) e ω ω − e ω ω δs · ds · G (cid:21) = e ω ω + e ω ω δsds (5.4.12)or by: dds (cid:20) G sin( α − β ) (cid:21) = cos( α − β ) . (5.4.13)The properties of concurrence in Myller sense can be taken from section2.8, Chapter 2. .5. Theory of curves in E E C in the nonholonomic manifold E one can uniquely associatea Myller configuration M t = M t ( C, α, π ) where π ( s ) is the orthogonal tothe normal versor I ( s ) of E at every point P ∈ C. Thus the geometry of curves in E is the geometry of associatedMyller configurations M t . It is obtained as a particular case taking ξ ( s ) = α ( s ) from the previous sections of this chapter.The Darboux frame of the curve C , in E is given by R D = { P ( s ); α ( s ) , µ ∗ ( s ) , I ( s ) } , µ ∗ = I × α and the fundamental equations of C in E are as follows: drds = α ( s )(5.5.1)and dαds = κ g ( s ) µ ∗ + κ n ( s ) I , (5.5.2) dµ ∗ ds = − κ g ( s ) α + τ g ( s ) I , (5.5.3) dI ds = − κ n ( s ) α − τ g ( s ) µ ∗ . (5.5.4)The invariant κ g ( s ) is the geodesic curvature of C at point P ∈ C, κ n ( s )is the normal curvature of curve C at P ∈ C and τ g ( s ) it geodesic torsion of C at P ∈ C in E .The geometrical interpretations of these invariants are exactly thosedescribed in the section 3.2, Chapter 3. Also, a fundamental theoremcan be enounced as in section 3.2, Theorem 3.2.2, Chapter 3.The calculus of κ g , κ n and τ g is obtained by the formulae (5.3.2), for ξ = α . E We have: κ g = (cid:28) α, dαds , I (cid:29) , (5.5.5) κ n = (cid:28) dαds , I (cid:29) = − (cid:28) α, dI ds (cid:29) , (5.5.6) τ g = (cid:28) α, I , dI ds (cid:29) . (5.5.7)By using the expressions (3.1.4), Ch. 3 we obtain κ g = κ sin ϕ ∗ , κ n = κ cos ϕ ∗ , τ g = τ + dϕ ∗ ds (5.5.8)with ϕ ∗ = ∢ ( α , I ), α being the versor of principal normal of curve C at point P .A theorem of Meusnier can be formulates as in section 3.1, Chapter3. The line C is geodesic (or autoparallel ) line of the nonholonomic man-ifold E if κ g ( s ) = 0, ∀ s ∈ ( s , s ). Theorem 5.5.1
Along a geodesic C of the nonholonomic manifold E the normal curvature κ n is equal to ± κ and geodesic torsion τ g is equalto the torsion τ of C .The differential equations of geodesics are as follows κ g ≡ ω ds (cid:20) dds (cid:16) ω ds (cid:17) + r (cid:21) − ω ds (cid:20) dds (cid:16) ω ds (cid:17) − r (cid:21) = 0 . (5.5.9)This equations are equivalent to the system of differential equations dds (cid:16) ω ds (cid:17) − r ( s ) = h ( s ) ω ds , (5.5.10) dds (cid:16) ω ds (cid:17) + r ( s ) = h ( s ) ω ds , (5.5.11)where h ( s ) = 0 is an arbitrary smooth function.If we consider σ = ∢ ( I , ν ), where ν the versor of binormal is of the .6. The fundamental forms of E field ( C, I ) and χ is the torsion of ( C, I ), then the κ g ( s ), the geodesiccurvature of the field ( C, α ) is obtained by the formulas of Gh. Gheo-rghiev: κ g = dσds + χ . (5.5.12)It follows that: the geodesic lines of the nonholonmic manifold E arecharacterized by the differential equations: dσds + χ ( s ) = 0 . (5.5.13)By means of this equations we can prove a theorem of existence ofuniqueness of geodesic on the nonholonomic manifold E , when σ = ∢ ( I ( s ) , ν ( s )) is a priori given. E φ, ψ and χ of the non-holonomic manifold E are defined as in the theory of surfaces in theEuclidean space E . The tangent vector dr to a curve C at point P ∈ C in E is given by: dr = ω ( s ) I + ω ( s ) I , ω = 0 . (5.6.1)The first fundamental form of E at point P ∈ E is defined by: φ = h dr, dr i = ( ω ( s )) + ( ω ( s )) , ω = 0 . (5.6.2)Clearly, φ has geometric meaning and it is a quadratic positive definedform.The arclength of C is determined by the formula (5.2.10) and thearclength element ds is given by: ds = φ = ( ω ) + ( ω ) , ω = 0 . (5.6.3) E The angle of two tangent vectors dr and δr = e ω I + e ω I is expressedby cos θ = h δr, dr i δsds = e ω ω + e ω ω p ( ω ) + ( ω ) p ( e ω ) + ( e ω ) . (5.6.4)The second fundamental from ψ of E at point P is defined by ψ = −h dr, dI i = pω − qω = p ω + ( p − q ) ω ω − q ω . (5.6.5)The form ψ has a geometric meaning. It is not symmetric.The third fundamental form χ of E at point P is defined by χ = h dr, θ i , ω = 0 , (5.6.6)where θ is given by (5.1.7).As a consequence, we have χ = pω + qω = p ω + ( p + q ) ω ω + q ω (5.6.7) χ has a geometrical meaning and it is not symmetric.One can introduce a fourth fundamental form of E , byΘ = h dI , dI i = p + q ( mod ω ) . But Θ linearly depends by the forms φ, ψ, χ . Indeed, we haveΘ =
M ψ − K t φ − T m χ, where M is mean curvature, T m is mean torsion and K t is total curvatureof E at point P. The expression of K t is K t = p q − p q . (5.6.8)The formulae (5.5.6) and (5.5.7) and the fundamental forms φ, ψ, χ allow to express the normal curvature and geodesic torsion of a curve C .7. Asymptotic lines. Curvature lines 97 at a point P ∈ C as follows κ n = ψφ = pω − qω ω + ω = p ω + ( p − q ) ω ω − q ω ω + ω (5.6.9)and τ g = χφ = pω + qω ω + ω = p ω + ( p + q ) ω ω + q ω ω + ω . (5.6.10)The cases when κ n = 0 and τ g = 0 are important. They will be investi-gated in the next section. An asymptotic tangent direction dr to E at a point P ∈ E is defined bythe property κ n ( s ) = 0 . A line C in E for which the tangent directions dr are asymptotic directions is called an asymptotic line of the nonholonomicmanifold E .The asymptotic directions are characterized by the following equationof degree 2. p ω + ( p − q ) ω ω − q ω = 0 . (5.7.1)The realisant of this equation is the invariant K g = K t − T m , (5.7.2)called the gaussian curvature of E at point P ∈ E .We have Theorem 5.7.1
At every point P ∈ E there are two asymptotic direc-tions:- real if K g < - imaginary if K g > - coincident if K g = 0The point P ∈ E is called planar if the asymptotic directions of E at P are nondeterminated. E A planar point is characterized by the equations p − q = 0 , p = q = 0 . (5.7.3)The point P ∈ E is called elliptic if the asymptotic direction of E at P are imaginary and P is a hyperbolic point if the asymptotic directionsof E at P are real.Of course, if P is a hyperbolic point of E then, exists two asymptoticline through the point P , tangent to the asymptotic directions, solutionsof the equations (5.7.1).The curvature direction dr at a point P ∈ E is defined by the prop-erty τ g ( s ) = 0 . A line C in E for which the tangent directions dr arethe curvature directions is called a curvature line of the nonholonomicmanifold E .The curvature directions are characterized by the following secondorder equations. p ω + ( p + q ) ω ω + q ω = 0 . (5.7.4)The realisant of this equations is T t = K t − M . (5.7.5)We have Theorem 5.7.2
At every point P ∈ E there are two curvature direc-tionsreal, if T t < imaginary, if T t > coincident, if T t = 0The curvature lines on E are obtained by integrating the equations(5.7.1) in the case T t ≤ Remark 5.7.1
In the case of surfaces ( T m = 0) , the curvature linescoincides with the lines of extremal normal curvature. .8. The extremal values of κ n . Euler formula. Dupin indicatrix 99 κ n . Euler for-mula. Dupin indicatrix The extremal values at a point P ∈ E , of the normal curvature κ n = ψφ when ( ω , ω ) are variables are given by φ ∂ψ∂ω i − ψ ∂φ∂ω i = 0 , ( i = 1 , ∂ψ∂w i − κ n ∂φ∂ω i = 0 , ( i = 1 , . (5.8.1)Taking into account the form (5.6.5) of ψ and (5.6.3) of φ , the system(5.8.1) can be written: − ( q + κ n ) ω + 12 ( p − q ) ω = 0(5.8.2) 12 ( p − q ) ω + ( p − κ n ) ω = 0 . (5.8.3)It is a homogeneous and linear system in ( ω , ω )-which gives thedirections ( ω , ω ) of extremal values of normal curvature.But, the previous system has solutions if and only if the followingequations hold (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ( q + κ n ) 12 ( p − q )12 ( p − q ) p − κ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0equivalent to the following equations of second order in κ n : κ n − ( p − q ) κ n + p q − p q −
14 ( p + q ) = 0 . (5.8.4)This equation has two real solutions 1 R and 1 R because its realisant is∆ = ( p + q ) + ( p − q ) . (5.8.5)We have ∆ ≥ . Therefore the solutions 1 R , R are real, different or
00 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E equal, 1 R , R are the extremal values of normal curvature κ n .Let us denote H = 1 R + 1 R = p − q (5.8.6)called the mean curvature of the nonholonomic manifold E at point P ,and K g = 1 R R = p q − p q −
14 ( p + q ) . (5.8.7)called the Gaussian curvature of E at point P .Substituting 1 R , R in the equations (5.8.2) we have the directionsof extremal values of the normal curvature-called the principal directions of E at point P ∈ E . These directions are obtained from (5.8.2) for κ n = 1 R , κ n = 1 R . Thus, one obtains the following equations whichdetermine the principal directions:( p − q ) ω + 2( p + q ) ω ω − ( p − q ) ω = 0 . (5.8.8)Let ( ω , ω ) , ( e ω , e ω ) the solution of (5.8.8). Then dr = ω I + ω I and δr = e ω I + e ω I are the principal directions on E at point P. The principal directions dr, δr of E in every point P are real andorthogonal.Indeed, because ∆ >
0, and if 1 R = 1 R we have h δr, dr i = 0.The curves on E tangent to δr and dr are called the lines of extremalnormal curvature . So, we have Theorem 5.8.1
At every point P ∈ E there are two real and orthogonallines of extremal normal curvature. Assuming that the frame R = ( P ; I , I , I ) has the vectors I , I in the principal directions δr, dr respectively, thus the equation (5.8.8)implies p − q = 0(5.8.9) .8. The extremal values of κ n . Euler formula. Dupin indicatrix 101 and the extremal values of normal curvature κ n are given by κ n − ( p − q ) κ n − p q = 0 . (5.8.10)We have 1 R = p , R = − q . (5.8.11)Denoting by α the angle between the versor I and the versor ξ of anarbitrary direction at P , tangent to E we can write the normal curvature κ n from (5.6.8) in the form κ n = 1 R cos α + 1 R sin α, α = ∢ ( ξ, I ) . (5.8.12)The formula (5.8.12) is called the Euler formula for normal curvatureson the nonholonomic manifold E .Consider the tangent vector −→ P Q = | κ n | − / ξ , i.e. −→ P Q = | κ n | −
12 (cos αI + sin αI ) . The cartesian coordinate ( x, y ) of the point Q , with respect to the frame( P ; I , I ) are given by x = | κ n | −
12 cos α, y = | κ n | −
12 sin α. (5.8.13)Thus, eliminating the parameter α from (5.8.12) and (5.8.13) are getsthe geometric locus of the point Q (in the tangent plan ( P ; I , I )): x R + y R = ± the Dupin ′ s indicatrix of the normal curvature of E at point P .It is formed by a pair of conics, having the axis in the principaldirections δr, dr and the invariants: H -the mean curvature and K g - the
02 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E gaussian curvature of E at P .The Dupin indicatrix is formed by two ellipses, one real and anotherimaginary, if K g > . It is formed by a pair of conjugate hyperbolae if K g <
0, whose asymptotic lines are tangent to the asymptotic lines of E at point P . It follows that the asymptotic directions of E at P aresymmetric with respect to the principal directions of E at P. In the case K g = 0, the Dupin indicatrix of E at P is formed by apair of parallel straight lines - one real and another imaginary. τ g . Bonnet for-mula. Bonnet indicatrix The extremal values of the geodesic torsion τ g = χφ at the point P , on E can be obtained following a similar way as in the previous paragraph.Such that, the extremal values of τ g are given by the system of equa-tions ∂χ∂ω i − τ g ∂φ∂ω i = 0 , ( i = 1 , . Or, taking into account the expressions of φ = ( ω ) + ( ω ) and χ = pω + qω = ( p ω + p ω ) ω + ( q ω + q ω ) ω , we have:( p − τ g ) ω + 12 ( p + q ) ω = 0(5.9.1) 12 ( p + q ) ω + ( q − τ g ) ω = 0 . But, these imply (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − τ g
12 ( p + q )12 ( p + q ) q − τ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0(5.9.2) .9. The extremal values of τ g . Bonnet formula. Bonnet indicatrix 103 or expanded: τ g − ( p + q ) τ g + ( p q − p q −
14 ( p − q ) ) = 0 . (5.9.3)The solutions 1 T , T of this equations are the extremal values of geodesictorsion τ g . They are real. The realisant of (5.9.3) is∆ = ( p − q ) + ( p + q ) (5.9.4)Therefore 1 T , T are real, distinct or coincident.We denote T m = 1 T + 1 T = p + q , (5.9.5) the mean torsion of E at point P. And T t = 1 T T = p q − p q −
14 ( p − q ) = K g − H (5.9.6)is called the total torsion of E at P. But, as we know the condition of integrability of the Pfaf equation ω = 0 is dω = 0 , modulo ω , which is equivalent to T m = p + q = 0 . Theorem 5.9.1
The nonholonomic manifold E of mean torsion T m equal to zero is a family of surfaces in E . The directions δr = e ω I + e ω I and dr = ω I + ω I of extremal geodesictorsion, corresponding to the extremal values 1 T and 1 T of τ g are givenby the equation obtained from (5.9.2) by eliminating τ g i.e.( p + q ) ω − ( p − q ) ω ω − ( p + q ) ω = 0 . (5.9.7)Thus δr and dr are real and orthogonal at every point P ∈ E for
04 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E which 1 T = 1 T . δr and dr are called the direction of the extremal geodesictorsion at point P ∈ E . Theorem 5.9.2
At every point P ∈ E , where T = 1 T there exist tworeal and orthogonal directions of extremal geodesic torsion. Now, assuming that, at point P , the frame R = ( P ; I , I , I ) has thevectors I and I in the directions δr , dr of the extremal geodesic torsion,respectively, from (5.9.7) we deduce p + q = 0 . (5.9.8)Thus, the equation (5.9.3) takes the form τ g − ( p + q ) τ g + p q = 0 . (5.9.9)Its solutions are 1 T = p , T = q (5.9.10)and, setting β = ∢ ( ξ, I ), from τ g = χφ one obtains [65] the so called Bonnet formula giving τ g for the nonholonomic manifold E : τ g = cos βT + sin βT , β = ∢ ( ξ, I ) . (5.9.11)If T m = 0 , (5.9.11) reduce to the very known formula from theory ofsurfaces: τ g = 1 T cos 2 β. (5.9.12)By means of the formula (5.9.9) we can determine an indicatrix ofgeodesic torsions.In the tangent plane at point P we take the cartesian frame ( P ; I , I ), I having the direction of extremal geodesic torsion δr correspondingto 1 T and I having the same direction with dr corresponding to 1 T . .9. The extremal values of τ g . Bonnet formula. Bonnet indicatrix 105 Consider the point Q ′ ∈ π ( s ) given by −−→ P Q ′ = | τ g | − ξ = | τ g | − (cos βI + sin βI )(5.9.13)with β = ∢ ( ξ, I ). The cartesian coordinate ( x, y ) at point Q ′ : −−→ P Q ′ = xI + yI , by means of (5.9.13) give us x = | τ g | − cos β, y = | τ g | − sin β. (5.9.14)Eliminating the parameter β from (5.9.14) and (5.9.11) one obtainsthe locus of point Q ′ : x T + y T = ± , (5.9.15)called the indicatrix of Bonnet for the geodesic torsions at point P ∈ E .It consists of two conics having the axis the tangents in the directionsof extremal values of geodesic torsion. The invariants of these conics are T m -the mean torsion and T t -the total torsion of E at point P. The Bonnet indicatrix is formed by two ellipses, one real and anotherimaginary, if and only if the total torsion T t is positive. It is composedby two conjugated hyperbolas if T t < . In this case the curvature lines of E at P are tangent to the asymptotics of these conjugated hyperbolas.Finally, if T t = 0 , the Bonnet indicatrix (5.9.15) is formed by two pair ofparallel straightlines, one being real and another imaginary. Remark 5.9.1
1. If T t < , the directions of asymptotics of the hyper-bolas (5 . .
15) are symmetric to the directions of the extremal geodesictorsion.2. The direction of extremal geodesic torsion are the bisectrices ofthe principal directions.3. In the case of surfaces, T m = 0 , the Bonnet indicatrix is formed bytwo equilateral conjugates hyperbolas.Assuming that the Dupin indicatrix is formed by two conjugate hyper-bolas and the Bonnet indicatrix is formed by two conjugate hyperbolas,we present here only one hyperbolas from every of this indicatrices as
06 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E follows (Fig. 1): .10. The circle of normal curvatures and geodesic torsions 107 Fig. 1
Consider a curve C ⊂ E and its tangent versor α at point P ∈ C. Denoting by α = ∢ ( α, I ) and using the formulae (5.3.8) we obtain κ n = p sin α + ( p − q ) sin α cos α − q c cos ατ g = p cos α + ( p + q ) sinα cos α + q sin α. (5.10.1)Eliminating the parameter α from (5.10.1) we deduce κ n + τ g − Hκ n − T m τ g − K g = 0(5.10.2)where H = p − q is the mean curvature of E at point P , T m = p + q is the mean torsion of E at P and K g = p q − p q is the Gaussian
08 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E curvature of E at point P . Therefore we have Theorem 5.10.1
1. The normal curvature κ n and the geodesic torsion τ g at a point P ∈ E satisfy the equations (5 . . .2. In the plan of variable ( κ n , τ g ) the equation (5 . . represents acircle with the center (cid:18) H , T m (cid:19) and of the radius given by R = 14 ( H + T m ) − K g . (5.10.3)Evidently, this circle is real if we have K g <
14 ( H + T m ) . It is a complex circle if K g >
14 ( H + T m ). This circle, for the non-holonomic manifolds E was introduced by the author [62]. It has beenstudied by Izu Vaisman in [41], [42].In the case of surfaces, T m = 0 and (5.10.3) gives us R = 14 H − K g = 14 (cid:18) R + 1 R (cid:19) − R R = 14 (cid:18) R − R (cid:19) . So, for surfaces the circle of normal curvatures and geodesic torsions isreal.If T m = 0, the total torsion is of the form T t = − (cid:18) R − R (cid:19) . (5.10.4)This formula was established by Alexandru Myller [35]. He provedthat T t is just “the curvature of Sophy Germain and Emanuel Bacaloglu”([35]). .11. The nonholonomic plane and sphere 109 The notions of nonholonomic plane and nonholonomic sphere have beendefined by Gr. Moisil who expressed the Pfaff equations, provided theexistence of these manifolds, and R. Miron [64] studied them. Gh.Vr˘anceanu, in the book [44] has studied the notion of non holonomicquadrics.The Pfaff equation of the nonholonomic plane Π given by Gr. Moisil[30], [71] is as follows ω ≡ ( q z − r y + a ) dx +( r x − p z + b ) dy +( p y − q x + c ) dz = 0 , (5.11.1)where p , q , r , a, b, c are real constants verifying the nonholonomy con-dition: ap + bq + cr = 0.While, the nonholonomic sphere Σ has the equation given by Gr.Moisil [30]: ω ≡ [2 x ( ax + by + cz ) − a ( x + y + z ) + µx + q z − r y + h ] dx +[2 y ( ax + by + cz ) − b ( x + y + z ) + µy + r x − p z + k ] dy +[2 z ( ax + by + cz ) − c ( x + y + z ) + µz + p y − q x + l ] dz. (5.11.2) µ, p , q , r , a, b, c, h, k, l being constants which verify the condition ω ∧ dω = 0 . In this section we investigate the geometrical properties of these spe-cial manifolds.1.
The nonholonomic plane Π The manifold Π is defined by the property ψ ≡ ψ being the secondfundamental form (5.6.3) of E .From the formula (5.6.3) is follows p = q = p − q = 0 . (5.11.3)
10 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E The nonholonomic plane Π has the invariants H, T m , . . . given by H = 0 , T m = 0 , T t = 14 T m , K t = T t , K g = 0 . (5.11.4)Conversely, (5.11.4) imply (5.11.3).Therefore, the properties (5.11.4), characterize the nonholonomic planeΠ .Thus, the following theorems hold: Theorem 5.11.1
1. The following line on Π are nondetermined:-The lines tangent to the principal directions.- The lines tangent to the directions of extremal geodesic torsion.- The asymptotic lines. Theorem 5.11.2
Also, we have- The lines of curvature coincide with the minimal lines, h δr, δr i = 0 .The normal curvature κ n ≡ . - The geodesic of Π are straight lines. Consequences: the nonholonomic manifold Π is totally geodesic.Conversely, a total geodesic manifold Π is a nonholonomic plane.If T m = 0 , Π is a family of ordinary planes from E . The nonholonomic sphere, Σ .A nonholonomic manifold E is a nonholonomic sphere Σ if the sec-ond fundamental from ψ is proportional to the first fundamental formΦ . By means of (5.6.3) it follows that Σ is characterized by the followingconditions p − q = 0 , p + q = 0 . (5.11.5)It follows that we have ψ = 12 Hφ, χ = 12 T m φ, Θ = K t φ. (5.11.6)Thus, we can say .11. The nonholonomic plane and sphere 111 Proposition 5.11.1
1. The normal curvature κ n , at a point P ∈ Σ isthe same (cid:18) = 12 H (cid:19) in all tangent direction at point P .2. The geodesic torsion τ g at point P ∈ Σ is the same in all tangentdirection at P. Proposition 5.11.2
1. The principal directions at point P ∈ E arenondetermined.2. The directions of extremal geodesic torsion at P are non determi-nated, too. Proposition 5.11.3
At every point P ∈ Σ the following relations hold: K g = 14 H , T t = 14 T m , K t = 14 ( H + T m ) , (5.11.7)Conversely, if the first two relations (5.11.7) are verified at any point P ∈ E , then E is a nonholonomic sphere.Indeed, (5.11.7) are the consequence of (5.11.5). Conversely, from K g = 14 H , T t = 14 T m we deduce (5.11.5). Proposition 5.11.4
The nonholonomic sphere Σ has the properties K g > , T t > , K t > . (5.11.8) Proposition 5.11.5
1. If H = 0 , the asymptotic lines of Σ are imagi-nary.2. The curvature lines of Σ , ( T m = 0) are imaginary. The geodesic of Σ are given by the equation (5.5.7) or by the equation(5.5.11). Theorem 5.11.3
The geodesics of nonholonomic sphere Σ cannot bethe plane curves.
12 Chapter 5. Applications of theory of Myller configurations in the geometry ofnonholonomic manifolds from E Proof.
By means of theorem 5.5.1, along a geodesic C we have κ n = | κ | ,τ g = τ , κ and τ being curvature and torsion of C . From (11.6) we get κ n = 12 H, τ g = 12 T m . So, the torsion τ of C is different from zero. It can not be a plane curve.We stop here the theory of nonholonomic manifolds E in the Eu-clidean space E , remarking that the nonholonomic quadric was investi-gated by G. Vr˘anceanu and A. Dobrescu [8], [44], [45]. eferences PART A (1966) [1] Bianchi, L.,
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Lect¸ii de geometrie , Bucure¸sti, Ed. didactic˘a¸si pedagogic˘a, 1965.[10] Gheorghiev, Gh.,
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20 References ontents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Preface of the book
Geometria configurat¸iilor Myller written in1966 by Octav Mayer, former member of the RomanianAcademy . . . . . . . . . . . . . . . . . . . . . . . . . . . 7A short biography of Al. Myller . . . . . . . . . . . . . . 7Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Versor fields. Plane fields in E . . . . . . . . . . . . . 19 C, ξ ) . . . . . . . . . . . . . . . . . . . . . 191.2 Spherical image of a versor field (
C, ξ ) . . . . . . . . . . 221.3 Plane fields (
C, π ) . . . . . . . . . . . . . . . . . . . . . . 23 M ( C, ξ, π ) . . . . . . . . . . . . . 25 G, K, T . . . . . . . . . . . . . 302.4 Relations between the invariants of the field (
C, ξ ) and theinvariants of (
C, ξ ) in M ( C, ξ, π ) . . . . . . . . . . . . . . 312.5 Relations between invariants of normal field (
C, ν ) andinvariants
G, K, T . . . . . . . . . . . . . . . . . . . . . . 322.6 Meusnier ′ s theorem. Versor fields ( C, ξ ) conjugated withtangent versor (
C, α ) . . . . . . . . . . . . . . . . . . . . 33
22 Contents
C, ξ ) with null geodesic torsion . . . . . . . 352.8 The vector field parallel in Myller sense in configurations M M . . . . . . . . . . . 40 M t . . . . . . . . . . . . 45 M t . . . . . . . . . . . . . 463.2 The invariants of the curve C in M t . . . . . . . . . . . . 473.3 Geodesic, asymptotic and curvature lines in M t ( C, α, π ) 493.4 Mark Krein ′ s formula . . . . . . . . . . . . . . . . . . . . 513.5 Relations between the invariants G, K, T of the versor field(
C, ξ ) in M t ( C, ξ, π ) and the invariants of tangent versorfield (
C, α ) in M t . . . . . . . . . . . . . . . . . . . . . . 52 M t E . . . . . . . . . 584.2 Gauss and Weingarten formulae . . . . . . . . . . . . . . 614.3 The tangent Myller configuration M t ( C, ξ, π ) associatedto a tangent versor field (
C, ξ ) on a surface S . . . . . . 634.4 The calculus of invariants G, K, T on a surface . . . . . . 654.5 The Levi-Civita parallelism of vectors tangent to S . . . 704.6 The geometry of curves on a surface . . . . . . . . . . . . 734.7 The formulae of O. Mayer and E. Bortolotti . . . . . . . 75 E . . . . . . 81 E . . . . . . . . . . . 825.2 Nonholonomic manifolds E . . . . . . . . . . . . . . . . 845.3 The invariants G, R, T of a tangent versor field (
C, ξ ) in E E . . . . . . . . . . . . . . . . . . . 935.6 The fundamental forms of E . . . . . . . . . . . . . . . 955.7 Asymptotic lines. Curvature lines . . . . . . . . . . . . . 97 ontents 123 κ n . Euler formula. Dupin indicatrix 995.9 The extremal values of τ g . Bonnet formula. Bonnet indi-catrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.10 The circle of normal curvatures and geodesic torsions . . 1075.11 The nonholonomic plane and sphere . . . . . . . . . . . . 109 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 113
24 Contents24 Contents