A submanifold $M\subseteq\mathbb{R}^n$ is said to have constant principal curvatures if the eigenvalues of the Weingarten operator along any (local) parallel normal vector field are constant.
My question: let $G\subseteq \operatorname{SO}(n)$ be a compact connected Lie subgroup and $v\in\mathbb{R}^n$ nonzero; does the orbit $Gv$ have constant principal curvatures? How would you show it? If needed, one might add the hypothesis that $G$ acts irreducibly on $\mathbb{R}^n$.