Consider a Riemannian manifold $M$ and two points $x,y \in U \subset M$ and a local diffeomorphic chart $\phi:U \rightarrow \mathbb{R}^n$.

Let $l_{uv}:[0,a] \rightarrow \mathbb{R}^n$ be the straight line segment that connects $u$ and $v$ in $\mathbb{R}^n$.

Hence $\gamma := \phi^{-1}(l_{\phi(x) \phi(y)})$ is a curve on $M$ that connects $x$ and $y$

Now consider $\psi:[0,T] \rightarrow \mathbb{R}^n$ the shortest path on $M$ that connects $x$ and $y$.

My questions is: What a is upper bound for the maximum distance between $\gamma$ and $\psi$?

Thanks in advance.


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