Self-isometry group

Definition

Symbol-free definition

The self-isometry group of a metric space (in particular, a Riemannian manifold) is the group of bijective isometries from the metric space to itself. Here, an isometry is a map that preserves distances between points.

For a Riemannian manifold, this is equivalent to requiring that the map be a diffeomorphism and that further, the induced map at the level of tangent spaces, be an isometry for the metric structure given to each tangent space.

The self-isometry group of a Riemannian manifold is thus a subgroup of its self-diffeomorphism group, which in turn is a subgroup of its self-homeomorphism group.