\begin{definition}[Definition:Riemannian Manifold Isotropic at Point]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $I_p$ be the isotropy representation at $p \in M$.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
Let $S_p \subseteq T_p M$ be the set of unit vectors in $T_p M$.
Suppose $I_p$ acts transitively on $S_p$.
Then $M$ is said to be '''isotropic at $p$'''.
\end{definition}