Totally geodesic submanifold

Definition

Let $M$ be a Riemannian manifold and $N$ be a Riemannian submanifold of $M$ (viz, $N$ is a submanifold with the induced Riemannian metric). Then, $N$ is said to be totally geodesic in $M$ if the following equivalent conditions hold:

• Geodesic lines in $N$ are also geodesic in the whole $M$
• For any normal vector to $N$ in $M$ the corresponding second fundamental form vanishes