nLab
totally geodesic submanifold

Contents

Definition

A submanifold ΣX\Sigma \subset X of a Riemannian manifold XX is called totally geodesic if every geodesic in Σ\Sigma is also a geodesic when regarded in XX.

Examples

Proposition

(fixed loci of smooth proper actions are submanifolds)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

Then the GG-fixed locus X GXX^G \hookrightarrow X is a smooth submanifold.

If in addition XX is equipped with a Riemannian metric and GG acts by isometries then the submanifold X GX^G is a totally geodesic submanifold.

(e.g. Ziller 13, theorem 3.5.2)

References