Definition:Isometry Group of Riemannian Manifold

Definition
Let $\struct {M, g}$ be a Riemannian manifold.

Let $\phi : M \to M$ be an isometry of $\struct {M, g}$.

The set of all $\phi$ is called the isometry group of $\struct {M, g}$ and is denoted by $\map {\text {Iso}} {M, g}$.