Let $M$ be a Riemannian $n$-manifold and $x$ be a point on $M$. Let $\varepsilon>0$ and $B(0, \varepsilon)$ be the $\varepsilon$-ball centered at the origin in $\mathbf R^n$. Suppose that there is a diffeomorphism $e:B(0, \varepsilon)\to U$, where $U$ is a an open subset of $M$, with $e(0)=x$. Further assume that the following two conditions are satisfied.
a) For all $y\in B(0, \varepsilon)$, and for all $v\in \mathbf R^n$, we have $|de_y(v)|\geq |v|$.
b) $|de_y(v)|=|v|$ whenever $v$ is along the line joining $0$ and $y$.
Then the following implications hold:
1) The image under $e$ of the line joining the origin and a point $y\in B(0, \varepsilon)$ is a geodesic in $M$.
2) The geodesic segment in (1) is the unique geodesic segment joining $e(0)$ and $e(y)$.
I do not see how $(2)$ holds.
(1) can be easily proved by noting that if $\gamma$ is the unit speed straight line segment in $\mathbf R^n$ joining $0$ and $y$, then $e\circ \gamma$ is a minimizing curve joining $e(0)$ and $e(y)$ in $M$. Since minimizing curves are geodesics, we have $e\circ \gamma$ is a geodesic.
But I am unable to see why the image of $e\circ \gamma$ is the unique geodesic segment in $M$ joining $e(0)$ and $e(y)$.