An Index Theory for Paths that are Solutions of a Class of Strongly Indefinite Variational Problems

2001/08/06 by Paolo Piccione, Piccione, Paolo, Daniel V. Tausk +1
Computer Science · Mathematics · #53C22 #53C50 #58E05 #58E10 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Optimization and Variational Analysis #math.DG #math.FA #msc:53C22 #msc:53C50 #msc:58E05 #msc:58E10

paper · pdf · doi:10.48550/arxiv.math/0108044

LaTeX2e, amsart.cls, 20 pages

arxiv created 2001/08/06 · openalex publication_date 2001/08/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we consider semi-Riemannian manifolds (M,\mathfrak g) admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for \mathfrak. In particular we obtain Morse relations for stationary semi-Riemannian manifolds and for the \em Gödel-type manifolds.

Citations

• An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

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