A submanifold of a Riemannian manifold is called totally geodesic if every geodesic in is also a geodesic when regarded in .
(fixed loci of smooth proper actions are submanifolds)
Let be a smooth manifold, a Lie group and a proper action by diffeomorphisms.
Then the -fixed locus is a smooth submanifold.
If in addition is equipped with a Riemannian metric and acts by isometries then the submanifold is a totally geodesic submanifold.
(e.g. Ziller 13, theorem 3.5.2)
Created on February 7, 2019 at 10:56:43. See the history of this page for a list of all contributions to it.