Genericity of nondegenerate geodesics with general boundary conditions
Keywords
Generic properties, semi-Riemannian geodesic flows, nondegenerate geodecics, general endpoints conditionsAbstract
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.Downloads
Published
2010-04-23
How to Cite
1.
BETTIOL, Renato G. and GIAMBÒ, Roberto. Genericity of nondegenerate geodesics with general boundary conditions. Topological Methods in Nonlinear Analysis. Online. 23 April 2010. Vol. 35, no. 2, pp. 339 - 365. [Accessed 4 November 2024].
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