Let $(M,g)$ be a Riemanian manifold. Let $p\in M$ and $(U,\phi)$ is a local chart centered at $p$ such that any two points in $U$ can join by a distance minimizing curve lying in $U$.
I wonder whether the following property is true?:
"For any points $M$ and $N$ in $U$, the image $\phi(\gamma)$ is a straight line in $\mathbb{R}^{n}$ with also minimizing distance between $\phi(M)$ and $\phi(N)$."
Here $\gamma$ is any distance minimizing curve joining $M$ and $N$ in $U$.
Thanks for any hint.
This is not true. The curve $\phi(\gamma)$ is distance minimizing for the metric $(\phi^{-1})^*g$ on $\phi(U)$ and not for the flat metric coming from $\mathbb R^n$. It would only be true if $\phi$ were an isometry to the flat metric, which would imply that $M$ is flat on $U$.