Geodesics and Distance
We are now ready to introduce the important concepts of parallel transport and geodesics. This will help us to define and understand Riemannian manifolds as metric spaces. One is led to two types of completeness. The first is of standard metric completeness, and the other is what we call geodesic completeness, namely, when all geodesics exist for all time. We shall prove the Hopf-Rinow Theorem, which asserts that these types of completeness for a Riemannian manifold are equivalent. Using the metric structure we can define metric distance functions. We shall study when these distance functions are smooth and therefore show the existence of the kind of distance functions we worked with earlier. In the last section we give some metric characterizations of Riemannian isometries and submersions.
KeywordsRiemannian Manifold Distance Function Integral Curve Riemannian Submersion Coordinate Chart
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