Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion $f\text{\hspace{0.17em}⁣}:M^n\to {\mathbb{R}}^{n+1}$ 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 $\Phi\colon Iso^0(M^n)\to SO(n+1)$suchthat$f\circ g=\Phi(g)\circ f$forevery$g\in Iso^0(M^n)$\mathrm{.}If$G$isaclosedconnectedsubgroupof$Iso(M^n)$ acting locally polarly on $M^n$, we prove that $\mathrm{\Phi }(G)$ acts polarly on ${\mathbb{R}}^{n+1}$, and we obtain that $f(M^n)$ is given as $\mathrm{\Phi }(G)(L)$, where $L$ is a hypersurface of a section which is invariant under the Weyl group of the $\mathrm{\Phi }(G)$-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.