Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $M$?

share|improve this question
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 '13 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 '13 at 0:13
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 '13 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 '13 at 3:25

3 Answers 3

up vote 10 down vote accepted

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 $\{(s,1)\\,\vert\\,s\in[0,2\pi]\}$.

share|improve this answer
Nice example; thanks! –  Oliver Jones Jan 17 '13 at 3:26
Hi, Carlo! –  Pietro Majer Jan 17 '13 at 7:29
Ciao Pietro!$ $ –  Carlo Mantegazza Jan 17 '13 at 12:33

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.

share|improve this answer
Thanks for the example. –  Oliver Jones Jan 17 '13 at 0:16

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

share|improve this answer
I'll add the condition that $N$ is geodesically complete. –  Oliver Jones Jan 17 '13 at 0:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.