Functions on the sphere with critical points in pairs and orthogonal geodesic chords
FFUNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS ANDORTHOGONAL GEODESIC CHORDS
ROBERTO GIAMB `O, FABIO GIANNONI, AND PAOLO PICCIONEA
BSTRACT . Using an estimate on the number of critical points for a
Morse-even functionon the sphere S m , m (cid:62) , we prove a multiplicity result for orthogonal geodesic chordsin Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. Thisyields also a multiplicity result for brake orbits in a potential well.
1. I
NTRODUCTION
The topology of spheres does not allow good estimates on the number of critical pointsof smooth functions. In fact, given the fact that any C -function on a compact manifold ad-mits at least two critical points, Reeb’s theorem characterizes spheres as the only compactmanifold that admits functions with exactly critical points (see for instance [14, Theo-rem 4.1, p. 25]). However, functions having some type of symmetry tend to have more thantwo critical points. For instance, if f : S m → R is even, i.e., f ( x ) = f (− x ) for all x , then f defines a function (cid:101) f on the projective space R P m , that must have at least m + distinctcritical points, by Lusternik–Schnirelman (or Morse) theory. Thus, the original function f must have at least m + distinct pairs of antipodal critical points.1.1. Morse-even functions.
Motivated by an example in classical Riemannian geometry,that will be described below, in this paper we prove that the same estimate on the number ofcritical points holds in a slightly more general situation, when resorting to the Lusternik–Schnirelman category of the projective space is not possible. More precisely, we willconsider functions whose critical points come in pairs with the same Morse index.
Definition.
Let M m be a compact m -dimensional manifold. A Morse function f : M → R is said to be Morse-even if for every k =
0, . . . , m , the set of critical points of f havingMorse index equal to k is an even number.The starting observation for Morse-even functions on spheres is the following: Theorem A.
Let M be a smooth manifold that is homeomorphic to an m -sphere, and let f : M → R be a Morse-even function. Then, f has critical points of arbitrary Morse indexin {
0, 1, . . . , m } . In particular, f admits at least ( m + ) distinct pairs of critical points.Proof. The Poincar´e polynomial (with integer coefficients) of the m -sphere is given by P m ( λ ) = + λ m . By Morse theory, there exists a polynomial Q ( λ ) = a + a λ + . . . + a m − λ m − with nonnegative integer coefficients such that:(1.1) + λ m +( + λ ) Q ( λ ) = ( + a )+( a + a ) λ + . . . +( a m − + a m − ) λ m − +( + a m − ) λ m is the Morse polynomial M f ( λ ) of f . Recall that M f ( λ ) = (cid:80) mk = κ k λ k , where κ k is thenumber of critical points of f whose Morse index is equal to k . Since f is Morse-even, theneach κ k is an even number. Using (1.1), one shows by an elementary argument that all the a i are non-zero and odd, i =
0, . . . , m − . Thus, all the κ k are positive, k =
0, . . . , m . (cid:3) Date : March 6th, 2015.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . D S ] M a r R. GIAMB `O, F. GIANNONI, AND P. PICCIONE
Remark . The above result has a natural generalization to Morse-even functions definedon a connected, orientable compact manifold M whose Betti numbers β k ( M ) are even for k =
1, . . . , m − . If f : M → R is a Morse-even function on such a manifold, then for all k =
1, . . . , m − , f admits a number strictly larger than β k ( M ) of pairs of critical pointshaving Morse index equal to k (this statement holds trivially also for k = and k = m ).A proof of this is obtained readily from the Morse relations, given by the equality: m (cid:88) k = β k x k + ( + x ) m − (cid:88) k = a k x k = m (cid:88) k = κ k x k , i.e.: κ = + a , κ k = β k + a k + a k − , for k =
1, . . . , m −
1, κ m = + a m , for some integer coefficients a k (cid:62) . The parity of the coefficients implies a , a m (cid:62) ,and a k + a k − (cid:62) for k =
1, . . . , m − .The condition on the parity of the Betti numbers is satisfied by a large class of manifolds.For instance, when m = , the condition β even is satisfied by every compact orientedsurface Σ . Namely, in this case β ( Σ ) = gen ( Σ ) (here gen ( Σ ) denotes the genus of Σ ). It is not hard to classify the homeomorphism classes of simply connected manifoldssatisfying the condition in low dimensions. When m = , if M is simply connectedthen β ( M ) = (and thus also β ( M ) = ). By the Poincar´e conjecture, the uniquesimply connected compact manifold that satisfies the assumptions is the -sphere. Manyinteresting cases of even Betti numbers are found in dimension . In this case, for a simplyconnected manifold it suffices to require that β is even. Recall that in dimension , thesecond Betti number β is additive by connected sums, i.e., given -manifolds M and M ,then β ( M M ) = β ( M ) + β ( M ) . Recall also that β ( C P ) = and β ( S × S ) = . Thus, connected sums of any number of copies of S × S and any even numberof copies of C P have even β .1.2. Orthogonal geodesic chords.
Our motivation for the result of Theorem A comesfrom a classical problem in Riemannian geometry, which consists in finding lower esti-mates on the number of geodesics departing and arriving orthogonally to the boundaryof a compact Riemannian manifold. These objects are called orthogonal geodesic chords (OGC). It is interesting to observe that there are manifolds diffeomorphic to Euclideanballs that have no OGC’s, see for instance [3]. Orthogonal geodesics chords for metricsin a ball have a special interest in the case where the boundary S m of B m + is strictlyconcave. Namely, in this situation, a multiplicity result for OCG’s yields an analogousmultiplicity result for brake orbits of natural Hamiltonians or Lagrangian in a potentialwell (see for instance [5]).In order to apply Theorem A to obtain information on the number of OGC’s, let us con-sider the following situation. Given a compact Riemannian manifold ( M, g ) with boundary ∂M , let exp denote the corresponding exponential map. Let (cid:126) ν be the unit normal field along ∂M pointing inwards. The metric g will be said to be regular (with respect to ∂M ) if thereexists a (necessarily smooth) function s g : ∂M → ] + ∞ [ such that, for any p ∈ ∂M ,the geodesic (cid:2)
0, s g ( p ) (cid:3) (cid:51) t (cid:55)→ exp p ( t · (cid:126) ν p ) meets transversally ∂M at t = s g ( p ) . In thissituation, we will call s g the crossing time function of the metric g .It is easy to see that the set of regular metrics on a given manifold with boundary M , thatwill be denoted by Reg ( M ) , is open in the C -topology. We also define non-focal a metricon M for which there are no ∂M -focal points along ∂M . Also in this case, it is not hard toshow that non-focal metrics form an open subset in the C -topology, see Proposition 3.2.Let us denote by Reg ∗ ( M ) the set of non-focal regular metrics on M . Our main interestis in the case when M is diffeomorphic to the unit ball B m + in the Euclidean space R m + .In this case, the set of regular and non-focal metrics is an open subset of all Riemannianmetrics containing, for instance, the set of radially symmetric metrics, see Corollary 3.3. UNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS 3
As an application of Theorem A, we prove the following:
Theorem B.
For a generic set of metrics g in Reg ∗ ( B m + ) , there are at least ( m + ) distinct orthogonal geodesic chords in ( B m + , g ) . More precisely, the result of Theorem B holds for all metrics in Reg ∗ ( B m + ) for whichevery OGC is nondegenerate in an appropriate sense, see the discussion after Corollary 2.2and Section 3.2 for details.When M is convex and homeomorphic to the m + -dimensional disk the multiplicityproblem for OGC’s is studied in [3, 12]. If M is concave there is an existence result anda multiplicity result of two OGC’s (see [7] and the references therein). To the authors’knowledge, Theorem B is the first result about multiplicity of OGC’s without convexity orconcavity assumption.For the Morse-theoretical aspects in the proof of Theorem B, one of the key ingredientswill be an index theorem for orthogonal geodesic chords (see Corollary 2.4 and Corol-lary 2.5), in the formulation given in [15]. This result, together with a stability result forfocal points proved in [13], is used to prove that the crossing time function s g is even-Morse, providing the desired link between even-Morse functions and orthogonal geodesicchords.1.3. Brake orbits of Lagrangian systems.
The result of Theorem B can be applied toprove a new multiplicity result for brake orbits, as illustrated below. We will present herea Lagrangian formulation of the brake orbits problem. An equivalent formulation can begiven for periodic solutions of Hamiltonian systems, via Legendre transform.Let (M m + , g) be a Riemannian manifold (without boundary), representing the con-figuration space of some dynamical systems, and let V : M → R be a smooth function,representing the potential energy of some conservative force acting on the system. Onelooks for periodic solutions x : [
0, T ] → M of the Lagrangian systems:(1.2) Dd t ˙ x = − ∇ V, where Dd t denotes the covariant derivative of the Levi–Civita connection of g for vectorfields along x , and ∇ V is the gradient of V . Solutions of (1.2) satisfy the conservation ofenergy law g( ˙ x, ˙ x )+ V ( x ) = E , where E is a real constant called the energy of the solution x . It is a classical problem to give estimate of periodic solutions of (1.2) having a fixedvalue of the energy E . This problem has been, and still is, the main topic of a large amountof literature, also for autonomous Hamiltonian systems, see for instance [8, 9, 11, 16] anthe references therein. We will give here a very short account of a geometric approach toperiodic solutions of (1.2)By the classical Maupertuis principle, solutions of (1.2) having energy E are, up to aparameterization, geodesics in the conformal metric:(1.3) g E = (cid:0) E − V ( p ) (cid:1) · g , defined in the closed E -sublevel M E = V − (cid:0) ]− ∞ , E ] (cid:1) of V . Observe that, in fact, g E degenerates on the boundary ∂M E = V − ( E ) . Among all periodic solutions of (1.2), his-torical importance is given to a special class called brake orbits ; these are “pendulum-like”solutions, that oscillate with constant frequency along a trajectory that joins two endpointslying in V − ( E ) . Thus, brake orbits correspond to g E -geodesics in M E with endpoints in ∂M E , or, more precisely, to g E -geodesics γ : ]
0, T [ → V − (cid:0) ]− ∞ , E [ (cid:1) , with lim t → − γ ( t ) and lim t → T − γ ( t ) in ∂M E .For such degenerate situation, it has been proved in [5] that, if E is a regular value ofthe function V (which implies in particular that ∂M E is a smooth hypersurface of M ), then Here generic is in meant in the topological sense. A subset of a topological set is generic if it contains aresidual set (countable intersection of open dense subsets).
R. GIAMB `O, F. GIANNONI, AND P. PICCIONE g E defines a distance-to-the-boundary function dist E : M E → [ + ∞ [ which is smoothin the interior of M E and extends continuously to on the boundary ∂M E . Moreover, if δ > 0 is small enough, then any OGC in the Riemannian manifold M = dist − (cid:0) [ δ, + ∞ [ (cid:1) endowed with the metric g E (which is now non-singular) can be extended uniquely to a g E -geodesic γ in M E with endpoints in ∂M E , as above. In conclusion, any result onmultiplicity of OGC’s can be reformulated to a multiplicity result for brake orbits at levela fixed regular energy level of a conservative dynamical system.A very famous conjecture due to Seifert, see [17], asserts that, given a Lagrangiansystem as in (1.2), if the sublevel V − (cid:0) ]− ∞ , E ]] (cid:1) is homeomorphic to an ( m + ) -ball B m + , then there are at least m + distinct brake orbits. This estimate is known to besharp, i.e., there are examples of Lagrangian systems having energy levels homeomorphicto an ( m + ) -ball and admitting exactly m + distinct brake orbits. A proof of Seifert’sconjecture in its full generality is still open, but the question has been solved affirmativelyin some cases. When V is even and convex, multiplicity results are obtained in [9, 10, 18,19, 20]. In particular in [9] there is the proof of the Seifert conjecture for euclidean metricsand even and convex potentials.When the E -sublevel V − (cid:0) ]− ∞ , E ] has the topology of the annulus, the multiplicity ofbrake orbits is studied in depth in [4] and [6].Theorem B yields the following contribution to Seifert’s conjecture: Theorem C.
Let E be a regular value of V , such that V − (cid:0) ]− ∞ , E ] (cid:1) is homemorphic toan ( m + ) -ball B m + . Assume that there exists δ > 0 sufficiently small such that theRiemannian manifold M = dist − (cid:0) [ δ, + ∞ [ (cid:1) endowed with the metric g E satisfies theassumptions of Theorem B. Then there are at least m + distinct brake orbits of energy E for the Lagrangian system (1.2) . (cid:3) In particular, from Proposition 3.2 below we obtain the following:
Corollary D.
Seifert conjecture is generically true in a C -open set of potentials V thatcontains the ones that are rotationally symmetric at level E . (cid:3) By a potential V rotationally symmetric at level E we mean that there is a continuousaction of the rotation group SO ( m + ) on V − (cid:0) ]− ∞ , E ] (cid:1) that makes the sublevel equiv-ariantly diffeomorphic to the Euclidean ball B m + with the canonical SO ( m + ) -action,and such that V is constant along the orbits of this action.2. V ARIATIONAL T HEORY FOR
OGC’ S Let us show how obtain multiple orthogonal geodesic chords in a compact Riemannianmanifold with boundary ( M, g ) , using a function on ∂M whose critical points are OGC’s.Next Proposition shows that, for a regular metric g , such function is precisely the crossingtime function s g . In order to prove this, let us introduce some notations.For p ∈ ∂M , let S p : T p ( ∂M ) → T p ( ∂M ) denote the shape operator of ∂M at p inthe normal direction (cid:126) ν p . Let exp ⊥ : U ⊂ T ( ∂M ) ⊥ → M denote the normal exponentialmap of g along ∂M ; for p ∈ ∂M , let γ p : (cid:2)
0, s g ( p ) (cid:3) → M denote the geodesic t (cid:55)→ exp p ( t · (cid:126) ν p ) . Recall that a point q ∈ M is a singular value of exp ⊥ exactly when q is focal to ∂M . If q ∈ M is a singular value of exp ⊥ and v ∈ T p ( ∂M ) ⊥ is the correspondingcritical point, so that γ p ( t ∗ ) = q for some t ∗ ∈ ]
0, s g ( p )] , then the kernel of d exp ⊥ ( v ) consists of ∂M -Jacobi fields along the geodesic γ p that vanish at t ∗ . Recall that a Jacobifield along γ p is called a ∂M -Jacobi field if it satisfies the initial conditions:(2.1) J ( ) ∈ T p ( ∂M ) , J (cid:48) ( ) + S p (cid:0) J ( ) (cid:1) ∈ T p ( ∂M ) ⊥ , where J (cid:48) denotes the covariant derivative of J along γ p . UNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS 5 M ∂ M F IGURE
1. The picture represents a situation in which critical pointsof the function s g do not correspond to orthogonal geodesic chords (seeRemark 2). Here, M is a compact subset of a Euclidean space. A portionof the boundary of M (in red) is a spherical hypersurface, whose center,which is a ∂M -focal point, lies on ∂M . Orthogonal geodesic chordsstarting from this spherical surfaces converge and meet at the center;thus the function s g is constant in this spherical region of the boundary.However, these critical points do not correspond to OGC’s in M .For p ∈ ∂M , denote by q p = γ p (cid:0) s g ( p ) (cid:1) ∈ ∂M , and let E p ⊂ T q p M the image of thelinear map d exp ⊥ (cid:0) s g ( p ) (cid:126) ν p (cid:1) . Equivalently: E p = (cid:10) J (cid:0) s g ( p ) (cid:1) : J is a ∂M -Jacobi field along γ p (cid:11) . When q p is not ∂M -focal, then E p = T q p M . Proposition 2.1.
Assume g regular. A point p ∈ ∂M is critical for s g : ∂M → ] + ∞ [ if and only if the vector ˙ γ p (cid:0) s g ( p ) (cid:1) ∈ T q p M is orthogonal to the intersection E p ∩ T q p ( ∂M ) . In particular, if q p is not ∂M -focal, then γ p is an orthogonal geodesic chordin M .Proof. Since s g > 0 , the critical points of s g coincide with those of s . For p ∈ ∂M ,denote by σ p : [
0, 1 ] → M the affinely reparameterized geodesic σ p ( t ) = γ p (cid:0) s g ( p ) · t (cid:1) , t ∈ [
0, 1 ] . Integration on [
0, 1 ] gives:(2.2) (cid:90) g ( ˙ σ p , ˙ σ p ) d t = s g ( p ) , i.e., σ can be thought of the geodesic action functional of g applied to the geodesic σ p . Let p ∈ ∂M be fixed, and let ρ : ]− ε, ε [ → ∂M be a C -curve satisfying ρ ( ) = p and ˙ ρ ( ) = w ∈ T p ( ∂M ) . Then, p is critical for s iff for any such ρ one has dd t (cid:12)(cid:12) t = s ( ρ ( t )) = .Using (2.2) one has:(2.3) dd t (cid:12)(cid:12) t = s ( ρ ( t )) = (cid:90) g ( ˙ σ p , J (cid:48) w ) d t, where J w is the Jacobi field along σ p given by: J w ( s ) = dd t (cid:12)(cid:12) t = σ ρ ( t ) ( s ) , s ∈ [
0, 1 ] . We observe that, under the assumptions of Proposition 2.1, q p is ∂M -focal if and only if E p ∩ T q p ∂M (cid:54) = T q p ∂M . This follows easily from the fact that E p always contains ˙ γ p (cid:0) ( s g ( p ) (cid:1) which is transversal to T q p ∂M . R. GIAMB `O, F. GIANNONI, AND P. PICCIONE
Keeping in mind that J w ( ) = w ∈ T p ( ∂M ) and that ˙ σ p ( ) ∈ T p ( ∂M ) ⊥ , integration byparts in (2.3) gives: dd t (cid:12)(cid:12) t = s ( ρ ( t )) = ( ˙ σ p ( ) , J w ( )) . It is easily seen that the map T p ( ∂M ) (cid:51) w (cid:55)→ J w ( ) ∈ E p ∩ T q p ( ∂M ) is surjective, andfrom this observation the thesis follows readily. (cid:3) Remark . It is not hard to give examples of regular metrics that do not satisfy the non-focal property, and in which critical points of s g do not correspond to orthogonal geodesicchords. See Figure 1.From Proposition 2.1, we obtain immediately the following: Corollary 2.2.
Let g be regular and non-focal. Then, a point p is critical for s g if andonly if the point q p = γ p (cid:0) s g ( p ) (cid:1) is critical for s g . In this case, s g ( p ) = s g ( q p ) , and γ q p = γ p up to orientation. (cid:3) Proposition 2.1 gives a first order variational principle relating orthogonal geodesicchords in M to critical points of a smooth function on ∂M . For our purposes, we needa related second order variational principle, relating nondegeneracy and Morse index ofOGC’s in M and critical points of s g . Let us recall a few facts from the variational theoryof OGC’s.Assume that γ p : (cid:2)
0, s g ( p ) (cid:3) → M is an OGC in M , i.e., that ˙ γ p (cid:0) s g ( p ) (cid:1) ∈ T q p ( ∂M ) ⊥ .The index form along γ p is the symmetric bilinear form I p defined on the vector space V p of (piecewise smooth) vector fields V along γ p satisfying V ( ) ∈ T p ( ∂M ) and V (cid:0) s g ( p ) (cid:1) ∈ T q p ( ∂M ) , defined by:(2.4) I p ( V, W ) = (cid:90) s g ( p ) g ( V (cid:48) , W (cid:48) ) + g (cid:0) R ( γ (cid:48) p , V ) γ (cid:48) p , W (cid:1) d s − (cid:104) g (cid:0) S p ( V ( )) , W ( ) (cid:1) + g (cid:0) S q p ( V ( s g ( p ))) , W ( s g ( p )) (cid:1)(cid:105) , where R is the curvature tensor of g , chosen with the sign convention R ( X, Y ) = [ ∇ X , ∇ Y ] − ∇ [ X,Y ] . It is well known that I p is the second variation of the geodesic action functional, defined inthe set of paths with endpoints in ∂M , at the critical point γ p . The OGC γ p is said to be nondegenerate if I p is a nondegenerate bilinear form on V p , i.e., if γ p is a nondegeneratecritical point of the geodesic action functional of M in the space of paths with endpoints in ∂M . The kernel of I p is the space of ∂M -Jacobi fields J along γ p that satisfy, in additionto (2.1), the following boundary condition at t = s g ( p ) :(2.5) J (cid:0) σ g ( p ) (cid:1) ∈ T q p ( ∂M ) , J (cid:48) (cid:0) s g ( p ) (cid:1) + S q p (cid:0) J ( s g ( p )) (cid:1) ∈ T q p ( ∂M ) ⊥ . Thus, γ p is nondegenerate if and only if there exists no non-trivial Jacobi field J along γ satisfying (2.1) and (2.5). The Morse index of γ p is the index of the symmetric bilinearform I p , which is the dimension of a maximal subspace of V p on which I p is negativedefinite. Recall that this is a (finite) nonnegative integer, that can be computed in terms ofsome focal invariants of ∂M , see [15] for details. Proposition 2.3.
Assume g regular. Let p ∈ ∂M be a critical point of s g , and assume that q p ∈ ∂M is not ∂M -focal along γ p . Then: (a) p is a nondegenerate critical point of s g if and only if γ p is a nondegenerate OGC; Observe that we have a different sign convention from the standard literature in the second boundaryterm of I p , because S q p has been defined as the shape operator in the normal direction (cid:126) ν q p , and (cid:126) ν q p =− ˙ γ p (cid:0) s g ( p ) (cid:1) . Similarly, the second condition in (2.5) has a sign different from the standard literature. UNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS 7 (b) the Hessian of s at the critical point p is identified with the symmetric bilinearform: (2.6) J p × J p (cid:51) ( J , J ) (cid:55)−→ g (cid:16) J (cid:48) ( s g ( p )) − S q p (cid:0) J ( s g ( p )) (cid:1) , J ( s g ( p )) (cid:17) ∈ R , where J p is the vector space of all ∂M -Jacobi fields J along γ p that satisfy J (cid:0) s g ( p ) (cid:1) ∈ T q p ( ∂M ) . Its index is less than or equal to the Morse index of the OGC γ p .Proof. As in Proposition 2.1, both statements are obtained by identifying the points p of ∂M with the curve γ p , as an element of the space of curves with endpoints in ∂M . Usingthis identification, the tangent space T p ( ∂M ) is identified with the space of variations of γ p by geodesics γ q that start orthogonally to ∂M and arrive onto ∂M . This space ofvariations is given by the vector space J p . The function s is the restriction to the set { γ p : p ∈ ∂M } of the geodesic action functional of M . We have proved that, underour assumptions, γ p is a critical point of this restriction, but also a critical point of the full geodesic action functional. Hence, the second derivative of s at p is given by therestriction of the index form I p to the space J p ; this implies, in particular, that the Morseindex of p is less than or equal to the Morse index of the OGC γ p . Formula (2.6) isobtained easily using partial integration in (2.4), the Jacobi equation and (2.1). This proves(b).The assumption that q p is not ∂M -focal along γ p implies that, as J runs in J p , thevector J ( b ) is an arbitrary vector in T q p ( ∂M ) . Hence, I p ( J , J ) = for all J if andonly if J satisfies (2.5), i.e., the kernel of the second derivative of s at the critical point p ∈ ∂M coincides with the kernel of the index form I p . This proves statement (a). (cid:3) Using the Morse index theorem for geodesics between two fixed submanifold, see forinstance [15], one proves the following more precise result on the Morse index of criticalpoints of s g : Corollary 2.4.
Under the assumption of Proposition 2.3, the Morse index of s at p isequal to the Morse index of the OGC γ p minus the number of ∂M -focal points along γ p ,counted with multiplicity.Proof. It follows readily from Proposition 2.3 and the result of [15, Theorem 2.7]. Moreprecisely, [15, Theorem 2.7] proves that, when q p is not ∂M -focal along γ p , then thespace V p (recall that this is the domain of the index form I p ) is the direct sum of: • the space V of (piecewise smooth) vector fields V along γ p satisfying V ( ) ∈ T p ( ∂M ) , • the space J p of ( ∂M ) -Jacobi fields J along γ p satisfying J (cid:0) s g ( p ) (cid:1) ∈ T q p ( ∂M ) .Such a direct sum decomposition is I p -orthogonal. Hence, the index of I p on V p , which isthe Morse index of the OGC γ p , is equal to the sum of the indices of the restriction of I p to each one of the two spaces above. The index of the restriction of I p to J p is preciselythe index of the bilinear form (2.6), i.e., the Morse index of s at p . The index of therestriction of I p to the space V , by the Morse index theorem for geodesics between asubmanifold and a fixed point (see [15, Theorem 2.5]), is given by the number of ∂M -focal points along γ p counted with multiplicity. (cid:3) Corollary 2.5.
Let M be a compact manifold with connected boundary ∂M , and let g bea regular and non-focal Riemannian metric on M . If every OCG in M is nondegenerate,then s g is an even-Morse function on ∂M .Proof. From Proposion 2.3, part (a), it follows that if every OGC is nondegenerate, thenalso every critical point of s g is nondegenerate, i.e., s g is a Morse function. Let us show R. GIAMB `O, F. GIANNONI, AND P. PICCIONE M ∂ M F IGURE
2. The picture illustrates an example of a non-degenerate OGCwhich has one ∂M -focal point, but its backward reparameterization hasno ∂M -focal point. Here M is a compact subset of a Euclidean space.The boundary of M has a portion which is a spherical surface. An or-thogonal geodesic chord departs from this surface, and it passes throughthe center of the sphere, which is a focal point. At the final endpoint, thegeodesic meets orthogonally a portion of ∂M which is planar (i.e., flat).Thus, there are no focal points along the backward reparameterization ofthe geodesic. Note that this phenomenon only occurs when the metricfails to be non-focal, so that the picture shows also that the non-focalproperty is not generic. Namely, any sufficiently small perturbation ofthe metric preserves the existence of a non-degenerate OGC, which hasdifferent Morse indexes when run in the two directions, by the stabilityof the Morse index. Therefore, any perturbed metric fails to be non-focal.that, given a critical point p of s g , then q p is a critical point with the same Morse index.From Corollary 2.4, the difference between the Morse indices of p and q p equals thedifference of the number of ∂M -focal points along γ p and along γ q p (which incidentallyis the same geodesic as γ p , with the opposite orientation, see Corollary 2.2). We claim thatthe number of ∂M -focal points along any geodesic γ r : [
0, s g ( r )] → M is constant for al r ∈ ∂M . This follows easily from the assumption that no γ r has any ∂M -focal point in M ,and an argument of stability of the number of focal points in Riemannian geometry, see [13]for details. Namely, as r runs in ∂M , the number of focal points (counted with multiplicity)along γ r changes continuously, and by connectedness, if such number is not constant in ∂M , then there would be a ∂M -focal point in ∂M along some γ r . This concludes theproof. (cid:3) Remark . It is not hard to show that in all the results of this section, the assumptionscannot be relaxed. As we observed in Remark 2, Figure 1 shows an example of a metricwhich is not non-focal, and where the critical points of s g do not correspond to OGC’s (cf.Proposition 2.1 and Corollary 2.2). Figure 2 illustrates an example where the function s g is a Morse function, but not even-Morse (cf. Corollary 2.5). Figure 3 gives an example ofa regular and non-focal metric on a manifold of arbitrary dimension, whose boundary isnot connected, and admitting only OGC’s.
Proposition 2.6.
Under the assumptions of Corollary 2.5, there are at least ( m + ) distinctorthogonal geodesic chords in ( B m + , g ) . Observe that the fact that γ q p is the same as γ p with the opposite orientation does not imply in principlethat the number of ∂M -focal points along the two geodesics is the same, see Figure 2. UNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS 9 M ∂ M γ γ F IGURE
3. When the boundary of M is not connected, one cannot ex-pect the existence of more than orthogonal geodesic chords, regardlessof the dimension of M . Consider for instance an annular region of R m + bounded by two non-concentrical spheres. In this situation, the metric isregular and, if the inner sphere contains the center of the outer sphere inits interior, then the metric is also non-focal. Thus, s g is an even-Morsefunction on a non-connected manifold. However, there are only two or-thogonal geodesic chords, and thus s g admits only two pairs of criticalpoints with the same Morse index. Observe that in Remark 1 on page 2only connected manifolds were considered. Proof.
It follows immediately from Theorem A and Corollary 2.5. (cid:3)
In next section we will discuss the assumptions of the results proved above, and we willprove the genericity of the nondegeneracy condition for OGC’s.3. P
ROOF OF T HEOREM
B. A
DISCUSSION OF THE ASSUMPTIONS .3.1.
On the genericity of metrics without degenerate OGC’s.
Recall that a metric on acompact Riemannian manifold is called bumpy if every one of its closed geodesic (possiblyiterated) is nondegenerate, i.e., it does not admit non-trivial periodic Jacobi fields. A clas-sical result due to Anosov [1] asserts that bumpy metric are generic. Genericity of metricsall of whose geodesics satisfying more general boundary condition are nondegenerate isproved in [2].Let us introduce the following terminology. Given a compact manifold M with bound-ary, a metric g ∈ Reg ∗ ( M ) will be called ∂ -bumpy if every OGC in M is nondegenerate.The following is what we need for the conclusion of Theorem B: Proposition 3.1.
Let M be a compact manifold with boundary. The set: (3.1) (cid:103) Reg ( M ) := (cid:10) g ∈ Reg ∗ ( M ) : g is ∂ -bumpy (cid:11) is C k -generic in Reg ∗ ( M ) for all k =
2, . . . , + ∞ .Proof. It suffices to show that (cid:103)
Reg ( M ) is C k -generic in the set of all metrics on M . Thisis an application of the result in [2]. In order to obtain the desired conclusion, one mustobserve that the proof of [2, Theorem 5.10] yields the genericity of metrics having onlynondegenerate geodesics satisfying arbitrary boundary conditions, provided that one canexclude a priori the existence of strongly degenerate periodic geodesics in this class. This The notion of strong degeneracy for a periodic geodesic has been introduced in [2], and it is stated in terms ofthe existence of a non-trivial periodic Jacobi field along an iterated periodic geodesic, satisfying some additionalproperties. A detailed discussion of this notion is irrelevant in the context of the present paper. is always the case for orthogonal geodesic chords, because this class does not contain anyperiodic geodesic at all. (cid:3) ∂∂∂MMM -focal points and non-focal metrics.
Unlike bumpy metrics, the set of non-focalRiemannian metrics on a compact manifold with boundary ∂M is not generic in general. Example . For instance, one can construct examples of hypersurfaces Σ in R n , diffeomor-phic to an ( n − ) -disk, whose set of focal points contains another hypersurface F . Now,consider the closure Ω of a bounded open subset whose smooth boundary ∂Ω contains Σ ,and such that ∂Ω intercepts F transversally. In this situation, transversality implies thatany C -perturbation of the flat metric on Ω will produce ∂Ω -focal points along ∂Ω . Thesituation is illustrated concretely in Figure 2. The picture provides an example of a metricwhich fails to be non-focal, and the same holds also for small perturbations of the metric.However, the absence of ∂M -focal points on ∂M is still a quite general assumptions,that holds in a large variety of circumstances. Intuitively, concavity of the boundary (whichis a property satisfied by Jacobi metric) and non-positive curvature have an effect of keep-ing focal points away from ∂M . More precisely, we can prove that when the sectionalcurvature of g and the eigenvalues of the shape operator of ∂M are not too large comparedto the size of M , then there are no ∂M -focal points in M . In order to give basis to thisassertion, let us introduce the following constants.Let g be a regular metric on M ; the maximum of the function s g on ∂M , denoted by L ( M, g ) will be called the maximal crossing time of ( M, g ) . Clearly, L ( M, g ) is somewhatrelated to, however in general larger than, the diameter of ( M, g ) . Moreover, denote by Λ the maximum, for p ∈ ∂M , of the largest eigenvalue of S p , and by K the maximum valueof the sectional curvatures of all -planes tangent to M . Proposition 3.2.
Let M be a compact manifold with smooth boundary. (a) The set of non-focal metrics on M is open with respect to the C -topology. (b) A sufficient condition for a metric g to be non-focal is that the following inequalityholds: (3.2) K + · L ( M, g ) + Λ + · L ( M, g ) − < 0, (where a + denotes the positive part of the real numner a .) Condition (3.2) implies indeedthat there is no focal point to ∂M in M .Proof. The set of focal points to a given compact hypersurface along a normal geodesicis stable by C -perturbations of all data. A formal proof of this fact can be found inreference [13]. From this stability, (a) follows easily.For the proof of (b), let us show that, under the assumption (3.2), given any p ∈ ∂M , theindex form I p is positive definite on the space on the vector space U p of (piecewise smooth)vector fields V along γ p satisfying V ( ) ∈ T p ( ∂M ) , V (cid:0) s g ( p ) (cid:1) = , and g ( V, ˙ γ p ) ≡ .Recall that γ p : (cid:2)
0, s g ( p ) (cid:3) → M is a unit speed geodesic.Let V ∈ U p \ { } be fixed; up to normalization, we can assume:(3.3) (cid:90) s g ( p ) g ( V, V ) d s = Since V (cid:0) s g ( p ) (cid:1) = , for all t ∈ (cid:2)
0, s g ( p ) (cid:3) we have: − g (cid:0) V ( t ) , V ( t ) (cid:1) = (cid:90) s g ( p ) t g ( V (cid:48) , V ) d s, which gives(3.4) g (cid:0) V ( t ) , V ( t ) (cid:1) (cid:54) (cid:90) s g ( p ) g ( V (cid:48) , V (cid:48) ) g ( V, V ) d s. UNCTIONS ON THE SPHERE WITH CRITICAL POINTS IN PAIRS 11
Integrating (3.4) on [
0, s g ( p )] and keeping in mind (3.3), Schwarz’s inequality gives:(3.5) = (cid:90) s g ( p ) g ( V, V ) d s (cid:54) g ( p ) (cid:90) s g ( p ) g ( V (cid:48) , V (cid:48) ) d s. Setting t = in (3.4) and using again Schwarz’s inequality, we obtain:(3.6) g (cid:0) V ( ) , V ( ) (cid:1) (cid:54) g ( p ) (cid:90) s g ( p ) g ( V (cid:48) , V (cid:48) ) d s. Let us now estimate I p ( V, V ) , using formula (2.4); note that g (cid:0) ˙ γ p , V ) ˙ γ p , V (cid:1) is equal to − K ( ˙ γ p , V ) g ( V, V ) , where K ( ˙ γ p , V ) is the sectional curvature of the -plane spanned by˙ γ p and V . Then, using the very definition of the constants Λ and K , we get:(3.7) I p ( V, V ) (cid:62) − Λg (cid:0) V ( ) , V ( ) (cid:1) + (cid:90) s g ( p ) (cid:2) g ( V (cid:48) , V (cid:48) ) − Kg ( V, V ) (cid:3) d s by (3.6) (cid:62) − + s g ( p ) (cid:90) s g ( p ) g ( V (cid:48) , V (cid:48) ) d s + (cid:90) s g ( p ) (cid:2) g ( V (cid:48) , V (cid:48) ) − Kg ( V, V ) (cid:3) d s by (3.5) (cid:62) (cid:104) − + s g ( p ) + − + s g ( p ) (cid:105) · (cid:90) s g ( p ) g ( V (cid:48) , V (cid:48) ) d s. Note that if V (cid:54) = , then (cid:82) s g ( p ) g ( V (cid:48) , V (cid:48) ) d s > 0 , because V (cid:0) s g ( p ) (cid:1) = , and therefore V cannot be parallel along γ . Thus, from (3.7) it follows that I p ( V, V ) > 0 if:(3.8) − + s g ( p ) − + s g ( p ) > 0. Note that (3.2) coincides with (3.8) when s g ( p ) = L ( M, g ) . On the other hand, if (3.2) issatisfied, it is easy to see that (3.8) holds for every p ∈ ∂M . (cid:3) For instance, when M is a ball, from part (a) of Proposition 3.2 we deduce the following: Corollary 3.3. If M is diffeomorphic to a ball, then the set of regular and non-focal metricson M is a C -open set that contains all the rotationally symmetric metrics.Proof. A rotationally symmetric metric in the ball is regular, because the geodesics startingorthogonally to the boundary are radial, and therefore they arrive transversally (orthogo-nally) to the boundary at the other endpoint. Moreover, for all these metrics the unique fo-cal point to the boundary is the center of symmetry, which is far from the boundary. Hence,sufficiently small perturbations of rotationally symmetric metrics are non-focal. (cid:3) R EFERENCES[1] D. V. A
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