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


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? 

