Genericity of nondegenerate geodesics with general boundary conditions

Details Genericity of nondegenerate geodesics with general boundary conditions Genericity of nondegenerate geodesics with general boundary conditions

2009-10-21

arxiv.org/abs/0910.4175v1

Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 339-365

Mathematics

Differential Geometry

LaTeX2e, 21 pages, no figures

Scientific paper

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of the product MxM that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal of MxM. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C^k-generic are given.

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