Definition:Riemannian Manifold Isotropic at Point

Definition
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 $p$.