Genericity of nondegenerate geodesics with general boundary conditions

Abstract

Let $M$ be a possibly noncompact manifold. We prove, generically
in the $C^k$-topology ($2\leq k\leq \infty$), 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 L. Biliotti, M. A. Javaloyes and P. Piccione [< i> Genericity of nondegenerate critical
points and Morse geodesic functionals< /i> ,
Indiana Univ. Math. J. < b> 58< /b> (2009), 1797–1830]
for geodesics
with fixed endpoints to the case where endpoints lie on a compact
submanifold $\mathcal P\subset M\times M$ that satisfies an admissibility
condition. Such condition holds, for example, when $\mathcal P$ is transversal
to the diagonal $\Delta\subset M\times M$. Further aspects of these boundary
conditions are discussed and general conditions under which metrics without
degenerate geodesics are $C^k$-generic are given.