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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?

Edit: JasonDeVito suggested a positive answer by using the first variation formula. An additional question of mine is now:

Is there a proof without using the first the variation formula and Jacobi fields just by using techniques like Riemann normal coordinates and the expobential function?

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Do you allow manifolds with boundary? – Jason DeVito Apr 1 '13 at 0:39
No, without boundary. – xyz Apr 1 '13 at 0:48
Whatever the implications (or lack thereof), I suspect that the missing link is the variational definition of geodesic ( Observe, in particular, that being autoparallel w.r.t. the Levi-Civita connection is precisely satisfying the Euler--Lagrange equations for the Energy functional on curves. – Branimir Ćaćić Apr 1 '13 at 0:56
I don't have time to write up a complete answer, but the thing to google search is "first variation formula". The short answer - yes, they are they same. – Jason DeVito Apr 1 '13 at 1:24
I edited my answer by inserting an additional question regarding your answer. – xyz Apr 1 '13 at 7:17

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