具有一般边界条件的非退化测地线的通有性
Genericity of nondegenerate geodesics with general boundary conditions
摘要：
设M是可能非紧流形。我们证明，一般在C^k-拓扑（k=2，.，\infty），M上给定指标的半黎曼度量不具有任何满足适当边界条件的退化测地线。这将Biliotti，Javaloyes和Piccione的结果与固定端点的测地线的情况下，端点位于一个紧凑的子流形P的产品MXM，满足一个容许条件。例如，当P横截于M × M的对角线时，这样的条件成立。这些边界条件的进一步方面进行了讨论和一般条件下，没有退化测地线的度量是C^k-通用。
英文摘要：
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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