Let $M$ be a (without boundary and not necessarly complete) Riemannian manifold.

A map $c\colon [a,b]\rightarrow M$ is called **geodesic of type A** iff $c$ is piecewise smooth, parametrized proportional to arclength and for all $t\in[a,b]$ there exists an $\epsilon>0$ such that $L(c|_{[t-\epsilon,t+\epsilon]})=d(c(t-\epsilon),c(t+\epsilon))$. ($L$ is the lengthfunctional and $d$ the distance in $M$.)

A map $c\colon [a,b]\rightarrow M$ is called **geodesic of type B** iff $c$ is smooth and is autoparallel with respect to the levi-civita-connection of $M$.

**Are those two notions always to 100% equivalent? If not, why and under which precondition and/or changes are they?**