totally geodesic submanifold

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

**(fixed loci of smooth proper actions are submanifolds)**

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a *proper* action by diffeomorphisms.

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

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

(e.g. Ziller 13, theorem 3.5.2)