\begin{definition}[Definition:Isometric Group Action]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $G$ be a group.
Suppose $\forall \phi \in G$ the mapping $x \mapsto \phi \cdot x$ is an isometry.
Then the group action is said to be an '''isometric action'''
{{Research|group action on manifolds}}
\end{definition}