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The Maslov index and a generalized Morse index theorem for non-positive definite metrics. (English. Abridged French version) Zbl 0980.53095

An extension of the Morse index theorem in Riemannian geometry to the case of geodesics in pseudo-Riemannian manifolds \((M, g)\) is presented. The Maslov index of a geodesic with respect to a non-degenerate submanifold \(P\) of \(M\) is defined by using the first singular homology group of the Grassmannian of all Lagrangian subspaces of a symplectic space. The case when both endpoints of the geodesic are variable is considered and the pseudo-Riemannian Morse index theorem is proven supposing that the geodesic is not \(P\)-focal. A new term in the formula for the Maslov index is obtained in addition to the usual one in Riemannian geometry. Some particular cases corresponding to the Riemannian and Lorentzian manifolds are considered for a better understanding of the statement of the index theorem. It is suggested that the theory presented in this paper can be generalized to the case of linearized Hamiltonian systems on symplectic manifolds.

MSC:

53D12 Lagrangian submanifolds; Maslov index
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
57T25 Homology and cohomology of \(H\)-spaces
53C22 Geodesics in global differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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