Steepest descent curves of convex functions on surfaces of constant curvature
SSTEEPEST DESCENT CURVES OF CONVEX FUNCTIONSON SURFACES OF CONSTANT CURVATURE
C. GIANNOTTI AND A. SPIRO
Abstract.
Let S be a complete surface of constant curvature K = ± S or L , and Ω ⊂ S a bounded convex subset. If S = S , assumealso diameter(Ω) < π . It is proved that the length of any steepestdescent curve of a quasi-convex function in Ω is less than or equal tothe perimeter of Ω. This upper bound is actually proved for the class of G -curves, a family of curves that naturally includes all steepest descentcurves. In case S = S , it is also proved the existence of G -curves, whoselength is equal to the perimeter of their convex hull, showing that theabove estimate is indeed optimal. The results generalize theorems byManselli and Pucci on steepest descent curves in the Euclidean plane. Introduction
Let S be a complete surface of constant Gaussian curvature K = 0, +1 or −
1, i.e. the Euclidean plane E , the unit sphere S or the Lobachevskij plane L . An absolutely continuous curve γ : [ a, b ] → S (e.g. a rectifiable curveparameterized by arc length) is called G -curve if it satisfies the followingcondition (here and in the rest of the paper, we use the notation γ s = γ ( s )): for any s o ∈ [ a, b ] for which ˙ γ s o exists and is different from , allpoints γ s with s ≤ s o are in the same closed half space, bounded bythe normal to γ at γ s o .Notice that the class of G -curves naturally includes all steepest descentcurves of convex functions, that is the C -curves γ t satisfying equations ofthe form ˙ γ t = − grad f | γ t for some convex C -function f : U ⊂ S → R . The G -curves have been originally considered by Manselli and Pucci in [MP],where they determined an optimal upper bound for the length of G -curvescontained in a given bounded convex subset of E .Here we consider the problem of establishing similar upper bounds for thelengths of G -curves in S and L . Our first result consists of the followinggeneralization of the upper bound determined in [MP]: Theorem 1.1.
Let S = S , E or L and γ : [ a, b ] → S a G -curve. In case S = S , assume also that diameter( γ ([ a, b ])) < π . Then the length (cid:96) ( γ ) of γ is less than or equal to the perimeter p ( (cid:98) γ ) of the convex hull (cid:98) γ of γ . Date : November 6, 2018.2000
Mathematics Subject Classification. a r X i v : . [ m a t h . C A ] D ec C. GIANNOTTI AND A. SPIRO
Notice that, as in Euclidean geometry, also in S or L the perimeter ofa bounded convex set is less than or equal to the perimeter of any largerconvex set (Prop. 2.3). Due to this, all G -curves in a given convex set havelength less than or equal to the perimeter of that set.We also remark that the condition on diameter( γ ([ a, b ])) in case S = S isa technical hypothesis, needed in our analysis of the growth of the perimeters p ( s ) of the convex hulls (cid:91) γ | [ a,s ] of the arcs γ | [ a,s ] , s ∈ ( a, b ]. It can be replacedby other assumptions, applicable when diameter( γ ([ a, b ])) ≥ π (like e.g.that the tangent to γ at γ s is a tangent line also to the convex hull (cid:91) γ | [ a,s ] forany s ). But, at the moment, the previous statement is the best and mostgeneral one, which we have at our reach.Secondly, for checking the optimality of this estimate, we consider theexistence problem for G -curves γ : [ a, b ] → S with maximal length property ,i.e. such that the length (cid:96) ( γ | [ a,s ] ) equals to the perimeter of the convex hull (cid:92) γ ([ a, s ]) for any s ∈ [ a, b ]. The solution of this problem in case S = E wasthe second main result in [MP] (see also [MP1]), where the authors provedthat, up to rigid motion, for given L there exists a unique G -curve, of length L and C outside the starting point, with the maximal length property. Thiscurve is the unique curve in E which is self-involute , i.e. it is equal to theinitial arc of its own involute (see § S = S or L , at least for sufficientlysmall curves. Our second main result shows that for the case S = S thisexpectation is indeed correct. In fact, we first prove that there exists on S curves which are self-involute (Corollary 4.6) and then we prove that anyself-involute of S is also a G -curve with maximal length property , showingin this way the optimality of the upper bound of Theorem 1.1.The existence problem for G -curves with maximal length property on L remains open. It might be also interesting to know if the self-involutes of S share the same uniqueness properties of the self-involutes of E and whetherthere exists or not self-involutes on L .The structure of the paper is as follows. In §
2, we collect a few basic factson convex sets in surfaces of constant curvature. For reader’s convenience,we tried to make the exposition as much as possible self-contained. In § G -curves and prove Theorem 1.1. In §
4, we recallsome well-known facts of the Differential Geometry of curves on surfaces, weintroduce the definitions of involutes, almost self-involutes and self-involutesand prove our final results on the existence of self-involutes and G -curveswith maximal length property on S . Acknowledgment.
The authors would like to thank Paolo Manselli, to whomare very much indebted for suggesting the problem, giving most of the ideasfor the proof of Theorem 4.3 and for a multitude of very cheerful and fruitful
TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 3 discussions on this subject. We are also grateful to Gabriele Bianchi andthe referee for valuable comments and helpful remarks.2.
Preliminaries
Convex sets in surfaces of constant curvature.
Consider an (abstract) complete surface S of constant curvature K =+1 , −
1, i.e. a complete 2-dimensional Riemannian manifold locallyisometric to the unit sphere S , the Euclidean plane E or the Lobachevskijplane L . It is well-known that either S or E or L is also the universalcovering space of S .A line of S is the trace of a maximal geodesic and, for any two points x , x ∈ S , the (geodesic) segment joining x and x is an arc of minimal lengthbetween those points. We can now recall the definition of convex sets in S (see e.g. [Al]). Definition 2.1.
A subset Q ⊂ S is called convex if for any two points x, x (cid:48) ∈ Q there exists a unique segment joining x to x (cid:48) and such segmentis entirely in Q . Given a subset A ⊂ S , its convex hull is the intersection (cid:98) A = (cid:84) A (cid:48) of all convex sets A (cid:48) ⊂ S containing A .Notice that, according to this definition, there are sets with no convexhull. For instance, if A ⊂ S contains a pair of points with more than onesegment joining them, there is no convex set that contains A .We also recall that, according to how many segments might join two givenpoints and whether or not they are all included in the subset, other notionsof convex subsets in a surface can be given (see e.g. [Al, BZ, Ud]). Howeverall these notions coincide when S = E or L , while for the case S = S theabove choice turned out to be the most convenient one for establishing anupper bound for the length of steepest descent curves.Let π : (cid:101) S −→ S be the universal covering space of S . If U ⊂ S is an openconvex set, then it is simply connected and the restriction π | (cid:101) U : (cid:101) U −→ U of π to a connected component (cid:101) U of π − ( U ) is an isometry between the convexsubset (cid:101) U of (cid:101) S and U . Hence, for our purposes, with no loss of generality wemay always reduce to the cases S = S , E or L . In addition, the followinglemma shows that the case S = S can be always replaced by the assumption S = S = S ∩ { ( x , x , x ) : x > } . Lemma 2.2.
Let
U ⊂ S be open and convex. Then it is contained in ahemisphere. C. GIANNOTTI AND A. SPIRO
Proof.
Intuitively, the claim is a consequence of the fact that a convexsubset of the unit sphere cannot contain a pair of antipodal points. Buta short and precise proof can be obtained as follows. Let C ⊂ R be thecone given by the half-lines from the origin passing through the points of U .One can check that U is convex if and only if C is convex. If we denote by α ⊂ R a supporting plane of C through the origin, it follows immediatelythat U = S ∩ C is contained in a hemisphere bounded by α .Consider the upper hemisphere S . The map ϕ : S → { x = 1 } (cid:39) R ,sending any x ∈ S into its radial projection ¯ x ∈ { x = 1 } , maps the linesof S into the lines of the Euclidean space E = ( R , g o ) (here g o is thestandard Euclidean metric). In fact, the traces of geodesics in S are greatcircles, i.e. intersections between S and affine planes of R through theorigin, and are mapped by ϕ into straight lines, given by the intersectionsof those planes with { x = 1 } .Similarly, consider the Lobachevskij plane L = ( { ( x ) + ( x ) − ( x ) = − , x > } , g ) , where g is the Riemannian metric induced from the indefinite metric on R g = dx ⊗ dx + dx ⊗ dx − dx ⊗ dx . Also in this case, the lines of L are given by the intersections between L and the affine planes of R through the origin. So, the map ψ : L −→{ x = 1 } (cid:39) R , sending the points of L into their radial projections fromthe origin on that plane, maps L into the open unit disc ∆ ⊂ R and thelines of L into the straight lines (chords) of ∆.We recall that the abstract surface K = (∆ , (cid:101) g def = ψ − ∗ ( g )) is usuallycalled Klein disc .Using the maps ϕ and ψ , we may always identify S and L with (anopen subset of) R , endowed with a suitable metric g = g ij dx i ⊗ dx j , whosegeodesic segments coincide with the standard Euclidean segments. Underthis identification, a subset Q ⊆ S is convex if and only if it is convex in theEuclidean sense. For this reason, in the figures of this paper, the segments and convexsubsets of S will be drawn as Euclidean segments and Euclidean convexsubsets of R . However, the reader should be aware that, when S is not theEuclidean space, these picture can be misleading for what concerns lengthsand angles. For instance, for any v, w ∈ T x S (cid:39) T x E , the norm | v | and thecosine cos( (cid:99) vw ) are in general different from the Euclidean values, since theyare given in terms of the non-Euclidean metric g = g ij dx i ⊗ dx j of S by | v | = (cid:113) g ij ( x ) v i v j , cos( (cid:99) vw ) = g ij ( x ) v i w j | v || w | . TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 5
A closed simple curve
P ⊂ S is called polygonal if it is union of finitelymany segments, called sides . The endpoints of the sides are called vertices .The distance between two points x o , y o ∈ S ( ⊆ R ) is equal to the lengthof the segment between x o and y o w.r.t. g i.e. d S ( x o , y o ) = (cid:90) (cid:113) g ij ( x ( t )) ˙ x i ( t ) ˙ x j ( t ) dt , where x ( t ) is the line x ( t ) = ( x o (1 − t ) + ty o , x o (1 − t ) + ty o ) . In particular, if x o , y o are in a fixed convex compact subset K ⊂ S , thereare 0 < C K , C (cid:48) K such that C K | x o − y o | ≤ d S ( x o , y o ) ≤ C (cid:48) K | x o − y o | .The length (cid:96) S ( P ) of a polygonal curve P is the sum of the lengths of itssides. A curve C ⊂ S is called rectifiable if its length (cid:96) S ( C ) = sup { (cid:96) S ( P ) , P polygonal curve with vertices in C } is finite. For an absolutely continuous parameterization γ : [ a, b ] → C ⊂ S ofa rectifiable curve C , the tangent vector ˙ γ s o exists for almost any s o ∈ [ a, b ]and d(cid:96) S ( γ | [ a,s ] ) ds (cid:12)(cid:12)(cid:12)(cid:12) s o = | ˙ γ s o | = (cid:112) g ( ˙ γ s o , ˙ γ s o ) . By previous remarks, for any compact subset K ⊂ S ( ⊂ R ), a curve C ⊂ K is rectifiable if and only if it is rectifiable as curve in the Euclidean planeand C K · (cid:96) ( C ) ≤ (cid:96) S ( C ) ≤ C (cid:48) K · (cid:96) ( C ) (2.1)where (cid:96) ( C ) is the length of C in Euclidean sense.The following proposition generalizes two well-known properties of convexsets in E . Proposition 2.3. i) The boundary ∂Q of a bounded convex set Q ⊂ S is a rectifiablecurve. ii) If Q ⊂ Q (cid:48) are two bounded convex subsets of S , then (cid:96) S ( ∂Q ) ≤ (cid:96) S ( ∂Q (cid:48) ) .Proof. (i) Since Q ⊂ S ⊂ R is convex also in the Euclidean sense, theclaim follows immediately from known facts on Euclidean convex sets and(2.1).(ii) Let P be a polygonal curve with vertices x , . . . , x n , x n +1 = x ∈ ∂Q and P the (open) convex polygon with ∂ P = P . Denote also by s i the sideof P joining x i and x i +1 .Being Q (cid:48) bounded and convex, the line (cid:98) s containing s cuts Q (cid:48) into twoconvex subsets and the polygon P is entirely contained in one of them (seeFig.1). Call Q (cid:48) (1) this convex set and observe that ∂Q (cid:48) (1) is obtained byreplacing a portion of ∂Q (cid:48) with the segment joining the endpoints of such C. GIANNOTTI AND A. SPIRO portion. Hence (cid:96) S ( ∂Q (cid:48) ) ≥ (cid:96) S ( ∂Q (cid:48) (1) ). Next, consider the line (cid:98) s containingthe side s and the convex subset Q (cid:48) (2) ⊂ Q (cid:48) (1) , which is cut by (cid:98) s andcontains P . As before, we have that (cid:96) S ( ∂Q (cid:48) (1) ) ≥ (cid:96) S ( ∂Q (cid:48) (2) ). Repeating thesame construction for all the lines (cid:98) s i containing the sides s i , we end up witha nested sequence of convex sets Q (cid:48) ( i ) , 1 ≤ i ≤ n , with Q (cid:48) ( n ) = P and suchthat (cid:96) S ( ∂Q (cid:48) ) ≥ (cid:96) S ( ∂Q (cid:48) (1) ) ≥ (cid:96) S ( ∂Q (cid:48) (2) ) · · · ≥ (cid:96) S ( ∂ P ) = (cid:96) S ( P )By arbitrariness of P , it follows that (cid:96) S ( ∂Q (cid:48) ) ≥ (cid:96) S ( ∂Q ). Fig. 1
Remark 2.4.
By a refinement of the proof, it is not hard to check that if
Q, Q (cid:48) are as in Proposition 2.3 (ii) and if Q (cid:48) \ Q has non empty interior, then (cid:96) S ( ∂Q (cid:48) ) > (cid:96) S ( ∂Q ).Given a bounded convex set Q ⊂ S , we denote by p ( Q ) the perimeter (cid:96) S ( ∂Q ).2.2. An auxiliary lemma.
We give here a lemma, needed in the proof of Theorem 1.1. It couldbe derived from the Gauss Lemma on derivatives of distance functions andit could be proved for Riemannian manifolds. However, we provide a self-contained proof, using only basic facts on surfaces of constant curvature.In the following, given y, z ∈ S = S , E or L , we denote by [ y, z ] thesegment joining them and, given a curve η : [ a, b ] → S , we denote by d y,η : [ a, b ] → R , d y,η ( s ) = (cid:96) S ([ y, η s ]) = d S ( y, η s ) . Lemma 2.5.
Let η : [ a, b ] → S be a curve in S , s ∈ [ a, b [ , such that thetangent vector ˙ η s (cid:54) = 0 exists at η s , and { h n } a sequence of positive (negative)real numbers converging to and { y n } ⊂ S a sequence of points, all of themdifferent from η s and η s + h n , converging to a point y o .We denote by α n and γ n the angles, with vertices in η s and y n , respec-tively, formed by the tangent vectors of the oriented segments [ η s , η s + h n ] and TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 7 [ η s , y n ] and by the tangent vectors of the oriented segments [ y n , η s + h n ] and [ y n , η s ] . If y o (cid:54) = η s lim n →∞ d y n ,η ( s + h n ) − d y n ,η ( s ) h n = | ˙ η s | cos ϕ . (2.2) where ϕ = lim n →∞ ( π − α n ) . This equality holds also when y o = η s , providedthat lim n →∞ α n exists and lim n →∞ γ n = 0 .Moreover, d (cid:48) y o ,η ( s ) = | ˙ η s | cos ϕ for any y o ∈ S , where ϕ denotes the anglebetween ˙ η s and the tangent in η s of the oriented segment [ η s , y o ] when y o (cid:54) = η s and ϕ = 0 when y o = η s .Proof. First of all, let us recall the following well-known formulae ofSpherical and Hyperbolic Trigonometry (see e.g. [AVS] § I.3).
On the sphere S : cos α = − cos β cos γ + sin β sin γ cos a , sin α sin a = sin β sin b = sin γ sin c , On the Lobachevskij plane L : cos α = − cos β cos γ + sin β sin γ cosh a , sin α sinh a = sin β sinh b = sin γ sinh c , where, a , b , c are the sides of a triangle and α , β and γ the correspondingopposite angles. Now, consider the case h n > n (the case h n < n , consider the triangle of vertices y n , η s and η s + h n and let us denote by a n = (cid:96) S ([ y n , η s + h n ]) , b n = (cid:96) S ([ y n , η s ]) , c n = (cid:96) S ([ η s , η s + h n ])and by α n , β n and γ n the corresponding opposite angles. Then, both thehypotheses y o (cid:54) = η s and y o = η s with lim n →∞ α n = π − ϕ , lim n →∞ γ n = 0,imply that lim n →∞ ( a n − b n ) = 0 , lim n →∞ c n = 0 , (2.3)lim n →∞ γ n = 0 , lim n →∞ α n = π − ϕ , lim n →∞ β n = ϕ . (2.4)Observe thatlim n →∞ d y n ,η ( s + h n ) − d y n ,η ( s ) h n = lim n →∞ a n − b n c n · lim n →∞ c n h n . The second limit is equal to | ˙ η s | , while the first limit can be written aslim n →∞ a n − b n c n = lim n →∞ sin( a n − b n )sin( c n ) = lim n →∞ sin a n cos b n − cos a n sin b n sin c n (2.5)or aslim n →∞ a n − b n c n = lim n →∞ sinh( a n − b n )sinh( c n ) = lim n →∞ sinh a n cosh b n − cosh a n sinh b n sinh c n (2.6) C. GIANNOTTI AND A. SPIRO
Using (2.3), (2.4) and the above relations in (2.5) when S = S or in (2.6)when S = L , we obtain that in both caseslim n →∞ a n − b n c n = lim n →∞ (cos β n − cos α n ) (1 − cos γ n )sin γ n = cos ϕ . In case S = E , using the Sine Law of Euclidean trigonometry and thestandard property of Euclidean triangles α n + β n + γ n = π , we havelim n →∞ a n − b n c n = lim n →∞ sin α n − sin β n sin γ n == lim n →∞ (cid:16) α n − β n (cid:17) cos (cid:16) α n + β n (cid:17) sin γ n = lim n →∞ (cid:16) α n − β n (cid:17) sin (cid:0) γ n (cid:1) sin γ n = cos ϕ , where the last equality follows from (2.4).For what concerns the last claim, in case y o (cid:54) = η s it is a direct consequenceof (2.2) for the sequence { y n = y o } , while in case y o = η s it follows from theimmediate observation that d (cid:48) y o ,η ( s ) = lim h → d S ( η s ,η s + h ) h = | ˙ η s | .3. G -curves Let f : U ⊂ S −→ R be a map defined on a convex subset U of S . We saythat f is quasi-convex if any level set U ≤ c def = { x ∈ U : f ( x ) ≤ c } is convexin S . Notice that, under the identification of S with R or ∆ ⊂ R , f isquasi-convex as function on S if and only if it is quasi-convex in the usualEuclidean sense.Let f : U ⊂ S −→ R be a C quasi-convex function f with df (cid:54) = 0 at allpoints, so that the level sets U ≤ t have C -boundaries. Any curve γ s of steepestdescent for f (i.e. such that ˙ γ s = − grad f | γ s ) intersects orthogonally allthe boundaries of the level sets and, for given s o , all points γ s , s ≥ s o , areincluded in the level set U ≤ f ( γ so ) . This means that the class of steepestdescent curves is naturally included in the following class of curves, whichextends the class considered in [MP]. Definition 3.1.
We say that an absolutely continuous curve γ : [ a, b ] → S is in the class G if, for any s o ∈ [ a, b ] such that ˙ γ s o exists and is differentfrom 0, all points γ s with s ≤ s o are in a same closed half space that isbounded by the normal line (cid:96) to γ at γ s o .In the following, we denote by γ : [ a, b ] → S a G -curve of S and, for any s ∈ [ a, b ] we indicate by p ( s ) the perimeter of the convex hull of γ | [ a,s ] . ByProposition 2.3 (ii), the function p ( s ) is not decreasing and hence p (cid:48) ( s ) existsfor almost all s ∈ [ a, b ]. Warning . In [Ud] and other places, “quasi-convex function” means a function on atotally convex set with totally convex level sets. Notice that if S = E , the notion ofconvexity and total convexity coincide. TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 9
Moreover, for any x ∈ S and v, w ∈ T x S , we will denote by C ( x ; v, w )a closed convex sector bounded by the two geodesic rays originating from x and tangent to v and w . Identifying S with (an open subset of) R , thesector C ( x ; v, w ) is the convex angle with vertex x and sides parallel to v and w . It is clearly always uniquely determined except when it is a half-plane,i.e. when (cid:99) vw = π .For any point γ s of the curve γ , we call projecting sector of γ at γ s thesmallest closed convex sector containing γ | [ a,s ] . For any point γ s of γ , we willdenote by v i = v i ( s ) ∈ T γ s S , i = 1 ,
2, the tangent vectors of the boundaryrays of the corresponding projecting sector, which will be therefore indicatedby C ( γ s ; v , v ). Theorem 3.2.
Let γ : [ a, b ] → S be a G -curve with convex hull (cid:98) γ and, inthe case S = S , assume also that diameter( γ ) < π . Then (cid:96) S ( γ ) ≤ p ( (cid:98) γ ) , (3.1) where the equality occurs only if for almost all s ∈ [ a, b ] the projecting sector C ( γ s ; v , v ) of γ is such that (cid:100) v v = π and either v or v is tangent to γ at γ s .In particular, for any G -curve in a bounded convex set Q ⊂ S (and sat-isfying the above condition when S = S ) the length (cid:96) S ( γ ) is less than orequal to p ( Q ) . The proof of this result is based on the following two lemmata.
Lemma 3.3.
Let s be such that both p (cid:48) ( s ) and ˙ γ s exist with ˙ γ s (cid:54) = 0 . Then p (cid:48) ( s ) | ˙ γ s | ≥ cos φ + cos φ , with φ i def = π − (cid:100) ˙ γ s v i , where v , v are the vectors of the projecting sector C ( γ s ; v , v ) .Proof. For s ∈ [ a, b [ and h >
0, consider the following notations:– A s = γ | [ a,s ] and (cid:99) A s is its convex hull;– A s ( h ) = A s ∪ { γ s + h } with convex hull (cid:92) A s ( h );– (cid:95)A s ( h )= C ( γ s ; v , v ) ∩ (cid:92) A s ( h ) . Fig. 2
Clearly, (cid:99) A s ⊆ (cid:95)A s ( h ) ⊆ (cid:92) A s ( h ) ⊆ (cid:91) A s + h and hence, by Proposition 2.3 (ii), p ( (cid:99) A s ) ≤ p ( (cid:95)A s ( h )) ≤ p ( (cid:92) A s ( h )) ≤ p ( (cid:91) A s + h )and p ( s + h ) − p ( s ) = p ( (cid:91) A s + h ) − p ( (cid:99) A s ) ≥ p ( (cid:92) A s ( h )) − p ( (cid:95)A s ( h )) . (3.2)The boundary of (cid:92) A s ( h ) contains two segments, lying on two rays coming outfrom γ s + h , and these segments necessarily intersect the sides of C ( γ s ; v , v )in two distinct points, which we call x h and x h (one of them might be γ s ).Moreover, since the set of points { x hi , h > } is bounded, there exists asequence { h n } with lim n →∞ h n = 0 and lim n →∞ x h n i = x i for some x i in oneside of C ( γ s ; v , v ).Using the notation of Lemma 2.5, we may write p ( (cid:92) A s ( h n )) − p ( (cid:95)A s ( h n )) = (cid:88) i =1 (cid:16) (cid:96) S ([ x h n i , γ s + h n ]) − (cid:96) S ([ x h n i , γ s ]) (cid:17) == (cid:88) i =1 (cid:16) d x hni ,γ ( s + h n ) − d x hni ,γ ( s ) (cid:17) . (3.3)Recall that, for fixed i = 1 ,
2, all x h n i lie in one of the two sides (call them r i , i = 1 ,
2) of C ( γ s ; v , v ). Hence, if there exists a subsequence { x h nk i } converging to a point y o (cid:54) = γ s , then y o ∈ r i and the angles α n k , formedby the tangents in γ s of the oriented segments [ γ s , γ s + h nk ] and [ γ s , x h nk i ],converge to π − φ i . By Lemma 2.5 we get (after replacing { x h n i } by theabove subsequence)lim n →∞ d x hni ,γ ( s + h n ) − d x hni ,γ ( s ) h n = | ˙ γ s | cos φ i . (3.4)The same conclusion holds also if there exists a subsequence { x h nk i } of points,all different from γ s , converging to y o = γ s , because in such case, by possibly TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 11 taking another subsequence, the angles in x h nk i , formed by the orientedsegments [ x h nk i , γ s + h nk ] and [ x h nk i , γ s ], tend to 0 and Lemma 2.5 applies.To check this claim, consider the rays r n k i with origin in γ s + h nk and con-taining [ γ s + h nk , x h nk i ]. By construction, any such ray lies in a line of supportfor γ | [ a,s ] and contains a sub-ray, with origin in x h nk i , included in C ( γ s ; v , v ).By taking a suitable subsequence, the rays r n k i tend to a ray r oi , with origin in γ s , entirely included in C ( γ s ; v , v ) and lying in a line of support (cid:96) o for γ | [ a,s ] .Due to this, if r oi intersected the interior of C ( γ s ; v , v ), this would not bethe smallest convex sector containing γ | [ a,s ] . So r oi ⊂ ∂C ( γ s ; v , v ), that is r oi = r i , and the angles in x h nk i , formed by [ x h nk i , γ s + h nk ] and [ x h nk i , γ s ], tendto 0 as claimed.Now, we claim that (3.4) is true also if there is no subsequence { x h nk i } ,made of points all different from γ s . In fact, in this case we may assumethat x h n i = γ s for any n and hence that the projecting sector C ( γ s ; v , v )lies in the intersection of half spaces, bounded by the lines (cid:96) n , which containthe boundary segment [ γ s + h n , γ s ] ⊂ ∂ (cid:92) A s ( h n ). We remark thata) the lines (cid:96) n tend to the tangent line (cid:96) o of γ at γ s and C ( γ s ; v , v ) iscontained in a half-space bounded by (cid:96) o ;b) since the rays, with origin in γ s and containing [ γ s , γ s − h ], h >
0, areincluded in C ( γ s ; v , v ) and tend to a ray r o ⊂ (cid:96) o when h →
0, itfollows that r o ⊂ C ( γ s ; v , v ).From this, we infer that r o = r i ⊂ ∂C ( γ s ; v , v ) ∩ (cid:96) o and φ i = π − (cid:100) ˙ γ s v i = 0.So, by Lemma 2.5,lim n →∞ d x hni ,γ ( s + h n ) − d x hni ,γ ( s ) h n = d (cid:48) γ s ,γ ( s ) = | ˙ γ s | cos 0 = | ˙ γ s | cos φ i as claimed. From (3.2) and (3.3), p (cid:48) ( s ) ≥ (cid:88) i =1 lim n →∞ d x hni ,γ ( s + h n ) − d x hni ,γ ( s ) h n = | ˙ γ s | (cos φ + cos φ ) . (cid:3) Lemma 3.4.
Let s be such that ˙ γ s exists with ˙ γ s (cid:54) = 0 and let C ( γ s ; v , v ) be the projecting sector of γ at γ s . If the curvature of S is K = +1 , assumealso that diameter( γ | [ a,s ] ) < π . Then (cid:100) v v ≤ π and hence, if φ i = π − (cid:100) ˙ γ s v i , cos φ + cos φ ≥ . The equality holds if and only if (cid:100) v v = π and one of the φ i ’s is equal to .Proof. First of all, we claim that, for any s o ∈ [ a, b [, the function d γ so ,γ : [ s o , b ] → R , d γ so ,γ ( s ) = d S ( γ s o , γ s )is non-decreasing. In fact, since γ is a G -curve, the vector ˙ γ s and the tangentvector in γ s to the oriented segment [ γ s o , γ s ] points towards the same side w.r.t. the normal line of γ in γ s . In particular, the angle ϕ between them isless than or equal to π . By Lemma 2.5, d (cid:48) γ so ,γ ( s ) is non-negative for almostall s and the claim follows.Secondly, we claim that for any a ≤ s < s < s ≤ b , the angle α formedby the oriented segments [ γ s , γ s ] and [ γ s , γ s ] is less than or equal to π .In case S = E or L , it can be checked as follows. In the triangle withvertices γ s , γ s and γ s , the sum of inner angles is less than or equal to π and hence α > π only if it is the largest of these three angles. But thiscannot be because the side [ γ s , γ s ], opposite to α , is not the largest one(by the previous claim, it is shorter or equal to [ γ s , γ s ]), in contrast with awell known fact of Euclidean and Hyperbolic Geometry.Also in case the curvature of S is K = +1 and diameter( γ | [ a,s ] ) < π , if α were larger than π , its opposite side in the triangle with vertices γ s , γ s and γ s would be the largest one, as it can be checked using the sphericallaw of cosines cos a = cos b cos c + sin a sin b cos α . Hence, also in this casewe conclude that α ≤ π by the same argument as above.Now, the first statement of the lemma follows immediately from the obser-vation that (cid:100) v v = φ + φ is limit of angles delimited by segments [ γ s , γ s ]and [ γ s , γ s ] for some a ≤ s < s < s ≤ b . To check the second state-ment, just look for the minimum of f ( φ , φ ) = cos φ + cos φ in the regionΩ = { ≤ φ i ≤ π , φ + φ ≤ π } .Combining the results of these lemmata, p (cid:48) ( s ) ≥ | ˙ γ s | for almost all s ∈ [ a, b ], with equality only if the amplitude of the projecting sector is π , andthis implies the theorem.4. Self-Involutes on spheres
Basic facts on curves of surfaces of constant curvature.
In this section S is a simply connected, complete surface of constantcurvature with a curvature K that might assume any real value. Recallthat, when K = ± /R , the surface S is either S R = { x ∈ R : x T · x = R } or L R = (cid:110) x ∈ R : x > x T · I , · x = − R where I , = (cid:16) − (cid:17) (cid:111) , respectively endowed with the Riemannian metric g , which is induced eitherby the standard Euclidean metric g E = dx ⊗ dx + dx ⊗ dx + dx ⊗ dx or by the Lorentzian metric g L = dx ⊗ dx + dx ⊗ dx − dx ⊗ dx . Foruniformity of notation, one can consider also E as a surface in R , namelysetting E = { x ∈ R : x = 1 } ⊂ R , with the metric induced by thestandard Euclidean metric g E of R .Given a C curve η : I ⊂ R → S ⊂ R , parameterized by arc length, the Frenet frame of η at s = s o is the orthonormal basis ( t s , n s ) ⊂ T η s S , given TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 13 by t s = ˙ η s and the unit vector n s , tangent to S , orthogonal to t s and so that( η s , t s , n s ) is a positively oriented basis for R . By a simple generalizationof the classical theory of curves in E , one can derive the following “Frenetformulae” ∇ ˙ η s t s = κ s n s , ∇ ˙ η s n s = − κ s t s , (4.1)for some smooth function κ s , called (geodesic) curvature of η . We also recallthe Levi-Civita connection ∇ of S is such that, for any smooth curve η andany vector field Y η s , tangent to S and defined at the points of η , ∇ ˙ η s Y = ˙ Y η s + Kg ( ˙ η s , Y η s ) η s (here, ˙ Y η s and plus sign denote the standard first order derivative and thesum of maps with values in R ). Also, given a point x o ∈ S ⊂ R and a unitvector v ∈ T x o S , the geodesic γ s with γ = x o and ˙ γ s = v is of the form γ s = cos (cid:16) sR (cid:17) x o + sin (cid:16) sR (cid:17) ( Rv ) or γ s = cosh (cid:16) sR (cid:17) x o + sinh (cid:16) sR (cid:17) ( Rv )or γ s = x o + sv (4.2)according to the value of K .We conclude this subsection, stating the following simple generalization ofthe classical “Fundamental Theorems of Plane Curves”. It is an immediateconsequence of the Existence and Uniqueness Theorem for O.D.E.’s. Theorem 4.1. Let η : [0 , L ] ⊂ R → S and η (cid:48) : [0 , L (cid:48) ] ⊂ R → S be two C curvesparameterized by arc length and with curvature functions κ and κ (cid:48) ,respectively. There exists an isometry g : S → S such that η (cid:48) = g ◦ η if and only if L = L (cid:48) and κ s = κ (cid:48) s for any s ∈ [0 , L ] . Let κ : [0 , L ] → R be C . For any x o ∈ S , v ∈ T x o S with | v | = 1 and (cid:101) L ≤ L sufficiently small, there exists a unique C curve η : [0 , (cid:101) L ) → S parameterized by arc length and with curvature function κ , such that η = x o and ˙ η = v . Involutes and (almost) self-involutes.Definition 4.2.
Let η : [0 , L ] → S ⊂ R a curve, parameterized by arclength (hence ˙ η s = t s ) and C on (0 , L ). The involute of η is the curve (cid:101) η : [0 , L ] → S such that, for any s ∈ (0 , L ), the point (cid:101) η s ∈ S is determinedby the formula (cid:101) η s = γ ( η s , − t s ) s , (4.3)where γ ( η s , − t s ) is the geodesic of S with γ ( η s , − t s )0 = η s and ˙ γ ( η s , − t s )0 = − t s .The curve η s is called almost self-involute if there exists an isometry of S , which maps η into the initial arc of length L of its own involute (cid:101) η (re-parameterized by arc length). In case η s coincides with such initial arc of (cid:101) η ,we call it self-involute . Fig. 3 - An almost self-involutes (left) and a self-involute (right)
According to the value of K , one immediately obtains that (cid:101) η s = cos (cid:16) sR (cid:17) η s − sin (cid:16) sR (cid:17) ( R ˙ η s ) or (cid:101) η s = cosh (cid:16) sR (cid:17) η s − sinh (cid:16) sR (cid:17) ( R ˙ η s )or (cid:101) η s = η s − s ˙ η s . (4.4)Simple computations shows that, in case η is C , the Frenet frames ( t (cid:101) s s , n (cid:101) s s ),the arc length parameter (cid:101) s and the curvature (cid:101) k (cid:101) s s at the points (cid:101) η (cid:101) s s = (cid:101) η s are(only for s < π/ K > (cid:101) s → − (cid:101) s ) t (cid:101) s s = − n s , n (cid:101) s s = sin (cid:0) sR (cid:1) η s R + cos (cid:0) sR (cid:1) ˙ η s if K = R sinh (cid:0) sR (cid:1) η s R + cosh (cid:0) sR (cid:1) ˙ η s if K = − R ˙ η s if K =0 , (4.5)˙ (cid:101) s s = R sin (cid:0) sR (cid:1) κ s if K = R R sinh (cid:0) sR (cid:1) κ s if K = − R sκ s if K =0 , (cid:101) κ (cid:101) s s = R cot (cid:0) sR (cid:1) if K = R R coth (cid:0) sR (cid:1) if K = − R s if K =0 Using Theorem 4.1, one can infer that a curve η : [0 , L ] → S R , parameterizedby arc length, smooth and with strictly positive curvature function κ s on(0 , L ), is congruent to the initial arc of length L of its own involute if andonly if there exists a strictly increasing smooth function τ : (0 , (cid:101) L ) ⊂ (0 , L ) → (0 , L ), with lim s → + τ s = 0 and satisfying the following equations ˙ τ s = R sin (cid:0) sR (cid:1) κ s κ τ s = R cot (cid:0) sR (cid:1) . (4.6)Moreover, for any pair of smooth functions κ : (0 , L ) → R and τ : (0 , (cid:101) L ) ⊂ (0 , L ) → (0 , L ) with κ s , ˙ τ s > s ∈ (0 , L ), lim s → + τ s = 0 andsatisfying (4.6), there exists a curve η : [0 , L ] → S R with curvature κ s and TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 15 congruent to the initial arc of its own involute. Similar claims hold for thecurves in L R or E , where (4.6) is replaced by ˙ τ s = R sinh (cid:0) sR (cid:1) κ s κ τ s = R coth (cid:0) sR (cid:1) or ˙ τ s = sκ s κ τ s = s , (4.7)respectively.The curvature functions κ s of almost self-involutes are exactly those whichsatisfy (4.6) or (4.7) with suitable τ s and the cardinality (up to congruences)of the self-involutes of a given length is clearly bounded by the cardinalityof the solutions to those systems.4.3. An existence result for the almost self-involutes on spheres.Theorem 4.3.
For any a > and any s o > sufficiently small, there existsan almost self-involute η : [0 , s o ] → S R of class C , with curvature κ s andrelated function τ s , which satisfy (4.6) on (0 , s o ] and such that lim s → + τ s = 0 , lim s → + ˙ τ s = a , lim s → + κ s = + ∞ . (4.8) Proof.
By Theorem 4.1 and the remarks in the previous subsection, theclaim is proved if we show the existence of a pair of functions κ and τ ofclass C satisfying (4.6) on (0 , s o ]. To prove this, consider the equation andthe initial conditions ϑ (cid:48) ( t ) = tan( ϑ ( ϑ ( t )))sin( ϑ ( t )) , lim t → + ϑ ( t ) = 0 , lim t → + ϑ (cid:48) ( t ) = A (4.9)on a C -function ϑ : (0 , t o ] → R and notice that if ϑ is a solution to (4.9)with ϑ (cid:48) ( t ) > (cid:0) τ s = R ϑ − (cid:0) sR (cid:1) , κ s = R cot (cid:0) ϑ (cid:0) sR (cid:1)(cid:1)(cid:1) is a C -solution of (4.6) on (0 , s o = R ϑ ( t o )] satisfying (4.8) with a = A . Hencethe lemma is proved if we show that there exists a solution of (4.9) for any0 < A < t o sufficiently small.Given 0 < A <
1, consider the following sequence of functions, definedby recursions on the same interval [0 , t o ] ⊂ R for some t o to be determinedlater ϑ ( t ) def = At , ϑ n ( t ) def = (cid:90) t tan ( ϑ n − ( ϑ n − ( u )))sin ( ϑ n − ( u )) du . (4.10)We want to show that it is possible to determine t o such that the functions ϑ n converge uniformly on [0 , t o ] to a solution of (4.9). This is a directconsequence of the following claims. Claim 1.
There exist t o ∈ (0 , and K < such that ϑ n ( t ) is defined forany n and any t ∈ [0 , t o ] and satisfies < ϑ n ( t ) ≤ Kt for any t > . (4.11) Claim 2. lim t → + ϑ (cid:48) n ( t ) = A for any n . Claim 3. ϑ (cid:48)(cid:48) n ( t ) ≥ for any n and any t ∈ (0 , t o ) . Claim 4. ϑ n − ( t ) ≤ ϑ n ( t ) for any n ≥ and any t ∈ [0 , t o ] . Claim 5.
The sequence { ϑ n } converges uniformly to a function ϑ : [0 , t o ] → R which is a solution of (4.9).Let us now proceed with the proofs of such claims. Proof of Claim 1.
Consider a value
B > A + A A B − B < F ( x ) = tan (cid:18) Ax + B x (cid:19) − (cid:18) A + B A x (cid:19) sin ( x )is such that F (0) = F (cid:48) (0) = F (cid:48)(cid:48) (0) = 0 and F (cid:48)(cid:48)(cid:48) (0) = 2 A (cid:18) A + A A B − B (cid:19) < I = (0 , ε ), on which F ( x ) < (cid:0) Ax + B x (cid:1) sin ( x ) < (cid:18) A + B A x (cid:19) . (4.14)Now, let t o > t o ≤ ε ;ii) + B A t o + B A t o < A + Bt o < n and t ∈ (0 , t o ]0 < ϑ n ( t ) ≤ At + B t . (4.15)First of all, for n = 0, we have that for any t > < ϑ ( t ) = At ≤ At + B t . (4.16)Secondly, assume that ϑ n − ( t ) is defined for any t ∈ [0 , t o ] and that 0 <ϑ n − ( t ) ≤ At + B t for any t ∈ (0 , t o ]. Then, by (iii), we have that ϑ n − ( t ) ≤ At + Bt ≤ t in [0 , t o ] and hence that ϑ n − ([0 , t o ]) ⊂ [0 , t o ]. It follows thatthe function ϑ n is well defined in [0 , t o ] and C with ϑ (cid:48) t > t > t ∈ (0 , t o ) ϑ (cid:48) n ( t ) = tan ( ϑ n − ( ϑ n − ( t )))sin ( ϑ n − ( t )) ≤ tan (cid:0) Aϑ n − ( t ) + B ϑ n − ( t ) (cid:1) sin ( ϑ n − ( t )) ≤≤ A + B A ϑ n − ( t ) ≤ A + B A (cid:18) A t + 23 ABt + B t (cid:19) ≤≤ A + Bt (cid:18)
12 + 13 Bt A + B t A (cid:19) ≤ A + Bt (4.17) TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 17
From this, the inequality (4.15) follows by integration. Now, by (4.15) andthe assumption (iii) on t o and setting K = A + Bt o <
1, we obtain theinequality (4.11), namely ϑ n ( t ) ≤ t (cid:16) A + Bt o (cid:17) = Kt . Proof of Claim 2.
By construction, ϑ (cid:48) (0) = A and if lim t → + ϑ (cid:48) n − ( t ) = A ,then lim t → + ϑ (cid:48) n ( t ) = lim t → + tan ( ϑ n − ( ϑ n − ( t )))sin ( ϑ n − ( t )) = lim t → + ϑ n − ( ϑ n − ( t )) ϑ n − ( t ) == lim t → + ϑ n − ( ϑ n − ( t )) − ϑ n − (0) ϑ n − ( t ) = lim τ → + ϑ n − ( τ ) − ϑ n − (0) τ == lim τ → + ϑ (cid:48) n − ( τ ) = A .
The claim follows by induction on n . Proof of Claim 3.
Also this claim is proved by induction. When n = 0, theclaim is trivial since ϑ (cid:48)(cid:48) ( t ) = 0. Assume now that ϑ (cid:48)(cid:48) n − ≥ , t o ) andobserve that ϑ (cid:48)(cid:48) n ( t ) = ddt tan ( ϑ n − ( ϑ n − ( t )))sin ( ϑ n − ( t )) == ( ϑ n − ( ϑ n − ( t ))) ϑ (cid:48) n − ( ϑ n − ( t )) ϑ (cid:48) n − ( t )sin ( ϑ n − ( t )) −− cos ( ϑ n − ( t )) tan ( ϑ n − ( ϑ n − ( t ))) ϑ (cid:48) n − ( t )sin ( ϑ n − ( t )) == ϑ (cid:48) n − ( t ) cos ( ϑ n − ( t ))cos ( ϑ n − ( ϑ n − ( t ))) sin ( ϑ n − ( t )) · F ( n − ( ϑ n − ( s )) (4.18)where, for any m , we denote F ( m ) ( s ) def = ϑ (cid:48) m ( s ) tan ( s ) − sin ( ϑ m ( s )) cos ( ϑ m ( s )) . (4.19)Using the assumption ϑ (cid:48)(cid:48) n − ( s ) ≥ s ∈ (0 , t o ] and the fact that ϑ (cid:48) n − ( s ) >
0, we have that F ( n − (cid:48) ( s ) = ϑ (cid:48)(cid:48) n − ( s ) tan( s ) + ϑ (cid:48) n − ( s ) 1cos ( s ) −− ϑ (cid:48) n − ( s )(cos ( ϑ n − ( s )) − sin ( ϑ n − ( s ))) == ϑ (cid:48)(cid:48) n − ( s ) tan( s ) + ϑ (cid:48) n − ( s )cos ( s ) (cid:0) − cos ( s ) cos(2 ϑ n − ( s )) (cid:1) ≥ . Since F ( m ) (0) = 0 for any m , it follows that F ( n − ( s ) ≥ s ∈ (0 , t o ]and hence that F ( n − ( ϑ n − ( t )) ≥ t ∈ (0 , t o ]. From this and (4.18),we get ϑ (cid:48)(cid:48) n ( t ) ≥ Proof of Claim 4.
First of all, notice that for any t ∈ (0 , t o ] ϑ (cid:48) ( t ) = tan( ϑ ( ϑ ( t )))sin( ϑ ( t )) = tan( A t )sin( At ) ≥ A tAt = A = ϑ (cid:48) ( t ) . Hence, by integration, ϑ ( t ) ≥ ϑ ( t ) for any t ∈ (0 , t o ]. Let us assumethat ϑ n − ≤ ϑ n − at all points of [0 , t o ]. In order to prove the claim byan inductive argument, we only need to check that ϑ (cid:48) n − ≤ ϑ (cid:48) n on the sameinterval [0 , t o ]. To see this, we notice that ϑ (cid:48) n ( t ) − ϑ (cid:48) n − ( t ) = tan( ϑ n − ( ϑ n − ( t )))sin( ϑ n − ( t )) − tan( ϑ n − ( ϑ n − ( t )))sin( ϑ n − ( t )) ϑ n − ≥ ϑ n − ≥≥ tan( ϑ n − ( ϑ n − ( t ))sin( ϑ n − ( t )) − tan( ϑ n − ( ϑ n − ( t ))sin( ϑ n − ( t )) == dds tan( ϑ n − ( s )sin( s ) (cid:12)(cid:12)(cid:12)(cid:12) s = (cid:101) s ( ϑ n − ( t ) − ϑ n − ( t ))for some (cid:101) s ∈ ( ϑ n − ( t ) , ϑ n − ( t )). On the other hand dds tan( ϑ n − ( s ))sin( s ) == cos s cos ( ϑ n − ( s )) sin s (cid:8) ϑ (cid:48) n − ( s ) tan( s ) − sin( ϑ n − ( s )) cos( ϑ n − ( s )) (cid:9) == cos s cos ( ϑ n − ( s )) sin s F ( n − ( s )where F ( n − ( s ) is as defined in (4.19). In the proof of the previous claim,we showed that F ( m ) ≥ m ≥ ϑ (cid:48) n − ≤ ϑ (cid:48) n as needed. Proof of Claim 5.
For the proof of this claim, we need the following proper-ties which are consequences of the previous claims:i) ϑ (cid:48) n − ( ϑ n − ( t )) ≤ ϑ (cid:48) n − ( t );ii) cos ( ϑ n − ( ϑ n − ( t ))) ≥ cos ( ϑ n − ( t ));iii) ϑ (cid:48) n − ( t ) ≤ ( ϑ n − ( t )) ≤ ( ϑ n − ( t o )) = K .Using these relations, for any t ≤ t o we have that (here “ ϑ m ( t )” stands for“ ϑ m ( ϑ m ( t ))”): ϑ (cid:48)(cid:48) n ( t ) = ϑ (cid:48) n − ( t ) cos( ϑ n − ( t ))cos ( ϑ n − ( t )) sin ( ϑ n − ( t )) ·· (cid:8) ϑ (cid:48) n − ( ϑ n − ( t )) tan( ϑ n − ( t )) − sin( ϑ n − ( t )) cos( ϑ n − ( t )) (cid:9) ≤≤ K sin ϑ n − ( t ) (cid:8) ϑ (cid:48) n − ( ϑ n − ( t )) tan( ϑ n − ( t )) − sin( ϑ n − ( t )) cos( ϑ n − ( t )) (cid:9) == K (cid:26) ϑ (cid:48) n − ( ϑ n − ( t )) 1sin( ϑ n − ( t )) cos( ϑ n − ( t )) − ϑ (cid:48) n ( t ) cos ( ϑ n − ( t ))sin( ϑ n − ( t )) (cid:27) == K (cid:8) ϑ (cid:48) n − ( ϑ n − ( t )) − ϑ (cid:48) n ( t ) cos( ϑ n − ( t )) cos ( ϑ n − ( t )) (cid:9) sin( ϑ n − ( t )) cos( ϑ n − ( t )) ≤ ( ii ) ≤ K (cid:48) sin( ϑ n − ( t )) (cid:8) ϑ (cid:48) n − ( ϑ n − ( t )) − ϑ (cid:48) n ( t ) cos ( ϑ n − ( t )) (cid:9) ≤ TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 19 ϑ (cid:48) n − ( t ) ≤ ϑ (cid:48) n ( t ) ≤ K (cid:48) ϑ (cid:48) n ( t )sin( ϑ n − ( t )) (cid:8) − cos ( ϑ n − ( t )) (cid:9) ≤≤ K (cid:48) − cos( ϑ n − ( t ))sin( ϑ n − ( t )) = 3 K (cid:48) tan (cid:18) ϑ n − ( t )2 (cid:19) ≤ K (cid:48) tan (cid:18) t o (cid:19) , (4.20)i.e. the functions ϑ (cid:48)(cid:48) n | [0 ,t o ] are uniformly bounded. From this we get thatthe ϑ (cid:48) n | [0 ,t o ] are uniformly bounded and equicontinuous. Since the sequence { ϑ (cid:48) n | [0 ,t o ] } is monotone, the sequence { ϑ n | [0 ,t o ] } uniformly converges to a C -function ϑ : [0 , t o ] → R satisfyinglim n →∞ ϑ (cid:48) n ( t ) = ϑ (cid:48) ( t ) for any t ∈ [0 , t o ] . We leave to the reader the simple task of checking that the sequence ϑ n ( ϑ n ( t )) uniformly converges to ϑ ( ϑ ( t )), from which follows that the limitfunction ϑ ( t ) is indeed a solution to the differential problem (4.9).4.4. Existence of self-involutes on the spheres.Theorem 4.4.
Let η : [0 , L ] → S R be an almost self-involute on S R of class C , parameterized by arc length, such that lim s → + τ s = 0 , lim s → + ˙ τ s = a , lim s → + κ s = + ∞ . (4.21) Then η is self-involute if and only if a is the unique solution of the equation a = e π a . (4.22) Proof.
Given an almost self-involute η : [0 , L ] → S R , let (cid:101) η be the initial arcof the involute that is congruent to η . With no loss of generality, we assumethat η = (0 , , R ). By definition, we have that (cid:101) η = η = (0 , , R ) and thatthere exists an orthogonal matrix A ( η ) = cos φ − sin φ φ cos φ
00 0 1 or A ( η ) = cos φ sin φ φ − cos φ
00 0 1 (4.23)such that (cid:101) η s = A ( η ) · η s for any s ∈ [0 , L ].We also identify E with the plane { x = 1 } in R and also in this casefor any almost self-involute η : [0 , L ] → E = { x = 1 } with η = (0 , , (cid:101) η its involute and by A ( η ) the orthogonal matrix as in (4.23)such that (cid:101) η s = A ( η ) · η s .Finally, for any almost self-involute η on S R or E , we call fundamentalpair of η the couple ( A, a ) given by A = A ( η ) and a = lim s → + ˙ τ s , where τ isdefined in (4.6) or (4.7), respectively. Clearly, η is self-involute if and onlyif A = I .The proof is a consequence of the existence of a canonical correspondencebetween any almost self-involute η on S R with fundamental pair ( a, A ) and an almost self-involute η ( ∞ ) s on E , determined up to Euclidean isometries,having the same fundamental pair of η and congruent to the curve η ( ∞ ) s = (cid:18) s √ a cos (cid:18) a log (cid:18) s √ a (cid:19)(cid:19) , − s √ a sin (cid:18) a log (cid:18) s √ a (cid:19)(cid:19) , (cid:19) (4.24) which is a parameterization by arc length of the curve γ t = ( e − ta cos t, e − ta sin t, , studied by Manselli and Pucci in [MP1]. In that paper, it is proved that η ( ∞ ) is self-involute (i.e. with a fundamental pair ( I, a )) if and only if a issolution of (4.22). Since η ( ∞ ) and η have the same fundamental pair, theconclusion follows.Let us prove the existence of the correspondence η (cid:55)−→ η ( ∞ ) describedabove. For any 0 (cid:54) = λ ∈ R , let η ( λ ) : [0 , L ] → S λR , η ( λ ) s def = λ η (cid:16) sλ (cid:17) , (4.25)which is the initial arc of length L (parameterized by arc length) of thedilatation of η by λ . From definitions, one can check that η ( λ ) is an almostself-involute of S λR , with involute (cid:103) η ( λ ) = (cid:101) η ( λ ) = A · η ( λ ) and with functions κ ( λ ) s and τ ( λ ) s given by κ ( λ ) s = 1 λ κ sλ , τ ( λ ) s = λτ sλ . (4.26)In particular, lim s → ˙ τ ( λ ) s = lim s → ˙ τ s = a and the fundamental pair ( A, a )is the same for all curves η ( λ ) . For all λ sufficiently large, η ( λ ) : [0 , L ] → S λR is included in the upper hemisphere and it can be identified with its imagein { x = λR } by the projection π ( λ ) : S λR (+) → { x = λR } (cid:39) E of centerthe origin.The correspondence we are looking for is based on the following lemma. Lemma 4.5.
There exists a sequence λ n → + ∞ such that the curves η ( λ n ) converge on (0 , L ] , uniformly on compacta together with their first and sec-ond derivatives, to a C -curve η ( ∞ ) .Proof. To prove this, first of all notice that, being of length L on thesphere and obtained via the projection π ( λ ) , any curve η ( λ ) starts from x o =(0 , , λR )( (cid:39) (0 , , ∈ E ) and it is contained in the closed disk D r ( x o ) ofradius r = λR tan (cid:0) LλR (cid:1) < L for all λ sufficiently large. Secondly, let usdenote by g ( λ ) the metric on E (cid:39) { x = λR } defined as push-forwardby the projection π ( λ ) of the metric of S λR . Notice that on any closeddisc D r o ( x o ), the metric g ( λ ) converges uniformly to the Euclidean metric g o together with all derivatives. Now, by construction, for any λ and s ∈ [0 , L ], we have that g ( λ ) ( ˙ η ( λ ) s , ˙ η ( λ ) s ) = 1 and hence | ˙ η ( λ ) s | = (cid:114) g o (cid:16) ˙ η ( λ ) s , ˙ η ( λ ) s (cid:17) is TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 21 uniformly bounded for all λ sufficiently large. A similar argument shows thatalso the normal vectors n ( λ ) s of the curve η ( λ ) (orthogonal to the t ( λ ) s = ˙ η s w.r.t. g ( λ ) ) are uniformly bounded. On the other hand, from (4.6), the factthat τ is monotone and that lim s → + τ s = 0, one has that for any fixed0 < ε o < L and any s ∈ [ ε o , L ]lim λ → + ∞ κ ( λ ) s = lim λ → + ∞ λ κ sλ = lim λ → + ∞ λR cot (cid:32) τ − (cid:0) sλ (cid:1) R (cid:33) == lim µ → + µτ − ( µs ) l’Hˆopital = as ≤ aε o (4.27)and with similar computationslim λ → + ∞ ˙ κ ( λ ) s = − as ≥ − aε o . From this and (4.1), it follows that the covariant derivatives ∇ ˙ η ( λ ) t ( λ ) s and ∇ η ( λ ) t ( λ ) s are uniformly bounded in any given interval [ ε o , L ]. Consideringthe explicit expression of such covariant derivatives in terms of the Christof-fel symbols of g ( λ ) and of the derivatives ¨ η ( λ ) s and ... η ( λ ) s , one can directly checkthat on any given interval [ ε o , L ], the curves η ( λ ) are uniformly bounded in C -norm. From this, the lemma follows.Using definitions and convergence in C , one can check that if ( A, a ) isthe fundamental pair of η : [0 , L ] → S R , then the limit curve η ( ∞ ) and (cid:101) η ( ∞ ) = A · η ( ∞ ) satisfy the relation (4.4), i.e. η ( ∞ ) is almost self-involute.Moreover, by (4.4), we see that the curvature of η ( ∞ ) is given by κ ( ∞ ) s = as and the associated function is τ ( ∞ ) s = as . In particular, the fundamentalpair of η ( ∞ ) is ( A, a ), the same of η . Notice also that since the curve (4.24)has curvature function given by κ s = as , by the Fundamental Theorem ofPlane Curves, η ( ∞ ) is congruent to (4.24) and this concludes the proof thatthe correspondence η (cid:55)−→ η ( ∞ ) has all stated properties.From this and Theorem 4.3, the next corollary follows immediately. Corollary 4.6.
There exists L o > such that for any L < L o there existsa self-involute curve of length L on the unit sphere S . G -curves on spheres with the “maximal length property” This section is devoted to show the existence of G -curves γ : [0 , L ] → S = S realizing the “maximal length property”, i.e. such that for any s ∈ [0 , L ],the length (cid:96) S ( γ | [0 ,s ] ) = s coincides with the largest possible value accordingto Theorem 3.2, namely (cid:96) ( γ | [0 ,s ] ) = p ( s ). Theorem 5.1. If η : [0 , L ] → S = S is a self-involute, then it is a G -curvewith the maximal length property, i. e. such that p ( s ) = (cid:96) S ( γ | [0 ,s ] ) , for any s ∈ [0 , L ] . (5.1) Proof.
In the following, we constantly identify the self-involute η withits image in { x = 1 } (cid:39) E , determined by the projection π : S →{ x = 1 } of center the origin. We also denote by g the metric on E defined as push-forward by the projection π of the metric of S and by g o = dx ⊗ dx + dx ⊗ dx the standard Euclidean metric.A simple computation shows g = g ij dx i ⊗ dx j with (cid:16) g g g g (cid:17) = x x (cid:16) x ) − x x − x x x ) (cid:17) . From this, by well-known formulae, we may express in term of η s = ( η s , η s )and its derivatives the following objects:– the Christoffel symbols Γ kij (cid:12)(cid:12)(cid:12) η s = g km (cid:16) ∂g mj ∂x i + ∂g im ∂x j − ∂g ij ∂x m (cid:17)(cid:12)(cid:12)(cid:12) η s ;– the covariant derivatives ∇ t s t s = (cid:18) ¨ η is + Γ ijk (cid:12)(cid:12)(cid:12) η s ˙ η js ˙ η ks (cid:19) ∂∂x i ;– the geodesic curvature k s = (cid:112) g ( ∇ t s t s , ∇ t s t s ) = g ( ∇ t s t s , n s ).Since κ s >
0, also the Euclidean curvature κ Es of η s is positive. This canbe checked as follows. Recall that ¨ η s = g o ( ˙ η s , ˙ η s ) κ Es n Es + λ s ˙ η s for somefunction λ s and with n Es denoting the unit normal vector in the Euclideansense. Since g ( ˙ η s , n s ) = g ( t s , n s ) = 0 and using the expression for ∇ t s t s , g o ( ˙ η s , ˙ η s ) κ Es g ( n Es , n s ) = g (¨ η s , n s ) = κ s − g (cid:18) Γ ijk (cid:12)(cid:12) η s ˙ η js ˙ η ks ∂∂x i , n s (cid:19) . Using properties of projectively flat connections or just by a direct compu-tation, one can see that the vector v s = Γ ijk (cid:12)(cid:12)(cid:12) η s ˙ η js ˙ η ks ∂∂x i is proportional to˙ η s = t s . From this, it follows that κ Es = κ s g o ( ˙ η s , ˙ η s ) g ( n Es , n s ) . Since n Es and n s lie on the same side w.r.t. t s , g ( n Es , n s ) > κ Es is positive.Being κ E >
0, the “angle” function (i.e. the Euclidean angle between ˙ η s and the x -axis), computable by ϕ : [0 , L ] → R , ϕ def = (cid:90) sL κ Eu du + ϕ L , with ϕ L = (cid:92) ˙ η L ∂∂x , is monotone increasing.Now, for any s ∈ (0 , L ], let us consider the following notation:– (cid:96) ( s )1 denotes the tangent line to η at the point x = η s , while (cid:96) ( s )2 denotes the line through x and parallel to the vector n s ; recall thatthese lines, up to reparameterizations, are geodesics for both theEuclidean metric g o and the spherical metric g ; TEEPEST DESCENT CURVES ON SURFACES OF CONSTANT CVURVATURE 23 – γ ( s ) is the closed, piecewise C curve, formed by the arc η | [ τ − ( s ) ,s ] and the segment joining η s and η τ − ( s ) ;– ρ ( s ) is the total rotation of γ ( s ) , i.e. the multiple of 2 π defined by ρ ( s ) = ϕ s − ϕ τ − ( s ) + (cid:91) n s t s (5.2)and m s = ρ ( s ) / π . Being ϕ monotone increasing, ρ ( s ) coincides withthe total curvature of the curve γ ( s ) . From this, by Fenchel’s theoremfor piecewise differentiable curves ([Fe]; see also [Mi, Ae]), γ ( s ) is asimple convex curve if and only if m s = 1.We claim that for any s , (cid:96) ( s )1 and (cid:96) ( s )2 are support lines for the arc η | [0 ,s ] andthat γ ( s ) is a simple and convex curve. This immediately implies that η is a G -curve and that γ ( s ) is the boundary of the convex hull of η | [0 ,s ] . Being η s self-involute, it follows that p ( s ) = (cid:96) S ( η | [ τ − ( s ) ,s ] ) + (cid:96) S ([ η τ − ( s ) , η s ]) = s − τ − ( s ) + τ − ( s ) = s i.e. η s satisfies (5.1) at any s .The proof of the theorem is therefore a direct consequence of the followingthree claims. Claim 1: m s = 1 for any s , i.e. any closed curve γ ( s ) is simple andconvex .In fact, from (5.2), the map s (cid:55)→ m s is continuous and therefore constant.Moreover, for any λ >
1, the arc η | [0 , Lλ ] is homothetic to the curve η ( λ ) described in (4.25) (which we also identify with the corresponding projectedcurve on E ) and, by the proof of Theorem 4.4, we may choose λ so largethat η ( λ ) is arbitrarily close in C -norm to the self-involute η ( ∞ ) . By theresults in [MP], η ( ∞ ) is a G -curve of E . So, if we denote by γ ( s |∞ ) theclosed curved formed by the segment joining η ( ∞ ) s and η ( ∞ )( τ ( ∞ ) ) − ( s ) and thearc η ( ∞ ) | [( τ ( ∞ ) ) − ( s ) ,s ] , it is immediate to realize that γ ( s |∞ ) is simple andconvex, that is m ( ∞ ) s = 1. Since γ ( s ) is homothetic to the piecewise C closed curve γ ( s | λ ) , close to γ ( s |∞ ) in C -norm, it follows that also m s = 1for any 0 < s ≤ Lλ and hence for all values of s . Claim 2: (cid:96) ( s )1 is a support line for η | [0 ,s ] .To see this, notice that, being γ ( s ) closed and convex, (cid:96) ( s )1 is a support linefor γ ( s ) and hence for η [ τ − ( s ) ,s ] . On the other hand, the spherical distance d S ( τ − ( s ) , (cid:96) ( s )1 ) is equal to the length of the segment joining η τ − ( s ) and η s ,because it lies in a line which is g -orthogonal to (cid:96) ( s )1 . The length of thissegment is equal to τ − ( s ) by the definition of self-involute. This length isalso equal to the length of the arc η [0 ,τ − ( s )] . Hence, this arc lies in the samehalf-plane of η [ τ − ( s ) ,s ] and (cid:96) ( s )1 is a support line for the whole curve η | [0 ,s ] . Claim 3: (cid:96) ( s )2 is a support line for η | [0 ,s ] .As before, being γ ( s ) closed and convex, (cid:96) ( s )2 is a support line for η [ τ − ( s ) ,s ] .On the other hand, by definition of self-involute, (cid:96) ( s )2 = (cid:96) ( τ − ( s ))1 and hence,by Claim 2, it is also a support line for the arc η | [0 ,τ − ( s )] . The two arcslie in the same half-plane, because the Euclidean curvature of η is strictlypositive at all points and the claim follows. References [Ae] A. Aeppli,
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Cristina Giannotti and Andrea Spiro, Dip. Matematica e Informatica, ViaMadonna delle Carceri, I- 62032 Camerino (Macerata), ITALY
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