The Morse index theorem in semi-Riemannian geometry

Abstract

We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the Maslov index of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of algebraic count of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a natural splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of symplectic differential systems, that can be obtained as linearizations of the Hamilton equations. The main results proven in this paper were announced in [23].Date: November 2000. 2000 Mathematics Subject Classification. 34B24, 58E05, 58E10, 58F05, 70H20. The first author is partially sponsored by CNPq (Processo n. 301410/95), the second author is sponsored by FAPESP (Processo n. 98/12530-2).Proof. By standard regularity arguments (see [13, Lemma 2.3, Proposition 3.3 and Lemma 4.3]), formula (5.44) shows that I and I # are C 1 in [a, b], which obviously implies thatÎ is C 1 in ]a, b]. Similarly, formula (5.46) shows that F is of class C 1 on [a, b]; from Corollary 5.26 we deduce that F t is surjective for t ∈ ]a, b]. The regularity of the family K # t t∈]a,b] follows then from Lemma 4.2.As to the regularity of the family {K t } t∈ [c,d] , we have to show that q • F t | H is surjective for t ∈ [c, d]. For t = a it follows directly from the definition of F a in (5.47). For t > a, the surjectivity follows from Corollary 5.18 and Corollary 5.28.Corollary 5.36. Suppose that R is a map of class C 1 . If there are no focal instants of the Morse-Sturm system (5.7) and also of the reduced symplectic system (5.25) in the interval [c, d] ⊂ ]a, b], then the index function i is constant on [c, d].

On the distribution of conjugate points along semi-Riemannian geodesies

“…To each symplectic differential system is naturally associated the notion of Maslov index; this formalism is used in [11] to prove a Morse index theorem for non convex Hamiltonian systems and for semi-Riemannian geometry (see also [8,10]). In [11] it is also defined the notions of multiplicity and of signature of a conjugate instant of a symplectic differential system; these notions, as well as that of Maslov index, can be defined directly in the context of abstract symplectic systems.…”

On a formula for the spectral flow and its applications

“…This is a symplectic invariant, which is computed as an intersection number in the Lagrangian Grasmannian of a symplectic vector space. Details on the definition and the computation of the Maslov index for a given geodesic γ, that will be denoted by i Maslov (γ) can be found in [15,16,22].…”

Untitled

“…Using the result on the equality between the Maslov index of a geodesic and the Morse index of the restricted action functional, the authors have obtained in [5] the Morse relations for geodesics of any causal character in a stationary Lorentzian manifold M ; such relations imply for instance that the number of geodesics joining two non conjugate points in M and having a fixed Maslov index k is greater than or equal to the k-th Betti number of the loop space of M . The index theorem of [5] was generalized successively in references 2 [12,15] as follows; given a geodesic γ : [a, b] → M in a semi-Riemannian manifold (M, g) with metric g of arbitrary index, and given a maximal negative distribution D t ⊂ T γ(t) M , t ∈ [a, b], one defines spaces K D and S D of variational vector fields along γ, where S D consists of vector fields along γ taking values in D and K D consists of variational vector fields corresponding to variations of γ by geodesics in the directions of D. Then, the index n − I γ | K D of the restriction of I γ to K D and the the co-index n + I γ | S D = n − −I γ | S D are finite, and their difference equals the Maslov index of γ. In order to use this result to develop an infinite dimensional Morse theory for semi-Riemannian geodesics using the reduction argument mentioned above, one needs to restrict to the case that the distribution is spanned by commuting Killing vector fields (see Subsection 4.2).…”

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We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds (M, g) admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for g. In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the Gödel-type manifolds (see [3]).

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