全测地黎曼叶状结构中的 Hopf-Rinow 定理

Hopf-Rinow Theorem on Totally Geodesic Riemannian Foliations

本文研究全测地黎曼叶状结构中关于广义 Bott 联络的测地线理论, 并将部分 Hopf-Rinow 定理推广到全测地黎曼叶状结构上. 它已被推广到一般的可求长的度量空间和伪厄米流形上. 在我们研究的过程中, 高斯引理的不成立带来了一些困难. 从而我们引入了自然距离 δ, 并得到若 (M, δ) 完 备则测地线完备. 但由于条件的局限性, 另一面不成立.

In this paper, we study the theory of geodesics with respect to the generalized Bot-t connection on totally geodesic Riemannian foliations, and part of the Hopf-Rinow theorem is generalized to totally geodesic Riemannian foliations. It has been gener- alized to length-metric spaces and pseudo-Hermitian manifolds. In the course of our research, the invalidity of Gauss lemma poses some difficulties. Thus we introduce the natural distance δ, and state that if (M, δ) is complete, then the geodesic is complete. However, due to the limitations of the conditions, the other side is not true.