I am interested in complete Riemannian manifolds whose geodesics minimize length globally. Such manifolds must be non-compact (otherwise there is always a self-intersecting geodesic)
However, I suspect this property is much more restrictive.
Question: Assume $(M,g)$ is complete and has this property.
Must $M$ be simply connected? The exponential map has to be a diffeomorphism on all $T_pM$? Does the sectional curvature has to be non-positive? Are there unique geodesics between any two points?
Update: From John Ma's answer, it turns out that the $exp_p$ is a diffeomorphism, and in particular there are unique geodesics between any two points.
I would still like to know if anything intelligent can be said about the curvature though.
My guess is that it does not has to be non-positive everywhere, but only 'mostly everywhere' in some sense. (i.e I can imagine a surface with small regions of positive curvature which doesn't violate our condition )
I am looking in general for necessary and sufficient conditions (topological\curvature constraints) for this property to hold.
One sufficient conditions is provided by Hadamard's Theorem.