Totally Geodesic Submanifold

A totally geodesic submanifold of a Riemannian manifold is a submanifold whose geodesics, with the induced metric, are also ambient geodesics. Equivalently, its second fundamental form vanishes identically. It is therefore a minimal submanifold, although the converse need not hold. A linear subspace of Euclidean space and an equatorial sphere in a round sphere are examples. The minimal Clifford hypersurfaces are minimal without being totally geodesic submanifolds, which is the distinction in the Solomon-Yau least-volume conjecture.