Welcome to MathOverflow

Question

5

2

Suppose that $N$ is a totally geodesic submanifold of a complete Riemannian manifold $(M,g)$. Is it the case that a geodesic segment that minimizes length in the submanifold $N$ also minimizes length in the ambient manifold $M$?

flag	asked Jan 16 at 22:53 Oliver Jones 227●6

1

The answer is no. Consider a surface of revolution that looks like a cylinder with a spherical cap. The complete geodesic through the origin is a totally geodesic submanifold but it is not distance minimizing. – Igor Belegradek Jan 16 at 23:28

So it's not true in general that the cut locus of a point $p$ w.r.t. $N$ is the intersection of the cut locus of $P$ w.r.t. $M$ intersected with $N$? That is $C_p(N)=C_p(M)\cap N$? – Oliver Jones Jan 17 at 0:13

1

No, there is no such relation between the cut loci. Take $p$ be the the origin in the surface of revolution described above, then $p$ is a pole, so its cut locus is empty, while the cut locus of $N$ is nonempty and complicated. – Igor Belegradek Jan 17 at 1:20

@Igor: I think you mean $M$, not $N$. In your example, $C_p(N)=\phi$ and so $C_p(N)\subseteq C_p(M)\cap N$ trivially. However, this is also too much to hope for in general. For example, a geodesic segment in $N$ joining $p$ to a cut point $q$ may hit a cut point earlier than $q$ when considered as a segment in $M$. – Oliver Jones Jan 17 at 3:25

Carlo Mantegazza · Accepted Answer · 2013-01-17 00:04:10Z UTC

Let $M$ the flat cylinder $R\times S^1\subset R\times C$ and $N=\{(t,e^{it})\,\vert\,t\in R\}$ which is a geodesic (hence a complete totally geodesic submanifold of $M$) minimizing between any two points of $N$ (among the geodesics of $N$). But the minimizing geodesic in $M$ between the points $(0,1)$ and $(2\pi,1)$ is the segment $\{(s,1)\,\vert\,s\in[0,2\pi]\}$.

Deane Yang · Answer 2 · 2013-01-16 23:27:45Z UTC

5

As for an example where $N$ is complete: Slice a 2-sphere just above and below a great circle. Keep the piece containing the great circle. Glue flat disks along the resulting boundaries and smooth the surface near the boundaries.

answered Jan 16 at 23:27

Deane Yang
12.6k●1●33●80

Thanks for the example. – Oliver Jones Jan 17 at 0:16

Pietro Majer · Answer 3 · 2013-01-16 23:09:36Z UTC

2

What about $M$ an Euclidean sphere, and $N$ a great circle minus a point?

answered Jan 16 at 23:09

Pietro Majer
19.7k●1●24●69

I'll add the condition that $N$ is geodesically complete. – Oliver Jones Jan 17 at 0:10

Totally Geodesic Submanifolds

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.