\begin{definition}[Definition:Isometry of Riemannian Manifold]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\phi : M \to M$ be an isometry such that:
:$\phi^* g = g$
Then $\phi$ is called the '''isometry of $\struct {M, g}$'''.
\end{definition}