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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...

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Asaf Shachar
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  • Might I ask one stupid question, if there are conjugate points $p,~q$ along one geodesic, then can we say that this geomdesic is not minimal on the $p,~q$ segment part ? – DLIN Apr 19 '19 at 13:58
  • No, we can't. Example: $S^2, p=\text{north pole }, q=\text{south pole}$. – Mathy Jan 09 '20 at 19:55

1 Answers1

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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.

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Alex M.
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  • Thanks. However, Hadamard's theorem implies $exp_p$ is a diffeomorphism on $M$, so $M$ can be considered as a normal neighbourhood, so it follows that every gedoesic in this case is globally minimizing. (From the well known fact that geodesic is minimizing as long as it stays in a normal neighbourhood), and (like I wrote in the question) this is not applicable for my case. I am actually interested in Lie groups which are not simply connected. (Think on matrix groups...) – Asaf Shachar Nov 28 '15 at 00:04
  • So I guess I have no choice but to produce an argument which rules out existence of multiple minimizing geodesics. – Asaf Shachar Nov 28 '15 at 00:07