- Email: [email protected]
The Maslov index and a generalized Morse index theorem for non-positive definite metrics
C. R. Acad. Sci. Paris, t. 331, Série I, p. 385–389, 2000 Géométrie différentielle/Differential Geometry
The Maslov index and a generalized Morse index theorem for non-positive definite metrics Paolo PICCIONE, Daniel V. TAUSK Departamento de Matemática, Universidade de São Paulo, Brazil E-mail: [email protected], [email protected] (Reçu le 16 avril 2000, accepté le 19 juin 2000)
Abstract.
We present an extension of the celebrated Morse index theorem in Riemannian geometry to the case of geodesics in pseudo-Riemannian manifolds. It is considered the case that both endpoints are variable. The notion of Maslov index for pseudo-Riemannian geodesics replaces the notion of geometric index for Riemannian geodesics. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
L’index de Maslov et le théorème d’indice de Morse généralisé pour une métrique non définie positive Résumé.
On étend ici le théorème d’indice de Morse en géométrie riemannienne au cas où la métrique n’est pas définie positive. On considère le cas des géodésiques avec les deux extrémités mobiles. La notion d’indice de Maslov pour les géodésiques pseudoriemanniennes remplace celle d’indice géométrique pour les géodésiques riemanniennes. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soient (M, g) une variété pseudo-riemannienne, γ : [a, b] → M une géodésique et P ⊂ M une sousvariété non dégénérée avec γ(a) ∈ P et γ(a) ˙ ∈ TP ⊥ . L’indice de Maslov iMaslov (γ) de γ relatif a P est un nombre entier que l’on définit en utilisant le premier groupe d’homologie relatif de la Grassmannienne de tous les sous-espace lagrangiens d’un espace symplectique. Génériquement, l’indice de Maslov de γ est une somme algébrique des points P -focaux le long de γ. Dans le cas où (M, g) est riemannienne, ou bien (M, g) est lorentzienne et γ est causal (g(γ, ˙ γ) ˙ 6 0), l’indice de Maslov devient l’indice géométrique donné par la somme des multiplicités des points P -focaux le long de γ. Nous considérons maintenant une autre sous-variété Q ⊂ M avec γ(b) ∈ Q et γ(b) ˙ ∈ TQ⊥ . Nous R b ˙ z) ˙ dt, définie définissons la forme d’indice I comme la variation seconde de l’énergie f (z) = 12 a g(z, sur l’espace de courbes reliant P et Q. Alors I est une forme bilinéaire symétrique bornée dans l’espace de Hilbert H des champs de vecteurs v le long de γ de régularité H1 tels que v(a) ∈ Tγ(a) P et v(b) ∈ Tγ(b) Q. Dans le cas riemannien, I a un indice fini ; en revanche, dans les autres cas, I a toujours un indice infini. Note présentée par Charles-Michel M ARLE. S0764-4442(00)01630-X/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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Le choix d’une distribution D de sous-espaces maximaux négatifs relatifs à g le long de γ nous permet de définir deux sous-espaces fermés KD et SD de H avec les propriétés suivantes : (a) KD et SD sont I-orthogonaux ; (b) l’indice de I|KD et le co-indice de I|SD sont finis. La version pseudo-riemannienne du théorème de l’indice de Morse pour les géodésiques avec extrémités mobiles, est donnée par l’égalité suivante, dans le cas où γ(b) n’est pas P -focal : Q − S|Tγ(b) Q , iMaslov (γ) = n− I|KD − n+ I|SD − n− g|Tγ(a) P − n− Sγ(b) ˙ Q est la deuxième où n− et n+ désignent l’indice et le co-indice d’une forme bilinéaire symétrique, Sγ(b) ˙ forme fondamentale de Q, évaluée dans la direction normale γ(b) ˙ et S est une forme bilinéaire symétrique définie en Tγ(b) M associée à la géodésique γ.
We consider a pseudo-Riemannian manifold (M, g), i.e., M is a smooth n-dimensional manifold and g is a smooth nondegenerate symmetric (2, 0)-tensor on M . We denote by ∇ the Levi-Civita covariant derivative for g and by R its curvature tensor, chosen with the sign convention R(X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] . Given a smooth submanifold N of M , a point p ∈ N and a normal vector n ∈ Tp N ⊥ , we denote by SnN the second fundamental form of N at p in the direction n, which is a symmetric bilinear form in Tp N given by SnN (v, w) = g(∇v W, n), where W is any smooth vector field tangent to N such that W (p) = w. Let γ : [a, b] → M be a geodesic and let P and Q be smooth submanifolds of M such that γ(a) ∈ P , ˙ ∈ Tγ(b) Q⊥ . Then, γ is a critical point of the action functional γ(a) ˙ ∈ Tγ(a) P ⊥ , γ(b) ∈ Q and γ(b) R 1 b ˙ z) ˙ in the space of curves in M connecting P and Q. Denoting by v 0 the covariant f (z) = 2 a g(z, derivative of a vector field v along γ, the second variation of f at γ is given by the index form: Z I(v, w) =
b
˙ v) γ, ˙ w g(v 0 , w0 ) + g R(γ,
a
Q P dt + Sγ(b) v(b), w(b) − Sγ(a) v(a), w(a) , ˙ ˙
which is a bounded symmetric bilinear form on the Hilbert space H of vector fields v along γ, of H1 -Sobolev regularity, with v(a) ∈ Tγ(a) P and v(b) ∈ Tγ(b) Q. We will assume henceforth that P is nondegenerate at γ(a), i.e., g is nondegenerate on Tγ(a) P ; then, P can be seen as a g-symmetric linear endomorphism of Tγ(a) P . the bilinear form Sγ(a) ˙ ˙ J) γ˙ = 0; a P -Jacobi field A Jacobi field J along γ is a solution of the differential equation J 00 − R(γ, is a Jacobi field J satisfying: J(a) ∈ Tγ(a) P
P and J 0 (a) + Sγ(a) J(a) ∈ Tγ(a) P ⊥ . ˙
We denote by J the n-dimensional space of P -Jacobi fields along γ; for t ∈ ]a, b] we set: J [t] = J(t) : J ∈ J . A point γ(t) with t ∈ ]a, b] is said to be P -focal along γ if there exists a non zero J ∈ J with J(t) = 0. The multiplicity of γ(t) is the dimension of the space of such J ’s. Equivalently, γ(t) is P -focal when J [t] 6= Tγ(t) M , and the multiplicity of γ(t) is the codimension of J [t]. Given a symmetric bilinear form B on a real vector space V , the negative (resp. positive) type number n− (B) (resp. n+ (B)) of B is the supremum of the dimensions of subspaces of V on which B is negative
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(resp. positive) definite. The negative (resp. positive) type number is also called the index (resp. the coindex). The signature sgn(B) of B is defined as the difference n+ (B) − n− (B), provided that at least one of the two type numbers is finite. Given a P -focal point γ(t), its signature is defined to be the signature of the restriction of g to J [t]⊥ ; if such restriction is nondegenerate, then γ(t) is said to be a nondegenerate P -focal point. P -focal points may accumulate; if γ has only a finite number of P -focal points, then we define the focal index of γ as the sum of the signatures of its P -focal points. It is known that nondegenerate P -focal points are isolated and that there are no P -focal points γ(t) for t sufficiently close to a; moreover, if (M, g) is real-analytic, then the set of P -focal points is finite. Let us recall the definition of Maslov index (see [1,3] and [6]); there are different notions of Maslov index defined in the literature in different contexts and we will use the same notion of [3], Section 2. Let us consider a parallel trivialization of TM along γ; for each t ∈ [a, b] we have an isomorphism φt : Tγ(t) M → Rn . Denote by g0 the nondegenerate symmetric bilinear form in Rn corresponding to g by φt (observe that g0 does not depend on t since g is parallel). We set: `(t) =
φt J(t) , φt J 0 (t) : J ∈ J ⊂ Rn ⊕ Rn ,
t ∈ [a, b];
then `(t) is a Lagrangian subspace of the symplectic space Rn ⊕ Rn endowed with the symplectic form: ω (x1 , y1 ), (x2 , y2 ) = g0 (x1 , y2 ) − g0 (x2 , y1 ). Thus, ` is a smooth curve in the Grassmannian Λ of all Lagrangian subspaces of (Rn ⊕ Rn , ω); Λ is a real-analytic connected and compact 12 n(n + 1)-dimensional manifold. Set L0 = {0} ⊕ Rn ∈ Λ; it is easy to see that γ(t) is P -focal if and only if `(t) is not transversal to L0 , i.e., `(t) ∩ L0 6= {0}. If we denote by Λ0 (L0 ) the open subset of Λ consisting of those Lagrangians that are transversal to L0 , then the first singular relative homology group H1 (Λ, Λ0 (L0 )) with integer coefficients is isomorphic to Z; the symplectic form ω defines canonically a choice for this isomorphism. Suppose that γ(b) is not P -focal; there exists ε > 0 such that γ(t) is not P -focal for t ∈ ]a, a + ε]; hence, the restriction `|[a+ε,b] is a curve in Λ with endpoints in Λ0 (L0 ) and therefore it defines a homology class in H1 (Λ, Λ0 (L0 )). The corresponding integer number is called the Maslov index iMaslov (γ) of γ (with respect to P ). Obviously, iMaslov (γ) does not depend on the choice of ε; moreover, it can be proven that it is also independent on the choice of the parallel trivialization of TM along γ. If all the P -focal points along γ are nondegenerate, then iMaslov (γ) is equal to the focal index of γ; observe also that if g is Riemannian, i.e., positive definite, or if g is Lorentzian, i.e., n− (g) = 1, and γ is causal, i.e., g(γ, ˙ γ) ˙ 6 0, then the P -focal points are always nondegenerate, and the focal index of γ equals the sum of the multiplicities of the P -focal points along γ. Due to its topological nature, the Maslov index is stable by C0 -small perturbations of the data. If g is Riemannian, then n− (I) is finite, and the classical Morse index theorem states that such index is equal to the geometric index of γ, which is the sum of the multiplicities of the P -focal points along γ, and a convexity term corresponding to the final manifold Q. If g is not Riemannian, then n− (I) is always infinite, due to the fact that there exists infinitely many linearly independent variations of γ along which the value of the action functional decreases. The notion of Maslov index is the correct generalization to the case of non-positive definite metrics of the notion of geometric index in Riemannian geometry. The pseudo-Riemannian version of the Morse index theorem we aim at will give the equality between the Maslov index and the difference n− (I K ) − n+ (I S ), with K and S suitable subspaces of H, plus two additional terms given by the contributions of the endmanifolds P and Q. The spaces K and S are determined by a choice of a maximal negative distribution D along γ, which is a smooth distribution {Dt }t∈[a,b] of k-dimensional (k = n− (g)) subspaces Dt ⊂ Tγ(t) M such that g is
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negative definite on Dt for all t ∈ [a, b]. A frame for D is a set {Y1 , . . . , Yk } of smooth vector fields along γ that form a pointwise basis for D. Suppose that a maximal negative distribution D along γ is given; we define the following closed subspaces of H: KD = v ∈ H : g(v 0 , Yi ) is of Sobolev regularity H1 , and ˙ v) γ, ˙ Yi , i = 1, . . . , k g(v 0 , Yi )0 = g(v 0 , Yi0 ) + g R(γ, SD = v ∈ H : v(a) = 0, v(b) = 0, v(t) ∈ Dt , ∀ t ∈ [a, b] , where {Y1 , . . . , Yk } is a frame for D. The space KD does not indeed depend on the choice of the frame. The space KD can be thought of as the space of Jacobi fields along D, namely, if v ∈ H is of class C2 , ˙ v) γ˙ is orthogonal to D. It is easy to prove that KD and SD are then v ∈ KD if and only if v 00 − R(γ, I-orthogonal. In order to describe the contribution of the final manifold Q, we now introduce the symmetric bilinear form S on Tγ(b) M . Assume that γ(b) is not P -focal so that every vector in Tγ(b) M can be written uniquely in the form J(b), with J ∈ J . Then, we set: S J1 (b), J2 (b) = −g J10 (b), J2 (b) . We are ready to state the pseudo-Riemannian Morse index theorem: T HEOREM 1. – Suppose that γ(b) is not P -focal. Then Q − S|Tγ(b) Q , iMaslov (γ) = n− I|KD − n+ I|SD − n− g|Tγ(a) P − n− Sγ(b) ˙
(1)
where all the terms in the above equality are finite integer numbers. For a better understanding of the statement of Theorem 1 we take a closer look at some particular cases: – if (M, g) is Riemannian, then D = 0, KD = H, SD = {0}, n− (g|Tγ(a) P ) = 0, and Theorem 1 becomes the standard Morse index theorem for Riemannian geodesics with both endpoints variable (see [7]). – If (M, g) is Lorentzian and if γ admits a Jacobi field Y with g(Y, Y ) < 0, then one can take D to be the unidimensional distribution spanned by Y . In this case, it can be seen that n+ (I|SD ) = 0 and KD is the space of those v ∈ H with g(v 0 , Y ) − g(v, Y 0 ) constant. In particular, if (M, g) is stationary, i.e., there exists a Killing vector field Y with g(Y, Y ) < 0, then its restriction along any geodesic is Jacobi. ˙ Y ) constant. The corresponding space KD is the space of variations of γ by curves z with g(z, – If (M, g) is Lorentzian and g(γ, ˙ γ) ˙ < 0, then one can take Y = γ˙ in the above construction. In this ˙ Moreover, the term case, the space KD is given by those v ∈ H that are pointwise orthogonal to γ. n− (g|Tγ(a) P ) is zero. In this case, Theorem 1 becomes the timelike Morse index theorem of Beem and Ehrlich (see [2] and the references therein for a complete bibliography of the known index theorems for causal Lorentzian geodesics; see also [7]). Observe that the last term in (1) is the usual term corresponding to the contribution of the final manifold Q in the Morse index theorem for geodesics with variable endpoints in Riemannian geometry; on the other hand the term n− (g|Tγ(a) P ) is a totally new feature of pseudo-Riemannian geometry. In particular, the presence of this term in formula (1) implies that the number n− (I|KD ) may be non zero even for arbitrarily small initial segments of γ. We remark that both terms n− (I|KD ) and n+ (I|SD ) depend on the choice of the distribution D; to conclude we give a method for computing the term n+ (I|SD ). Consider the following second order linear differential system in Rk : k X 0 ˙ Yj ) γ, ˙ Yi fj = 0, g(Yi , Yj ) fj0 + g(Yj0 , Yi ) − g(Yj , Yi0 ) fj0 + g Yj00 − R(γ, j=1
388
i = 1, . . . , k.
(2)
The semi-Riemannian Morse index theorem
It is easy to see that KD ∩ SD = {0} if and only if every solution f = (f1 , . . . , fk ) : [a, b] → Rk of (2) with f (a) = f (b) = 0 is identically zero; in this case it can be proven that H = KD ⊕ SD . The following formula holds: X dim f : [a, b] → Rk solution of (2) : f (a) = f (t) = 0 , n+ I|SD = t∈ ]a,b[
where all but a finite number of the terms in the above sum are zero. The theory presented in this paper and Theorem 1 can be generalized to the case of linearized Hamiltonian systems on symplectic manifolds. Finally, the authors observe that a version of the Morse index theorem for pseudo-Riemannian geodesics based on the spectral theory of the Jacobi operator has already been stated in [5]. The formulation of [5], Theorem 7.1, is somewhat similar to that of Theorem 1. However, the following two important facts must be taken into consideration. In first place, the spaces K and S appearing in the statement of Theorem 1 are explicitly given by a geometrical construction. Such construction is suitable for the development of a global Morse theory for the pseudo-Riemannian action functional (see [4] for the case of stationary Lorentzian manifolds). The second point is that, as presented by the author, the proof of [5], Theorem 7.1, is not satisfactory. References [1] Arnol’d V.I., Characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967) 1–13. [2] Beem J.K., Ehrlich P.E., Easley K.L., Global Lorentzian Geometry, 2nd Edition, Marcel Dekker, Inc., New York– Basel, 1996. [3] Duistermaat J.J., On the Morse index in variational calculus, Adv. in Math. 21 (1976) 173–195. [4] Giannoni F., Masiello A., Piccione P., Tausk D., A generalized index theorem for Morse–Sturm systems and applications to semi-Riemannian geometry, Asian J. Math. (1999) (to appear) (LANL math.DG/9908056). [5] Helfer A.D., Conjugate points on spacelike geodesics or pseudo-self-adjoint Morse–Sturm–Liouville systems, Pacific J. Math. 164 (2) (1994) 321–340. [6] Mercuri F., Piccione P., Tausk D., Stability of the focal and the geometric index in semi-Riemannian geometry via the Maslov index, Technical Report RT-MAT 99-08, Mathematics Department, University of São Paulo, Brazil, 1999 (LANL math.DG/9905096). [7] Piccione P., Tausk D., A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry, J. Math. Phys. 40 (12) (1999) 6682–6688.
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The Maslov index and a generalized Morse index theorem for non-positive definite metrics Paolo PICCIONE, Daniel V. TAUSK Departamento de Matemática, Universidade de São Paulo, Brazil E-mail: [email protected], [email protected] (Reçu le 16 avril 2000, accepté le 19 juin 2000)
Abstract.
We present an extension of the celebrated Morse index theorem in Riemannian geometry to the case of geodesics in pseudo-Riemannian manifolds. It is considered the case that both endpoints are variable. The notion of Maslov index for pseudo-Riemannian geodesics replaces the notion of geometric index for Riemannian geodesics. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
L’index de Maslov et le théorème d’indice de Morse généralisé pour une métrique non définie positive Résumé.
On étend ici le théorème d’indice de Morse en géométrie riemannienne au cas où la métrique n’est pas définie positive. On considère le cas des géodésiques avec les deux extrémités mobiles. La notion d’indice de Maslov pour les géodésiques pseudoriemanniennes remplace celle d’indice géométrique pour les géodésiques riemanniennes. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soient (M, g) une variété pseudo-riemannienne, γ : [a, b] → M une géodésique et P ⊂ M une sousvariété non dégénérée avec γ(a) ∈ P et γ(a) ˙ ∈ TP ⊥ . L’indice de Maslov iMaslov (γ) de γ relatif a P est un nombre entier que l’on définit en utilisant le premier groupe d’homologie relatif de la Grassmannienne de tous les sous-espace lagrangiens d’un espace symplectique. Génériquement, l’indice de Maslov de γ est une somme algébrique des points P -focaux le long de γ. Dans le cas où (M, g) est riemannienne, ou bien (M, g) est lorentzienne et γ est causal (g(γ, ˙ γ) ˙ 6 0), l’indice de Maslov devient l’indice géométrique donné par la somme des multiplicités des points P -focaux le long de γ. Nous considérons maintenant une autre sous-variété Q ⊂ M avec γ(b) ∈ Q et γ(b) ˙ ∈ TQ⊥ . Nous R b ˙ z) ˙ dt, définie définissons la forme d’indice I comme la variation seconde de l’énergie f (z) = 12 a g(z, sur l’espace de courbes reliant P et Q. Alors I est une forme bilinéaire symétrique bornée dans l’espace de Hilbert H des champs de vecteurs v le long de γ de régularité H1 tels que v(a) ∈ Tγ(a) P et v(b) ∈ Tγ(b) Q. Dans le cas riemannien, I a un indice fini ; en revanche, dans les autres cas, I a toujours un indice infini. Note présentée par Charles-Michel M ARLE. S0764-4442(00)01630-X/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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P. Piccione, D.V. Tausk
Le choix d’une distribution D de sous-espaces maximaux négatifs relatifs à g le long de γ nous permet de définir deux sous-espaces fermés KD et SD de H avec les propriétés suivantes : (a) KD et SD sont I-orthogonaux ; (b) l’indice de I|KD et le co-indice de I|SD sont finis. La version pseudo-riemannienne du théorème de l’indice de Morse pour les géodésiques avec extrémités mobiles, est donnée par l’égalité suivante, dans le cas où γ(b) n’est pas P -focal : Q − S|Tγ(b) Q , iMaslov (γ) = n− I|KD − n+ I|SD − n− g|Tγ(a) P − n− Sγ(b) ˙ Q est la deuxième où n− et n+ désignent l’indice et le co-indice d’une forme bilinéaire symétrique, Sγ(b) ˙ forme fondamentale de Q, évaluée dans la direction normale γ(b) ˙ et S est une forme bilinéaire symétrique définie en Tγ(b) M associée à la géodésique γ.
We consider a pseudo-Riemannian manifold (M, g), i.e., M is a smooth n-dimensional manifold and g is a smooth nondegenerate symmetric (2, 0)-tensor on M . We denote by ∇ the Levi-Civita covariant derivative for g and by R its curvature tensor, chosen with the sign convention R(X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] . Given a smooth submanifold N of M , a point p ∈ N and a normal vector n ∈ Tp N ⊥ , we denote by SnN the second fundamental form of N at p in the direction n, which is a symmetric bilinear form in Tp N given by SnN (v, w) = g(∇v W, n), where W is any smooth vector field tangent to N such that W (p) = w. Let γ : [a, b] → M be a geodesic and let P and Q be smooth submanifolds of M such that γ(a) ∈ P , ˙ ∈ Tγ(b) Q⊥ . Then, γ is a critical point of the action functional γ(a) ˙ ∈ Tγ(a) P ⊥ , γ(b) ∈ Q and γ(b) R 1 b ˙ z) ˙ in the space of curves in M connecting P and Q. Denoting by v 0 the covariant f (z) = 2 a g(z, derivative of a vector field v along γ, the second variation of f at γ is given by the index form: Z I(v, w) =
b
˙ v) γ, ˙ w g(v 0 , w0 ) + g R(γ,
a
Q P dt + Sγ(b) v(b), w(b) − Sγ(a) v(a), w(a) , ˙ ˙
which is a bounded symmetric bilinear form on the Hilbert space H of vector fields v along γ, of H1 -Sobolev regularity, with v(a) ∈ Tγ(a) P and v(b) ∈ Tγ(b) Q. We will assume henceforth that P is nondegenerate at γ(a), i.e., g is nondegenerate on Tγ(a) P ; then, P can be seen as a g-symmetric linear endomorphism of Tγ(a) P . the bilinear form Sγ(a) ˙ ˙ J) γ˙ = 0; a P -Jacobi field A Jacobi field J along γ is a solution of the differential equation J 00 − R(γ, is a Jacobi field J satisfying: J(a) ∈ Tγ(a) P
P and J 0 (a) + Sγ(a) J(a) ∈ Tγ(a) P ⊥ . ˙
We denote by J the n-dimensional space of P -Jacobi fields along γ; for t ∈ ]a, b] we set: J [t] = J(t) : J ∈ J . A point γ(t) with t ∈ ]a, b] is said to be P -focal along γ if there exists a non zero J ∈ J with J(t) = 0. The multiplicity of γ(t) is the dimension of the space of such J ’s. Equivalently, γ(t) is P -focal when J [t] 6= Tγ(t) M , and the multiplicity of γ(t) is the codimension of J [t]. Given a symmetric bilinear form B on a real vector space V , the negative (resp. positive) type number n− (B) (resp. n+ (B)) of B is the supremum of the dimensions of subspaces of V on which B is negative
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(resp. positive) definite. The negative (resp. positive) type number is also called the index (resp. the coindex). The signature sgn(B) of B is defined as the difference n+ (B) − n− (B), provided that at least one of the two type numbers is finite. Given a P -focal point γ(t), its signature is defined to be the signature of the restriction of g to J [t]⊥ ; if such restriction is nondegenerate, then γ(t) is said to be a nondegenerate P -focal point. P -focal points may accumulate; if γ has only a finite number of P -focal points, then we define the focal index of γ as the sum of the signatures of its P -focal points. It is known that nondegenerate P -focal points are isolated and that there are no P -focal points γ(t) for t sufficiently close to a; moreover, if (M, g) is real-analytic, then the set of P -focal points is finite. Let us recall the definition of Maslov index (see [1,3] and [6]); there are different notions of Maslov index defined in the literature in different contexts and we will use the same notion of [3], Section 2. Let us consider a parallel trivialization of TM along γ; for each t ∈ [a, b] we have an isomorphism φt : Tγ(t) M → Rn . Denote by g0 the nondegenerate symmetric bilinear form in Rn corresponding to g by φt (observe that g0 does not depend on t since g is parallel). We set: `(t) =
φt J(t) , φt J 0 (t) : J ∈ J ⊂ Rn ⊕ Rn ,
t ∈ [a, b];
then `(t) is a Lagrangian subspace of the symplectic space Rn ⊕ Rn endowed with the symplectic form: ω (x1 , y1 ), (x2 , y2 ) = g0 (x1 , y2 ) − g0 (x2 , y1 ). Thus, ` is a smooth curve in the Grassmannian Λ of all Lagrangian subspaces of (Rn ⊕ Rn , ω); Λ is a real-analytic connected and compact 12 n(n + 1)-dimensional manifold. Set L0 = {0} ⊕ Rn ∈ Λ; it is easy to see that γ(t) is P -focal if and only if `(t) is not transversal to L0 , i.e., `(t) ∩ L0 6= {0}. If we denote by Λ0 (L0 ) the open subset of Λ consisting of those Lagrangians that are transversal to L0 , then the first singular relative homology group H1 (Λ, Λ0 (L0 )) with integer coefficients is isomorphic to Z; the symplectic form ω defines canonically a choice for this isomorphism. Suppose that γ(b) is not P -focal; there exists ε > 0 such that γ(t) is not P -focal for t ∈ ]a, a + ε]; hence, the restriction `|[a+ε,b] is a curve in Λ with endpoints in Λ0 (L0 ) and therefore it defines a homology class in H1 (Λ, Λ0 (L0 )). The corresponding integer number is called the Maslov index iMaslov (γ) of γ (with respect to P ). Obviously, iMaslov (γ) does not depend on the choice of ε; moreover, it can be proven that it is also independent on the choice of the parallel trivialization of TM along γ. If all the P -focal points along γ are nondegenerate, then iMaslov (γ) is equal to the focal index of γ; observe also that if g is Riemannian, i.e., positive definite, or if g is Lorentzian, i.e., n− (g) = 1, and γ is causal, i.e., g(γ, ˙ γ) ˙ 6 0, then the P -focal points are always nondegenerate, and the focal index of γ equals the sum of the multiplicities of the P -focal points along γ. Due to its topological nature, the Maslov index is stable by C0 -small perturbations of the data. If g is Riemannian, then n− (I) is finite, and the classical Morse index theorem states that such index is equal to the geometric index of γ, which is the sum of the multiplicities of the P -focal points along γ, and a convexity term corresponding to the final manifold Q. If g is not Riemannian, then n− (I) is always infinite, due to the fact that there exists infinitely many linearly independent variations of γ along which the value of the action functional decreases. The notion of Maslov index is the correct generalization to the case of non-positive definite metrics of the notion of geometric index in Riemannian geometry. The pseudo-Riemannian version of the Morse index theorem we aim at will give the equality between the Maslov index and the difference n− (I K ) − n+ (I S ), with K and S suitable subspaces of H, plus two additional terms given by the contributions of the endmanifolds P and Q. The spaces K and S are determined by a choice of a maximal negative distribution D along γ, which is a smooth distribution {Dt }t∈[a,b] of k-dimensional (k = n− (g)) subspaces Dt ⊂ Tγ(t) M such that g is
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negative definite on Dt for all t ∈ [a, b]. A frame for D is a set {Y1 , . . . , Yk } of smooth vector fields along γ that form a pointwise basis for D. Suppose that a maximal negative distribution D along γ is given; we define the following closed subspaces of H: KD = v ∈ H : g(v 0 , Yi ) is of Sobolev regularity H1 , and ˙ v) γ, ˙ Yi , i = 1, . . . , k g(v 0 , Yi )0 = g(v 0 , Yi0 ) + g R(γ, SD = v ∈ H : v(a) = 0, v(b) = 0, v(t) ∈ Dt , ∀ t ∈ [a, b] , where {Y1 , . . . , Yk } is a frame for D. The space KD does not indeed depend on the choice of the frame. The space KD can be thought of as the space of Jacobi fields along D, namely, if v ∈ H is of class C2 , ˙ v) γ˙ is orthogonal to D. It is easy to prove that KD and SD are then v ∈ KD if and only if v 00 − R(γ, I-orthogonal. In order to describe the contribution of the final manifold Q, we now introduce the symmetric bilinear form S on Tγ(b) M . Assume that γ(b) is not P -focal so that every vector in Tγ(b) M can be written uniquely in the form J(b), with J ∈ J . Then, we set: S J1 (b), J2 (b) = −g J10 (b), J2 (b) . We are ready to state the pseudo-Riemannian Morse index theorem: T HEOREM 1. – Suppose that γ(b) is not P -focal. Then Q − S|Tγ(b) Q , iMaslov (γ) = n− I|KD − n+ I|SD − n− g|Tγ(a) P − n− Sγ(b) ˙
(1)
where all the terms in the above equality are finite integer numbers. For a better understanding of the statement of Theorem 1 we take a closer look at some particular cases: – if (M, g) is Riemannian, then D = 0, KD = H, SD = {0}, n− (g|Tγ(a) P ) = 0, and Theorem 1 becomes the standard Morse index theorem for Riemannian geodesics with both endpoints variable (see [7]). – If (M, g) is Lorentzian and if γ admits a Jacobi field Y with g(Y, Y ) < 0, then one can take D to be the unidimensional distribution spanned by Y . In this case, it can be seen that n+ (I|SD ) = 0 and KD is the space of those v ∈ H with g(v 0 , Y ) − g(v, Y 0 ) constant. In particular, if (M, g) is stationary, i.e., there exists a Killing vector field Y with g(Y, Y ) < 0, then its restriction along any geodesic is Jacobi. ˙ Y ) constant. The corresponding space KD is the space of variations of γ by curves z with g(z, – If (M, g) is Lorentzian and g(γ, ˙ γ) ˙ < 0, then one can take Y = γ˙ in the above construction. In this ˙ Moreover, the term case, the space KD is given by those v ∈ H that are pointwise orthogonal to γ. n− (g|Tγ(a) P ) is zero. In this case, Theorem 1 becomes the timelike Morse index theorem of Beem and Ehrlich (see [2] and the references therein for a complete bibliography of the known index theorems for causal Lorentzian geodesics; see also [7]). Observe that the last term in (1) is the usual term corresponding to the contribution of the final manifold Q in the Morse index theorem for geodesics with variable endpoints in Riemannian geometry; on the other hand the term n− (g|Tγ(a) P ) is a totally new feature of pseudo-Riemannian geometry. In particular, the presence of this term in formula (1) implies that the number n− (I|KD ) may be non zero even for arbitrarily small initial segments of γ. We remark that both terms n− (I|KD ) and n+ (I|SD ) depend on the choice of the distribution D; to conclude we give a method for computing the term n+ (I|SD ). Consider the following second order linear differential system in Rk : k X 0 ˙ Yj ) γ, ˙ Yi fj = 0, g(Yi , Yj ) fj0 + g(Yj0 , Yi ) − g(Yj , Yi0 ) fj0 + g Yj00 − R(γ, j=1
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i = 1, . . . , k.
(2)
The semi-Riemannian Morse index theorem
It is easy to see that KD ∩ SD = {0} if and only if every solution f = (f1 , . . . , fk ) : [a, b] → Rk of (2) with f (a) = f (b) = 0 is identically zero; in this case it can be proven that H = KD ⊕ SD . The following formula holds: X dim f : [a, b] → Rk solution of (2) : f (a) = f (t) = 0 , n+ I|SD = t∈ ]a,b[
where all but a finite number of the terms in the above sum are zero. The theory presented in this paper and Theorem 1 can be generalized to the case of linearized Hamiltonian systems on symplectic manifolds. Finally, the authors observe that a version of the Morse index theorem for pseudo-Riemannian geodesics based on the spectral theory of the Jacobi operator has already been stated in [5]. The formulation of [5], Theorem 7.1, is somewhat similar to that of Theorem 1. However, the following two important facts must be taken into consideration. In first place, the spaces K and S appearing in the statement of Theorem 1 are explicitly given by a geometrical construction. Such construction is suitable for the development of a global Morse theory for the pseudo-Riemannian action functional (see [4] for the case of stationary Lorentzian manifolds). The second point is that, as presented by the author, the proof of [5], Theorem 7.1, is not satisfactory. References [1] Arnol’d V.I., Characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967) 1–13. [2] Beem J.K., Ehrlich P.E., Easley K.L., Global Lorentzian Geometry, 2nd Edition, Marcel Dekker, Inc., New York– Basel, 1996. [3] Duistermaat J.J., On the Morse index in variational calculus, Adv. in Math. 21 (1976) 173–195. [4] Giannoni F., Masiello A., Piccione P., Tausk D., A generalized index theorem for Morse–Sturm systems and applications to semi-Riemannian geometry, Asian J. Math. (1999) (to appear) (LANL math.DG/9908056). [5] Helfer A.D., Conjugate points on spacelike geodesics or pseudo-self-adjoint Morse–Sturm–Liouville systems, Pacific J. Math. 164 (2) (1994) 321–340. [6] Mercuri F., Piccione P., Tausk D., Stability of the focal and the geometric index in semi-Riemannian geometry via the Maslov index, Technical Report RT-MAT 99-08, Mathematics Department, University of São Paulo, Brazil, 1999 (LANL math.DG/9905096). [7] Piccione P., Tausk D., A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry, J. Math. Phys. 40 (12) (1999) 6682–6688.
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