I am interested in the concept of geodesic in metric space. In wikipedia I read
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve $γ : I → M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v ≥ 0$ such that for any $t ∈ I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $a$, $b ∈ J$ we have
$ d(\gamma (a),\gamma (b))=v\left|a-b\right|$.
However, I would have thought that a a curve is locally a distance minimizer if for any $t ∈ I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $a$, $b ∈ J$ and any curve $\tilde{\gamma} : I\to M$ we have $$\operatorname{length}_{a,b}\tilde{\gamma}\ge\operatorname{length}_{a,b}\gamma$$ where $$\operatorname{length}_{a,b}(\gamma):=\sup\left\{\sum\limits_{k=0}^{n-1}d(\gamma(t_k),\gamma(t_{k+1})),n\ge 1, a=t_0<t_1<\ldots<t_n=b \right\}.$$
Are these thes same?