An isometry fixed set is totally geodesic
= An isometry fixed set is totally geodesic
A smooth <nondegenerate> component of the <fixed-point set> of an <isometry> is a <totally geodesic submanifold>. If a <geodesic> starts tangent to the fixed component, its image under the <isometry> has identical position and tangent. Uniqueness of the <geodesic equation> makes the two <geodesics> coincide throughout their common domain, so the original <geodesic> stays in the fixed set.