\begin{definition}[Definition:Isotropy Representation]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\map {\text {Iso}} {M, g}$ be the set of all isometries from $M$ to itself.
Let $\map {\text {Iso}_p} {M, g}$ be the isotropy subgroup at $p \in M$.
That is, let $\map {\text {Iso}_p} {M, g}$ be the subgroup of $\map {\text {Iso}} {M, g}$ consisting of isometries that fix $p \in M$.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
For each $\phi \in \map {\text {Iso}_p} {M, g}$ let $\rd \phi_p$ be a linear mapping such that:
:$\rd \phi_p : T_p M \to T_p M$
Let $\GL {T_p M}$ be the general linear group over $T_p M$.
Let $I_p : \map {\text {Iso}_p} {M, g} \to \GL {T_p M}$ be a mapping such that $\map {I_p} \phi = \rd \phi_p$.
Then the mapping $I_p$ is a representation of $\map {\text {Iso}_p} {M, g}$ and is called the '''isotropy representation'''.
\end{definition}