Non conjugate points on a Riemannian manifold with two connecting minimizing geodesics

Asked Apr 27 '17 at 21:15

Active Apr 27 '17 at 21:15

Viewed 433 times

We know that for a point $q$ in the cut locus of a point $p$ in a complete Riemannian manifold there are two possibilities: the point $q$ is conjugate to $p$ or they are connected by (at least) two minimizing geodesics.

I have been told that there is an example of two conjugate points with a unique minimizing geodesic, and, conversely, of two non conjugate points connected by two minimizing geodesics.

Can someone provide me some hints about the latter? I suspect that something on the torus can help me, but I can't say anything precise.

asked Apr 27 '17 at 21:15

W.Rether

3,180

2

The flat torus (or even just a cylinder) is a good place to look, since the curvature is zero. Think about what $R=0$ implies about Jacobi fields vanishing at the endpoints of a geodesic. – Anthony Carapetis Apr 28 '17 at 03:06
Of course! Since $R=0$ the Jacobi equation implies that $J$ is "affine", so if it vanishes in two points then it is constant $0$. Right? – W.Rether Apr 28 '17 at 07:09
Yup, that's right. – Anthony Carapetis Apr 28 '17 at 07:57
2

https://math.stackexchange.com/questions/1607191/example-for-conjugate-points-with-only-one-connecting-geodesic?rq=1 – Moishe Kohan Apr 28 '17 at 14:37

Non conjugate points on a Riemannian manifold with two connecting minimizing geodesics

0 Answers0