### 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.

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.

#### Keywords

Generic properties; semi-Riemannian geodesic flows; nondegenerate geodecics; general endpoints conditions

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