A Note on the Morse Index Theorem for Geodesics between Submanifolds in semi-Riemannian Geometry

Abstract

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorems of Beem, Ehrlich and Kim.

12 pages, LaTeX2e, amsart style. To appear on the Journal of Mathematical Physics

References in corpus (1)

• A Generalized Index Theorem for Morse-Sturm Systems and Applications to semi-Riemannian Geometry

Cited by in corpus (6)

• Maslov index in semi-Riemannian submersions
• An Index Theorem for Non Periodic Solutions of Hamiltonian Systems
• Genericity of nondegenerate geodesics with general boundary conditions
• Pseudo focal points along Lorentzian geodesics and Morse index
• Functions on the sphere with critical points in pairs and orthogonal geodesic chords
• The Morse Index Theorem in semi-Riemannian Geometry