We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see [3]) imply that each non degenerate and non null conjugate (or P -focal) point along a semi-Riemannian geodesic is a bifurcation point. CONTENTS

ABSTRACT. We prove an extension of the Index Theorem for Morse-Sturm systems of the form −V ′′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P. Under generic circumstances, the Maslov index of γ is given by an algebraic count of the P-focal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.

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].

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