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...)


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...

  • $\begingroup$ 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 ? $\endgroup$ – DLIN Apr 19 '19 at 13:58
  • $\begingroup$ No, we can't. Example: $S^2, p=\text{north pole }, q=\text{south pole}$. $\endgroup$ – Mathy Jan 9 at 19:55

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.

  • $\begingroup$ 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...) $\endgroup$ – Asaf Shachar Nov 28 '15 at 0:04
  • $\begingroup$ So I guess I have no choice but to produce an argument which rules out existence of multiple minimizing geodesics. $\endgroup$ – Asaf Shachar Nov 28 '15 at 0:07

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