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Let $M$ be a complete Riemannian manifold and $N\subset M$ a closed submanifold. If codimension of $N$ is $0$ take $q\in\partial N$ and $v\in T_qN$, where $\partial N$ is the boundary of $N$ as a subset of $M$ and if codimension of $N$ is bigger than $0$, take $q\in N$ and $v\in T_qN$. Let $\gamma:(-\epsilon,\epsilon)\rightarrow M$ be a differentuable curve such that $\gamma(0)=q$ and $\gamma'(0)=v$. Let $d$ denote the geodesic distance in $M$.

How can one show that $d(\gamma(t),N)=o(t)$ for small $t$?


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If $M=\mathbb{R}^n$ and $\gamma$ is a line, then this is true. Thats the only thing i know @FlybyNight – Tomás Nov 21 '12 at 21:12
Could you put this question into context please? There are some things that worry me. For example, the usual definition of a closed manifold is a compact manifold without boundary, so $\partial N = \emptyset$. – Fly by Night Nov 21 '12 at 21:31
When i say $\partial N$, it is the boundary of $N$ as a subset of $M$. Im gonna edit it to make it clear. – Tomás Nov 21 '12 at 21:40
This problem appear in trying to solve this problem:… – Tomás Nov 21 '12 at 21:47

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