I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian manifold?
Let $M$ be a Riemannian manifold. A submanifold $N$ of $M$ is totally geodesic if every geodesic in $N$ is also a geodesic in $M$. $N$ is flat if its curvature vanishes.
Let me know if there is anything above that you would like further clarification on.