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 $M$?

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


As for an example where $N$ is complete: Slice a 2sphere 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. 


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

