Motivation: I am trying to prove a certian geodesic is minimizing. The only generic tool I know for doing that is the fact the a gedoesic is minimizing as long as it stays in a normal neighbourhood of it's starting point. (But this is too crude for my needs…)
Question:
It's well known that geodesics do not minimize length past conjugate points.
Are there any conditions which imply the converse holds?
,i.e certain Riemannian manifolds where a geodesic with no conjugate points is always length minimizing?
I am interested specifically in non-compact matrix groups with left invariant metrics (In particular complete manifolds).
To clarify, the manifolds I am interested in are not simply connected, and in particular it's not true that every geodesic is globally minimizing ($exp_p$ is not a diffeomorphism from $T_pM$ onto $M$)
The geodesics I am interested in do not intersect themselves and are not dense.
A counter example can be given by the cylinder $\mathbb{S}^1 \times \mathbb{R}$ (considered as a Riemannian submanifold of $\mathbb{R}^3$):
There are no conjugate points along any geodesic, but of course there are closed geodesics which stop minimizing at some point…
Best Answer
Let $p,q \in M$ and let $pq$ be a geodesic connecting them. $pq$ will be minimizing if and only if its intersection with the cut locus of $p$ is empty. The cut locus of $p$ is made of: (1) the conjugate points of $p$ and (2) those points $r$ such that there are multiple minimizing geodesics from $p$ to $r$. Since, by assumption, $pq$ contains no point conjugate to $p$, if we manage to also eliminate the points of type (2) from it we are done. A sufficient condition that guarantees the absence of those points (globally from $M$, not just from $pq$) is the Cartan-Hadamard theorem, that has the following consequence: if $M$ is connected, simply-connected, complete, with sectional curvature $\le 0$, then each two points are connected by a unique minimizing geodesic (note that this smells of the Hopf-Rinow theorem, but this one does not guarantee uniqueness, it only guarantees existence).
To conclude: if $M$ is connected, simply-connected, complete, with sectional curvature $\le 0$, and $pq$ contains no conjugate point of $p$ (or $q$), then it is minimizing.