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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 $M$?

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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
up vote 15 down vote accepted

Let $M$ be the flat cylinder $\mathbb{R} \times S^1 \subset \mathbb{R} \times \mathbb{C}$ and $N = \{(t,e^{it}) | t \in \mathbb{R}\}$, 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 $\{(s,1) | s\in[0,2\pi]\}$.

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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.

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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?

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

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