Definition:Isotropy Representation

Definition

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.

2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Symmetries of Riemannian Manifolds