Polar actions on compact Euclidean hypersurfaces

Abstract

Given an isometric immersion $f:M^n\to {\mathbb{R}}^{n+1}$ of a compact Riemannian manifold of dimension $n\ge 3$ into Euclidean space of dimension $n+1$, we prove that the identity component $Is{o}^0(M^n)$ of the isometry group $Iso(M^n)$ of $M^n$ admits an orthogonal representation $\Phi :Is{o}^0(M^n)\to SO(n+1)$ such that $f\circ g=\Phi (g)\circ f$ for every $g\in Is{o}^0(M^n)$. If $G$ is a closed connected subgroup of $Iso(M^n)$ acting locally polarly on $M^n$, we prove that $\Phi (G)$ acts polarly on ${\mathbb{R}}^{n+1}$, and we obtain that $f(M^n)$ is given as $\Phi (G)(L)$, where $L$ is a hypersurface of a section which is invariant under the Weyl group of the $\Phi (G)$-action. We also find several sufficient conditions for such an $f$ to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension $n\ge 3$ are characterized by their underlying warped product structure.

17 pages