Genericity of nondegenerate critical points and Morse geodesic functionals

Details Genericity of nondegenerate critical points and Morse geodesic functionals Genericity of nondegenerate critical points and Morse geodesic functionals

2008-04-23

arxiv.org/abs/0804.3724v2

Mathematics

Differential Geometry

23 pages; a new section (now number 5) about genericity in the C^\infty-topology has been added and other minor changes

Scientific paper

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--Smale theorem. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

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