An encounter in the realm of Structural Stability between a qualitative theory for geometric shapes and one for the integral foliations of differential equations
AAn encounter in the realm of Structural Stabilitybetween a qualitative theory for geometric shapes andone for the integral foliations of differential equations
Jorge Sotomayor ∗ September 15, 2020
To the memory of Maur´ıcio M. Peixoto (1921 - 2019), Carlos E. Harle(1937 - 2020), Carlos T. Guti´errez (1944 - 2008) and Daniel B. Henry (1945- 2002)
Abstract
This evocative essay focuses on some landmarks that led the author to the studyof principal curvature configurations on surfaces in R , their structural stability andgeneric properties. The starting point was an encounter with the book of D. Struikand the reading of the references to the works of Euler, Monge and Darboux foundthere. The concatenation of these references with the work of Peixoto, 1962, ondifferential equations on surfaces, was a crucial second step. The circumstancesof the convergence toward the theorems of Guti´errez and Sotomayor, 1982 - 1983,are recounted here. The above 1982 - 1983 theorems are pointed out as the firstencounter between the line of thought disclosed from the works of Monge, 1796,Dupin, 1815, and Darboux, 1896, with that transpiring from the achievements ofPoincar´e, 1881, Andronov - Pontrjagin, 1937, and Peixoto, 1962. Some mathemat-ical developments sprouting from the 1982 - 1983 works are mentioned on the finalsection of this essay. Keywords— umbilic point, principal curvature cycle, principal curvature lines.
MSC:
This story begins on a hot October night in 1970 in Rio de Janeiro. Suffering from insomnia, Idecided to snoop around the books that my wife had carefully accommodated in our shelf.My candid and relaxed attitude contrasted with a strange tension that emanated from theensemble of books, flooding the living room. Intrigued, I was impelled to find out the cause. ∗ The author is a fellow of CNPq, Grant: PQ-SR- 307690/2016-4. a r X i v : . [ m a t h . HO ] S e p Squeezed in a corner, wrapped in an elegant green cover, nervously pulsed the Aguilar,Spanish translation of the book “
Lectures on Classical Differential Geometry ” of D. Struik.My devotion to Geometry and to the language of Cervantes, drove me to, naively, open thatbook. And I did it just on a page from which, as if it had been lurking, the picture of the triaxialellipsoid, posted in Fig. 1, popped up.
Figure 1:
Monge’s Ellipsoid. Illustration of the ellipsoid with three different axes: x /a + y /b + z /c = 1 , a > b > c > , endowed with its Principal Curvature Configuration, whichconsists on its umbilic singular points and, outside them, its principal curvature lines. Theseparatrices are the curvature lines that connect the umbilic points.More than a century and a half had elapsed since the French mathematician Gaspard Mongeconceived it, calculating its principal curvature lines, and locating its four umbilic points. Mean-while, printed on books, restricted to an asphyxiating two-dimensional existence, the Ellipsoidhad traveled thousands of kilometers, transposed mountains and crossed wide seas, so that onthat singular tropical night, we could meet face to face. Figure 2:
Ellipsoid of Revolution. Illustration of the (oblate, left) ellipsoid with two equal axes: x /a + y /c + z /b = 1 , < a = c < b, its principal curvature lines and its pair of umbilicsingular points. The (prolate, center) ellipsoid corresponds to 0 < b < a = c. On the sphere,right, 0 < c = a = b, all the points are umbilic. As early as in 1961 I had browsed through Struik’s book, but I did not notice the picture.At that time, I also was fond to peruse the classic “
Geometry and the Imagination ” of Hilbertand Cohn Vossen, not so much to read it but for contemplating its pictures. From this book, asStruik annotates, was taken the alluded picture.How could I have ignored it?With meticulous attention I had studied two other excellent textbooks, of a comparable levelto that of Struik: Willmore’s “
Introduction to Differential Geometry ” as an undergraduatein 1961, and O’Neill’s, “
Elementary Differential Geometry ”, as a young instructor in 1967.However, none of these two remarkable models of geometric exposition contains not even anoutline of the fascinating picture. It should be mentioned, however, that O’Neil proposes acommented exercise in which he outlines how to locate the umbilic points of the ellipsoid, withno concern with their local principal configurations.It was like a love at first glance. The symmetry and beauty of its curves captivated meimmediately. As in a ritual of mutual measuring, we stared at each other for some minutes.I promptly read the adjacent text, as well as other relevant sections.The reader of that singular night, however, was somewhat distant from the naive student of1961. Having traveled along pathways of the Qualitative Theory of Differential Equations andDynamical Systems, he had his mathematical awareness enlarged and shaped by the study ofStructural Stability and Bifurcation Theory.In an hour of active reading I made a mathematical journey that, chronologically, hadspanned almost two centuries.There, the classical results concerning the principal curvatures on a surface were proved. Ireviewed Euler’s formula that expresses the normal curvature in terms of the principal curvaturesand directions.
Figure 3: Principal Directions at a point p on a Surface S oriented by the inward normalfield. The principal curvature lines are the curves, along a surface, whose tangent directions, denominated principal directions , make it to bend extremely in R . The scalar measures ofthese extreme curvatures, are called the principal curvatures . One of them, the minimum, isdenoted by k ; the other, the maximum, is designated by k . Their values are given by the normalcurvature (the second fundamental form of the surface), evaluated at the principal directions. So, k < k , except at the umbilic points , where k = k .Euler’s formula states that the normal curvature k n ( θ ) in the direction that makes angle θ with the minimum principal direction, is given by k n ( θ ) = k cos θ + k sin θ. This formula is equivalent to the diagonalization, by means of a rotation, of the SecondFundamental Form of Surface Theory.In
Recherches sur la courbure des surfaces,
M´emoires de l’Academie de Sciences de Berlin,16, (1760), Euler accomplished the first study of these mathematical objects, with which heinaugurated the use of Differential Calculus in the investigation of surface geometry. It isderived from this work that, in general, the principal directions are orthogonal.At umbilic points, the principal curvatures coincide. Only outside them, the principal di-rections are defined and determine a pair of mutually orthogonal tangent line fields, L and L ,called the principal line fields; L , corresponds to the minimal and L , to the maximal principalcurvatures. The surface is assumed to be oriented. An exchange in its orientation permutes L and L .With Struik I reviewed Dupin’s Theorem that allows to determine the lines of principalcurvature of surfaces that belong to a triply orthogonal family. To this end it is enough tointersect the surface under study with those of the two families orthogonal to it. In the caseof Monge’s Ellipsoid, one must take the hyperboloids of one and two sheets, which, togetherwith the ellipsoids, form the so-called homofocal system of quadrics. These quadrics are alsothe coordinate surfaces of the so-called Ellipsoidal coordinate system in R . I felt enraptured and rewarded. The browsing and reading session had allowed me to becomeaware of a remarkable qualitative jump in the evolution of the mathematical ideas, in threesteps.
Remark 1. A triple qualitative jump.
Firstly, Euler established the definitions and basic concepts.The second step was taken by Monge, calculating a key example.Thirdly, Dupin demonstrated a general theorem that unifies the particular cases, all integrable,known at that time.
From Dupin’s Theorem, a handy set of color pencils and numerous diagrams allowed me toreach the following conclusions:
Conclusion 1.
Consider the nine dimensional space Q of quadratic compact surfaces, endowedwith the topology defined by the, normalized, coefficients. The surfaces whose principal configu-rations are Structurally Stable (that is, they are not altered topologically by small perturbationsof their coefficients), are precisely the ellipsoids of Monge, whose three axes are distinct. Thisclass, denoted by E , is an open and dense subset in Q . Figure 4: Dupin’s Theorem. Illustration of Principal Configurations for HomofocalQuadrics: Ellipsoids and Hyperboloids, with one and two sheets.
Conclusion 2.
Inside Q = Q \ E , those whose principal configurations are First Order Struc-turally Stable (that is, they are not altered by small disturbances within Q ), are the ellipsoidsof non-spherical revolution; these form a sub-variety E , of codimension 2 in Q . Conclusion 3.
Inside the complement of E ∪ E in Q , the spheres form a submanifold ofdimension , that is of codimension . Analogous results could also be formulated for the non compact quadrics. While I wasquitting the long working session, the following inquiry struck me:
Problem 1.
How would it be, in general, the principally structurally stable configurations ofsmooth oriented compact non quadratic surfaces?
Two references from Struik had attracted my attention. One of them was a note in Volume III ofthe famous classical treatise of Gaston Darboux “
Le¸cons de la The´orie des Surfaces ,” Gauthier- Villars, 1888 - 1896, the other was an article in Acta Matematica, 1904, by Alvar Gullstrand.Having slept over Problem 1, next morning, I promptly perused them at the library of IMPA . National Institute of Pure and Applied Mathematics
Both were concerned with the possible configurations of the lines of principal curvature inthe vicinity of an umbilic point.The first one had a description of the three cases of generic umbilic points, characterizedby algebraic conditions, expressed as inequalities, involving the third derivatives of a functionrepresenting the surface in Monge coordinates centered at the umbilic point. See illustrationsin Fig. 5. Everything suggested that principal configurations of these types could be alsostructurally stable, locally at the umbilic point.As was usual, in works of around 1886, they were real analytic objects; surfaces in our case.The details of the proofs at the first readings seemed very hard to assimilate.
Figure 5:
Darbouxian Umbilic Points. The subscripts stand for the number of umbilic separa-trices for each of the principal foliations . Consider a surface in Monge form: z = ( k/ x + y ) + ( a/ x + ( b/ xy + ( c/ y + O [( x + y ) ] , in which the coefficient of the term x y has been eliminated by means of a rotation.The conditions that define the Darbouxian umbilic points are written as follows: Definition 1.
T) Transversality Condition: b ( b − a ) (cid:54) = 0;D) Discriminant Conditions:D : ab > (cid:16) c b (cid:17) + 2;D : (cid:16) c b (cid:17) + 2 > ab > a (cid:54) = 2 b ;D : ab < Back home, for a long while, I contemplated and tried to organize coherently the preciousantique pieces I had already collected.I quitted to rest earlier than usual.I woke up at dawn and, certain of being contributing to the fourth stage in the multiplemathematical qualitative jump witnessed the previous night, annotated in remark 1, withouthesitation, I wrote:
Problem 2.
The fundamental problem on the study of the Principal Curvature Configurationsof the differentiable, compact and oriented surfaces immersed into R , consists in establishingthe following two theorems: Theorem 4.
The necessary and sufficient conditions for a surface to have its Principal Config-uration Structurally Stable, with respect to C small deformations of its immersion into R , arethe following:4.a) The umbilic points must all be Darbouxian.4.b) The periodic curvature lines (principal cycles) must be all hyperbolic. That is, theirPoincar´e Transformation, or First Return Map, must have a derivative different of 1.4.c) It must not admit connections, or self-connections, of umbilical separatrices.4.d) The limit sets of any non-periodic principal curve must be an umbilic point or a principalcycle. Figure 6:
A periodic principal curvature line, i.e. a principal cycle, which is hyperbolic. Illus-tration of lines of curvature neighboring a principal curve, with first return map with derivativeless than 1. The arrows are placed to indicate a local conventional orientation. The lines ofcurvature, generally, are not globally orientable. See, for example, a neighborhood of the umbilicpoints in Fig. 5.
Theorem 5.
Within the space of immersions of compact oriented surfaces, endowed with the C topology, the surfaces that satisfy the conditions (4.a) to (4.d) form a set that is5.a) openand5.b) dense. A rough scheme of demonstration for Theorems 4 and 5, revealed that part (5.a) of Theorem5, as well as the sufficiency of conditions (4.a) to (4.d) for Theorem 4, seemed feasible. However,serious difficulties in establishing global aspects of the necessity of the conditions and, above all,the density, part (5.b) of Theorem 5, became apparent.At that time no example was known of a compact surface satisfying the conditions (4.a) to(4.d).The influence that Peixoto’s work on the genericity of the
Structurally Stable vector fields ontwo-dimensional manifolds,
Topology, 1962, was crucial in the formulation of the FundamentalProblem 2. This matter has been evoked in the author’s essay: • On a list of ordinary differential equations problems , S˜ao Paulo Journal of MathematicalSciences, https://doi.org/10.1007/s40863-018-0110-3 . Zbl 1417.01036 Zbl 1028.34001.MR3947401. • The article , in an OxfordUniversity webpage, contains a very objective outline of Differential Geometry, with pertinentlinks. Figure 7: Illustrations of principally structurally stable patterns.
A library search for examples led nowhere. Not a single clue for an isolated principal cycle,much less hyperbolic, was encountered.The same for surfaces with recurrent principal curvature lines, that is with those lines thatviolate condition (4d) in Theorem 4, as it is the case of integral curves of non singular vectorfields with irrational rotation number on the torus.The Qualitative Theory of Differential Equations and Dynamical Systems Theory, foundedby Poincar´e, almost 100 years before, had not penetrated this aspect of Principal Configurationsin Classical Differential Geometry, contrasting with the progress that dynamic ideas had in thestudy of geodesics.
A first round of calculations led to a formula for the derivative of the Poincar´e First ReturnMap, in terms of the Mean Curvature H = ( k + k ) /
2, the Gaussian Curvature: K = k k and the Christoffel Symbols. The elimination of these symbols was achieved later using Codazziintegrability conditions.The derivative, T (cid:48) , of the first return map, T , for a periodic curvature line γ , after simplifi-cation achieved the following expression:log( T (cid:48) ) = ± (cid:90) γ dH/ ( H − K ) / . I accumulated a large collection of bibliographic references on principal curvature lines andumbilic points, none of them elucidating Problem 2. Among them were several papers relativeto the Carath´eodory Conjecture:
Conjecture 6.
Every C convex and compact surface has at least two umbilic points.Years later I found out that this conjecture was considered demonstrated for the case ofanalytic surfaces. The conjecture is acknowledged to be open for C ∞ surfaces.For the generic case it is trivial. In fact, having the Darbouxian umbilic points indices ± / .I learned that Gullstrand, the author for me initially unknown, had been awarded the 1911Nobel Prize in Medicine, for his contribution to Ophthalmology. The Acta Mathematica paperthat I had browsed through was the mathematical part of the work that made him worthy ofthe award.Discussions about aspects of my project with colleagues, more experienced than me, inthe fields of Dynamical Systems and Differential Geometry, led to no mathematical reward.Unfortunately, Peixoto was traveling, outside Rio de Janeiro, during that crucial semester.I was lucky to meet the distinguished young American Geometer Herbert Blaine Lawson,visiting IMPA. He was very receptive and made the following thoughtful and stimulating com-ment:“ If what you propose works, it will be opening a new research area. ” In March 1971, I visited the University of S˜ao Paulo. I was asked by colleagues to deliver aseminar lecture presenting the subject of my research.This prompted me to organize the pertinent material I had gathered, starting with the classi-cal background: Euler, Monge, Dupin, then continuing with the geometric transformations that The biography of C. Carath´eodory by the historian Maria Georgiadou, Springer, 2004, suggests tohave elucidated the origin of this conjecture. However, the reference given leads to a work on umbilicpoints unrelated to the conjecture-problem 6. preserve the principal configurations: rigid motions, inversions and small parallel displacementsalong the normals to the surface, etc. Concluding with a discussion of the statements of Theo-rems 4 and 5, the Darbouxian Umbilics and, to include a personal contribution, the formula forthe derivative of the Poincar´e return map that I had found.Although the seminar lecture on Principal Configurations did not happen because of admin-istrative reasons, I could have rewarding discussions of local aspects of the project with WaldyrOliva and Edgard Harle (1937 - 2020).In particular, Edgard Harle, Geometer and fluent in German, helped me to understand thearticle of J. Fischer, Deutsche Math., 1935, written in the Gothic alphabet, giving an exampleof a principal configuration with spiraling behavior around an umbilic point.The surfaces that verified the conditions (4.b) and (4.d) would be of this sort, around prin-cipal cycles.Note that in the case of quadrics, surfaces of revolution and all the other examples establishedin Classical Differential Geometry, by the fact of deriving from a calculation, there is always afirst integral.The visit to S˜ao Paulo and the discussions, along the preparation for a seminar presentation,helped me to reinforce the conviction that the subject was not devoid of interest. The International Colloquium on Dynamical Systems of Salvador arrived, in July 1971. There Imet the French mathematician Ren´e Thom 1923 - 2002) with whom I discussed the expectationsI had around The Fundamental Problem 2.It was an interesting dialog.For him, the umbilics represented catastrophes, within the focal set: the envelope of thefamily of normal lines to the surface, that is the caustic of the surface.When I asked something about the umbilical separatrices associated with the points ofDarboux, he answered mentioning properties about the “ridges” associated with the hyperbolicand elliptic, (focal) umbilics of the generic surfaces.When I mentioned the nets of lines of principal curvature on surfaces, he answered me withthe focal set located in the ambient space.
In July 1972, just before traveling to Trieste, to the International Center for Theoretical Physics(ICTP), and to Paris, for a visit to the Institut de Hautes ´Etudes Scientifiques (IHES), CarlosGuti´errez, a doctoral candidate at IMPA, asked me for a suggestion of a thesis problem. Withoutthinking twice, I took out of my briefcase a copy of Darboux’s article (which always accompaniedme) and I proposed:
Problem 3.
Give a modern proof of this theorem, so that mortals can understand it.Prove that conditions (4.a) to (4.d) in the Fundamental Problem, Theorems 4 and 5, are generic.Pay special attention to the last condition (4.d).On that trip I carried with me a good part of the bibliographic material relevant to Problem3, in case I had to correspond with Guti´errez. To every Swedish mathematician I met in Trieste I commented about Gullstrand paper. Iinquired for the medical journal in which he published his contribution to Ophthalmology. Morethan one colleague promised to mail me a copy, which never arrived to my hands.By the end of November, I received a letter from Guti´errez. He had decided to considera more general form of Line Fields in two-dimensional manifolds. In that context, he alreadyhad a coherent theory involving a weak form of C density for a class of Line Fields defined byconditions resembling, topologically, those formulated in (4a) to (4d), Theorem 4.His topologically general theory, however, did not bring anything enlightening on the gener-icity of the principal line fields and their singularities (the umbilic points), as stated in theFundamental Problem 2.I encouraged him to continue on the path he had set out, which I also found interesting.When we met again in January 1973, we quickly discussed the difficulties encountered inattacking the problem incorporating the modification he suggested. We went on considering themore abstract approach, though quite distant form Classical Differential Geometry.In this direction he converged to writing a Doctoral Thesis, IMPA 1974. In this endeavor,Guti´errez became an exceptional expert in recurrent integral curves of line fields. A fact thatwill be crucial for the unfolding, in section 8, of the story recounted in this essay.For several years we did not speak again about umbilic points and principal curvature con-figurations. During the second semester of 1975 I visited Dijon for the first time. After perambulating severalhotels, I ended lodged at the small Monge Hotel , located on the street with the same illustriousname, in the heart of the charming neighborhood called “ Dijon Hist´orique” . I stayed there fora week.My host, Robert Roussarie from the University of Dijon, took me to visit the famous vine-yards of the Burgundy “
Cˆote d’Or”.
We extended the visit to the historic village of Beaune. There, after a tour by impressivemedieval buildings, I came upon, face to face, with an imposing statue of Gaspard Monge, https://commons.wikimedia.org/wiki/File:Beaune_-_Monument_de_Gaspard_Monge.jpg ,who was originally from that town.The picture of the ellipsoid encountered in 1970 in Rio de Janeiro, as evoked in section 1,reappeared in my memory. The cultural tour had excited my imagination. I was intrigued withthe coincidence of my encounter with the statue and with the name of my hotel. The memoryof the latent related problems 1, 2 and 3, came back.The coincidence of historical, cultural and mathematical circumstances intrigued me alongthe rest of my visit to Dijon, while being lodged at the Hotel Monge. I profited from the occasionto browse Monge’s book
Une application d’analyse `a la g´eom´etrie, 1795, at the University ofDijon Library, in the section of rare publications. There he studies the general properties ofprincipal curvatures and direction fields. After perusing some of its sections, I got convincedthat as a representative of the European Mathematics of the eighteenth century, Monge wouldhave had a hard time to understand my gibberish and problematics, in 1, 2 and 3, typical of ananonymous tropical redoubt of the decade of the sixties, in the 20th century. Presently the restaurant “Le Marrakech” operates in the building that hosted the hotel at 20, rueMonge. See and .. The link https://books.google.com.br/books?id=aSEOAAAAQAAJ&redir_esc=y , leads pre-sently to the above cited book of Monge.
Toward 1976, I had the help of mathematicians fluent in German, that translated for me sub-stantial portions of Gullstrand’s paper and helped me to interpret his results. It was clear thatthe author was strongly interested in the focal normal set, specially in how its ridge singularcurves approached the umbilic points.However, he did not address the analytic foundations to justify the principal configurationpictures of Darboux and, much less, the other more degenerate ones that he also considered inhis work. Subjects such as the uniqueness of the umbilical separatrices, for example, were nottouched at all.I packed the bibliographic material pertinent to the Fundamental Problem 2, Theorems 4and 5, and also my handwritten notes, in a plastic bag, which I carried up and down with me.For long periods, when my briefcase was too heavy, I used to leave the bag resting at home.Months later I would take it along with me again.The transparency of the plastic prompted me to locate and have the Fundamental Problempresent. The transportation ceremony ritual gave me a strange sensation of possession and ofbeing working on it, secretly.With uncertain periodicity, however, I engaged in longer intuitive exercises, without directlyattacking its fundamental mathematical difficulties.I particularly enjoyed extrapolating conditions 4.a to 4.d, weakening and adapting them to higherorder ones, predicting the generic bifurcations of principal curvature configurations on surfacesevolving subject to one parameter spatial deformations. This amounted to an additional stagein the sequence of qualitative jumps discussed in sections 1 and 2.A bolder, higher order, extrapolation to which the previous predictions pointed to were theprincipal configurations of 3 − dimensional manifolds immersed into R .In section 10 will be given references to works achieved later in the lines of research outlinedabove. At the beginning of 1980 we received a determination from the CNPq Central Administration:Researchers should be organized in teams, as in soccer, and present joint projects. This wasa novelty. We were accustomed to receiving requests to fill out new cadastre forms every timethere was a change in the administrative cadres.This naive administrative measure ended up being beneficial for me, for Carlos Guti´errezand also for the Fundamental Problem on Principal Curvature Configurations.I engaged Guti´errez to join forces with me to work in the Issue.The progress was surprisingly fast, taking into account that each of us worked simultaneouslyon individual projects. The two of us, and the Problem, had gained maturity.We updated the formula for the derivative, T (cid:48) , of the first return transformation, T , of a periodiccurvature line γ .In terms of the mean H and Gaussian K curvatures, the beautiful following expression,mentioned also in section 3, holds: log( T (cid:48) ) = ± (cid:90) γ dH/ ( H − K ) / . In fact, chronologically, it was obtained after its equivalent versionlog( T (cid:48) ) = ± (cid:90) γ dk / ( k − k ) , in terms of the principal curvatures k < k , since H = ( k + k ) / K = k k ,from which we derived a perturbation method to hyperbolize periodic curvature lines.Using a method of resolution (“blowing up”) of singularities, very close to the traditionalone, used to reduce the study of complicated singular points to simpler ones, we fully justified,for the case of surfaces of class C , the Darbouxian principal configuration pictures in Fig. 5.This was a novelty in relation to the analytic case considered by the French Geometer.On the recurrent principal lines of curvature, Carlos Guti´errez, formulated the followingdiagnosis: It will be possible to get an approximation in class C that eliminates them. The increasingfrom class C to C , however, will be very difficult. We produced an example of a Toroidal Surface in R without umbilic points and with denserecurrent curvature lines. Its immersion, however, was quite distant from the standard Torus ofrevolution.The pieces of the puzzle seemed to fit. We had a sustainable version!The time had come to make a broad communication of the results. The 1981 InternationalDynamical Systems Symposium inaugurating the new installations of IMPA, constituted anauspicious occasion. On the eve of the lecture, I stayed until later than usual discussing with Guti´errez what thepresentation would be like. Some delicate points emerged.It became explicit that we did not have examples of principal recurrences other than thatin the aforementioned, free of umbilics, Torus. In particular, we did not know immersions ofspheres, necessarily carrying umbilic points, exhibiting principal recurrences. Furthermore, atthat time the Torus example seemed too technical to be quickly explained geometrically duringthe lecture.We were in the uncomfortable position of being able to eliminate all the recurrent curvaturelines, replacing them after an approximation C -small, by periodic lines of curvature or, in somecases, by connections of umbilical separatrices. Specifically we knew only the example of theTorus.What if someone, interested in specific examples, raised the question? Worst still, what ifthere were no more examples of recurrences than those on the Torus?In that case, our results would be considerably weakened.The search for an example of principal recurrences on a surface of genus zero lasted a longwhile. It was then that Monge’s Ellipsoidal reappeared; this time it was more flexible than ever.It allowed more daring deformations and contortions, including non-analytic ones. So, bendingmore here, rotating there, we stumbled over an example.The lecture was ready! It would be the first one on the next morning.I started projecting Monge’s Ellipsoid (Fig. 1) and reviewed how to explain its principalconfiguration, using the Theorem of Dupin (Fig. 4). I commented on the absence of examplesof Global Principal Configurations, distinct from that one.After the preparatory setting concerning the space of Immersions of oriented two-manifoldsinto R , I proposed the problem of recognizing globally the immersion with Structurally Stableprincipal configurations and establishing their genericity. Then came the statement of our result,insisting on the limitation in class C for the density approach. I ended up summoning thelisteners to raise this class to C , thus solving the only problem that remained open in relationto the initial program, as stated in Theorems 4 and 5. Remark 2.
This problem remains open until now.
Reaching the end of the lecture, none of the experts in recurrences on Dynamical Systemsin the audience asked for specific examples. This was disappointing.Fortunately, Dan Henry (1945 - 2002), an expert in Differential Equations in Infinite Di-mensions, formulated the expected question:“ I do not know why you make the hypothesis (4.d). I do not know any surface that does notsatisfy it. ”I drew an ellipsoid of revolution (Fig. 2, left). I deformed it slightly so that, in both polarcaps, it was like one of Monge, with their three axes distinct, while around the equator itcontinued being one of revolution.Then, I rotated around the major axis only the upper hemisphere of the surface. Becauseof its equatorial rotational symmetry, this was a family of C ∞ surfaces E θ , depending on theparameter θ , designating the angle of rotation. Ellipsoidal Surface E θ with oscillatory recurrent principal curvature lines.It is evident that the curvature lines of E θ define in their second return to the equatorialcircle a rotation of angle 2 θ . So, for angles incommensurable with respect to 2 π , the lines ofcurvature become all dense in E θ . See illustration in Fig. 8.The written presentation of the work was divided in two parts.The first one, establishing that conditions (4.a) to (4.d) define a C open set consisting inStructural Stability immersed surfaces, was published in Asterisque , Vol. 98-99, 1982.The second part, which demonstrates that any compact and oriented surface can be arbi-trarily C approximated by one that verifies the four above mentioned conditions, appeared in Springer Lecture Notes in Mathematics , Vol. 1007, 1983.
These papers are pointed out as the documentation pertinent to the first encounter betweenthe line of thought disclosed from the works of Monge, 1796, Dupin, 1815, and Darboux, 1896,with that transpiring from the achievements of Poincar´e, 1881, Andronov - Pontrjagin, 1937,and Peixoto, 1962.
No photo of the lecture at IMPA, 1981, was found. Here is one from a lecture onPrincipal Configurations delivered in Montevideo, 2012. Photo by A. Chenciner. The ellipsoidpicture was borrowed from E. Ghys site.
In the first half of 1990, on the occasion of the “
Ann´ee Sp´eciale de Syst`emes Dynamiques”, sponsored by the CNRS (National Research Council) of France, I delivered a short course at theUniversity of Dijon. The subject was
Umbilic Points and Principal Curvature Lines.
During that visit I had access to the original work of Monge, entitled
Sur les lignes decourbure de la surface de l’Ellipsoide , published in
Journal de l’Ecole Polytechnique., II cah,1796 .The contemplation of Monge’s original pictures, supplemented with some reflections andadditional readings, led me to make explicit an intuitive observation about the history of math-ematical ideas, which, in an embryonic form, was already present in my initial motivation, afterthe first contact with the Ellipsoid and the subsequent formulation of the Fundamental Problem1. There is no bibliographic record that Euler, responsible for the conceptualization of the prin-cipal curvature line fields, would have integrated them, visualizing the Principal Configuration.Monge was the first mathematician to recognize the importance of this structure, providingthe first non-trivial example, for global integration of the differential equations of the lines ofcurvature (that is, of the line fields L and L ) in the case of the Ellipsoid.Let’s inquire: What would have led him to establish a result of this nature, which in ourdays fits perfectly within the Qualitative Theory of Ordinary Differential Equations, considering that this happened almost one hundred years before Poincar´e founded it and defined its goals? Monge’s motivation derived from a complex interaction of aesthetic and practical considerations,and also the explicit intention to apply the results of his mathematical research.At Monge’s time was more tenuous the artificial separation between Mathematics and Ap-plications. The lines of curvature were discovered studying the problem of transportation ofdebris in the construction of embankments and fortifications, called the “deblais et remblais” problem. His ellipsoidal picture was proposed in the architectural project for the constructionof the vault, over an elliptical terrain, of Legislative Assembly Building of the French RevolutionGovernment: the lines of curvature would be the guiding curves for the placement of the stonebricks, the umbilics would serve as supporting points for hanging the sources of illumination,under one of which would be located the rostrum for the speakers. The architectural projectwas never put into effect.
Remark 3.
If Poincar´e, for the scope and depth of its contribution, is recognized as the founder,Monge has the merit of being regarded as a precursor –a herald– of the Qualitative Theory ofDifferential Equations and of Foliations with Singularities Theory.
However, it should be emphasized that the novelty of the Qualitative Theory of Poincar´elies in the methods developed for the study of the phase portrait of general non-integrable dif-ferential equations, that he established for polynomial equations, in the generic case. Keepingin mind that this was a laboratory model for the far reaching and more complex problems ofCelestial Mechanics that he was investigating.The connection between the works of Monge and Poincar´e, outlined above, was overlookedin the accomplished historical work of Ren´e Tat´on “
L’Oeuvre Scientifique de Monge, ” PressesUniv. de France, 1951.
10 Books, later developments and updated refer-ences.
The project of writing an expository book based on the lectures delivered at the 1990 shortcourse in Dijon, was formulated on that occasion. See the beginning of section 9.In 1991, I engaged Guti´errez to collaborate with me in writing a small book of didacticvocation, proposed for the 18th Brazilian Mathematics Colloquium.Besides the results on Structural Stability and Approximation established in the originalpapers, the lecture notes included more details concerning the theoretical foundations and themotivation for the study of Principal Curvature Configurations. The examples of principal re-currences were improved and reformulated in more conceptual terms, especially the one devotedto the immersed Toroidal Surface . The bibliographic references were also updated. • C. Gutierrez and
J. Sotomayor,
Lines of Curvature and Umbilical Points on Surfaces,18th Brazilian Math. Colloquium, Rio de Janeiro, IMPA, (1991).Reprinted as Structurally Stable Configurations of Lines of Curvature and Umbilic Points onSurfaces, Lima, Monografias del IMCA, (1998). MR2007065. This example of 1991 has only one of the principal foliations with dense curves. In
R. Garcia and
J.Sotomayor,
Tori embedded in R with dense principal lines. Bull. Sci. Math., (2009), 348-354,was given an example in which both principal foliations have its lines dense. A book of broader scope than that of 1991, with more topics on the rich interaction of ClassicalDifferential Geometry and Differential Equations, was published in 2009. • R. Garcia and
J. Sotomayor , Differential Equations of Classical Geometry, a QualitativeTheory , Publica¸c˜oes Matem´aticas, 27 o Col´oquio Brasileiro de Matem´atica, IMPA, (2009). Zbl1180.53002. MR2532372. R . Without claiming to be complete, some additional works that are pertinent to the present essayare listed below in chronological order. • J. W. Bruce and
D. L. Fidal,
On binary differential equations and umbilic points,
Proc.Royal Soc. Edinburgh , 1989, 147-168. Zbl 0685.34004. • C. Gutierrez and
J. Sotomayor,
Principal lines on surfaces immersed with constant meancurvature.
Trans. Amer. Math. Soc. , 1986, no. 2, 751 - 766. MR816323. Zbl 0598.53007. • R. Garcia and
J. Sotomayor,
Lines of curvature near singular points of implicit surfaces .Bull. Sci. Math. 117 (1993), no. 3, 313 - 331. MR1228948. • R. Garcia and
J. Sotomayor,
Lines of curvature near hyperbolic principal cycles . Dynam-ical systems (Santiago, 1990), 255 - 262, Pitman Res. Notes Math. Ser., 285, Longman Sci.Tech., Harlow, 1993. MR1213951. • C. Gutierrez and
J. Sotomayor,
Periodic lines of curvature bifurcating from Darbouxianumbilical connections . Bifurcations of planar vector fields (Luminy, 1989), 196 - 229, LectureNotes in Math., 1455, Springer, Berlin, 1990. MR1094381. • R. Garcia and
J. Sotomayor,
Lines of Curvature on Algebraic Surfaces , Bull. SciencesMath. , (1996), 367-395. MR1411546. • J. Sotomayor,
Lines of curvature and an integral form of Mainardi-Codazzi equations.
An.Acad. Brasil. Ciˆenc. 68 (1996), no. 2, 133 - 137. MR1751266. • T. Maekawa, F. E. Wolter and
N. M. Patrikalakis,
Umbilics and lines of curvaturefor shape interrogation,
Comput. Aid. Geometr. Des. (1996), 133 –161. Zbl 0875.68858. • R. Garcia and
J. Sotomayor,
Structural stability of parabolic points and periodic asymp-totic lines , Matem´atica Contemporˆanea, , (1997), 83-102. MR163442. • C. Gutierrez and
J. Sotomayor,
Lines of Curvature, Umbilical Points and Carath´eodoryConjecture , Resenhas IME-USP, , 1998, 291-322. MR1633013. • R. Garcia , C. Gutierrez and
J. Sotomayor,
Lines of principal curvature around umbilicsand Whitney umbrellas.
Tohoku Math. J. (2) (2000), 163-172. MR1756092. • R. Garcia and
C. Gutierrez,
Ovaloids of R and their umbilics: a differential equationapproach. J. Differential Equations , 2000, no. 1, 200–211. MR1801351. • R. Garcia and
J. Sotomayor,
Structurally stable configurations of lines of mean curvatureand umbilic points on surfaces immersed in R , Publ. Matem´atiques. , (2001), 431-466.MR1876916. Zbl 0875.68858. • R. Garcia and
J. Sotomayor,
Umbilic and tangential singularities on configurations ofprincipal curvature lines.
An. Acad. Brasil. Ciˆenc. , 2002, no. 1, 1–17. MR1882514. • R. Garcia and
J. Sotomayor,
Lines of Geometric Mean Curvature on surfaces immersedin R , Annales de la Facult´e des Sciences de Toulouse, , (2002), 377-401. MR2015760. • R. Garcia and
J. Sotomayor,
Lines of Harmonic Mean Curvature on surfaces immersedin R , Bull. Bras. Math. Soc., , (2003), 303-331. MR1992644. • R. Garcia and
J. Sotomayor,
Lines of Mean Curvature on surfaces immersed in R , Qualit. Theory of Dyn. Syst. , 2004, 137-183. MR2129722. • R. Garcia , C. Gutierrez and
J. Sotomayor,
Bifurcations of Umbilic Points and RelatedPrincipal Cycles.
Journ. Dyn. and Diff. Eq.
16, 2 , (2004), 321-346. MR2129722. • R. Garcia and
J. Sotomayor,
On the patterns of principal curvature lines around a curveof umbilic points.
An. Acad. Brasil. Ciˆenc. 77 (2005), no. 1, 13 - 24. MR2114929 • R. Garcia and
J. Sotomayor , Lines of principal curvature near singular end points ofsurfaces in R . Singularity theory and its applications, 437-462, Adv. Stud. Pure Math., ,Math. Soc. Japan, Tokyo, (2006). MR2325150. • R. Garcia and
J. Sotomayor,
Tori embedded in R with dense principal lines. Bull. Sci.Math., (2009), 348-354. MR2503006. • R. Garcia, R. Langevin and
P Walczak , Foliations making a constant angle with principaldirections on ellipsoids.
Annales Polonici Mathematici, (2015), 165-173. MR3312099. • R. Garcia and
J. Sotomayor , Historical Comments on Monge’s Ellipsoid and the Config-urations of Lines of Curvature on Surfaces,
Antiquitates Mathematicae, (2016), 348-354.Zbl 1426.53008. MR3613151. • V. V. Ivanov,
An analytic conjecture of Carath´eodory.
Siberian Math. J. , no. 2, 2002,251–322. MR1902826. • B. Smyth and
F. Xavier,
A sharp geometric estimate for the index of an umbilic on asmooth surface.
Bull. London Math. Soc. , 1992, no. 2, 176–180. MR1148679. • B. Smyth and
F. Xavier,
Eigenvalue estimates and the index of Hessian fields.
Bull.London Math. Soc. , 2001, no. 1, 109–112. MR1798583. • B. Smyth,
The nature of elliptic sectors in the principal foliations of surface theory.
EQUAD-IFF 2003, 957–959, World Sci. Publ., Hackensack, NJ, 2005. R . An important achievement that must be mentioned is the extension of the generic properties ofthe lines of curvature associated to three-dimensional manifolds immersed in R , first obtainedby Ronaldo Garc´ıa in his Thesis (IMPA, 1989). See partial list below. • R. Garcia
Principal Curvature Lines near Partially Umbilic Points in hypersurfaces im-mersed in R , Comp. and Appl. Math., 20 , 121 - 148, 2001.In this direction, the following more recent developments should be added: • D. Lopes, J. Sotomayor and R.
Garcia , Umbilic and Partially umbilic singularities ofEllipsoids of R . Bulletin of the Brazilian Mathematical Society, , (2014), 453-483. • D. Lopes, J. Sotomayor and R.
Garcia , Partially umbilic singularities of hypersurfacesof R . Bulletin de Sciences Mathematiques, , (2015), 421- 472. • R. Garcia and
J. Sotomayor,
Lines of curvature on quadric hypersurfaces of R , LobachevskiiJournal of Mathematics , (2016), 288-306. R . The name axial refers to the association with the axes of the normal ellipse of curvature definedat the regular points of surfaces mapped into R . • R. Garcia and
J. Sotomayor,
Lines of axial curvature on surfaces immersed in R , Dif-ferential Geom. Appl. , (2000), 253–269. MR1764332. Zbl 0992.53010. • L. F. Mello,
Mean directionally curved lines on surfaces immersed in R , Publ. Mat. ,(2003), 415–440. • R. Garcia, L. Mello and
J. Sotomayor , Principal mean curvature foliations on surfacesimmersed in R , EQUADIFF (2003), 939-950, World Sci. Publ., Hackensack, NJ, 2005. Zbl1116.57024. • R. Garcia and
J. Sotomayor , Lines of axial curvature at critical points on surfaces mappedinto R , Sao Paulo J Math Sci , (2012), no. 2, 277- 300. • R. Garcia, J. Sotomayor and
F. Spindola , Axiumbilic Singular Points on SurfacesImmersed in R and their Generic Bifurcations, Journal of Singularities , (2014), 124-146. C to C , as inquired in Theorem 510.6.2 Recurrence on cubic deformations of Monge’s ellipsoid. From exercise3.6.3, p. 83, of the 2009, Garcia and Sotomayor book. For ρ small, consider the cubic surface S ρ = f − ρ (0) f ρ ( x, y, z ) = x a + y b + z + ρxyz − a > , b > , ( a − b − a − b ) (cid:54) = 0 . From simulations of the possible global behaviors of principal foliations of S ρ , formulate a con-jecture about the possibility of dense principal lines on algebraic surfaces of spherical type.The surface S ρ , for ρ (cid:54) = 0 is the simplest algebraic one that exhibits the twisting effectdepicted in the smooth example in Fig. 8. This effect, determined by the cubic term ρxyz , wasfirst studied in Garcia and Sotomayor, Bull. Sci. Math. 1993, for the case of the cubic surface x a + y b − z + ρxyz = 0; a > , b > , ( a − b − a − b ) (cid:54) = 0 , locally conoidal at its critical point (0 , , S ρ , for ρ (cid:54) = 0 has been publishedyet. Two lectures delivered by the author, overlapping with the subject of this essay have been postedin: • https://youtu.be/JX2pHiCvaxw , in Workshop International de Sistemas Dinˆamicos - 90Mauricio Peixoto, 2011, and in • https://youtu.be/IscUm7UHv50 , A lecture evocative of Carlos Guti´errez, 2018. At UNICAMP, 2000, three mathematical generations: right to left, C. T. Guti´errez,M. A. Teixeira, J. Sotomayor, M. M. Peixoto and R. A. Garcia . If we wish to foresee the future of Mathematics,the right approach is to study its history and present condition.For us mathematicians, is it not this procedure, to some extent, routine?We are accustomed to extrapolation, which is a methodof deducing the future from the past and the present;and, since we are well aware of its limitations, we run no riskof deluding ourselves as to the scope of the results it provides.
H. Poincar´e, in The Future of Mathematics (
L’Avenir des Math´ematiques ), speech read by G.Darboux at the ICM, Rome, 1908.
This work is an English version, which includes considerable revision, upgrading and adaptation,based on • O Elips´oide de Monge,
Portuguese, Mat. Univ., 15, 1993, and its Spanish translation inMaterials Matem`atiques, 2007: http://mat.uab.cat/matmat/PDFv2007/v2007n01.pdf . Acknowledgements.