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Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ two distinct points. I have the following conjecture:

Conjecture: If $M$ is a closed manifold, then there exists a path $\gamma:[0,1]\to M$ from $p$ to $q$ such that $L(\gamma)=d(p,q)$, where $L(\gamma)$ is the length of $\gamma$, and $d$ is the Riemannian distance. This statement does not hold for arbitrary Riemannian manifolds without boundary.

I know of no particular way to approach a proof to this statement, or to find a counterexample to prove the second statement.

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    see here: https://en.wikipedia.org/wiki/Hopf%E2%80%93Rinow_theorem – 2017-02-28
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    That proves the first part of the conjecture. How about the counterexample part? – 2017-02-28
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    well, $M=\mathbb R^2-\{0\}$ with the euclidean metric, $p=(1,0)$, $q=(-1,0)$ – 2017-02-28
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    A suggestion: You can learn Riemannian geometry by picking up a textbook (there are many, my favorite is do Carmo's "Riemannian Geometry") and reading. – 2017-03-01

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