\begin{definition}[Definition:Isometry Group of Riemannian Manifold]
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 $\map {\text {Iso}} {M, g}$.
\end{definition}