Totally-geodesic manifold

totally-geodesic submanifold

A submanifold $M^n$ of a Riemannian space $V^N$ such that the geodesic lines (cf. Geodesic line) of $M^n$ are also geodesic lines in $V^N$. A totally-geodesic submanifold is characterized by the fact that for every normal vector of $M^n$ the corresponding second fundamental form vanishes; this is equivalent to the vanishing of all normal curvatures of $M^n$.

Comments

The existence of totally-geodesic submanifolds in a general Riemannian manifold is exceptional. Conversely, the existence of many such totally-geodesic submanifolds is used in various recent work to characterize some special manifolds, e.g. symmetric spaces. See [a1].

References

[a1]	W. Ballmann, M. Gromov, V. Schroeder, "Manifolds of non-positive curvature" , Birkhäuser (1985)
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)