Totally geodesic submanifold
From Diffgeom
Template:Riemannian submanifold property
Definition
Let be a Riemannian manifold and
be a Riemannian submanifold of
(viz,
is a submanifold with the induced Riemannian metric). Then,
is said to be totally geodesic in
if the following equivalent conditions hold:
- Geodesic lines in
are also geodesic in the whole
- For any normal vector to
in
the corresponding second fundamental form vanishes