Totally-geodesic manifolds

Given a pseudo-Riemannian manifold $(M, g)$ , a submanifold $X$ is called totally-geodesic if every geodesic in $X$ (with respect to the induced metric $g_{i n d}$ ) is also a geodesic when regarded in $M$ .
In $X$ we can consider the second fundamental form induced by the ambient, defined as follows

I I (X, Y) = \nabla_{X} Y - {\nabla_{i n d}}_{Y} X

being $\nabla$ and $\nabla_{i n d}$ the corresponding Levi-Civita connection of $g$ and $g_{i n d}$ respectively, and $X, Y \in T_{p} X$ (see relationship parallel transport, covariant derivatives and metrics).

Being a totally-geodesic manifolds is equivalent to $I I = 0$ . See this webpage for more info and references.