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 ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso 0(M n ) of the isometry group Iso(M n ) of M n admits an orthogonal representation $$\Phi\colon\,Iso^0(M^n) \to SO(n + 1)$$ such that $$f \circ g = \Phi(g)\circ f$$ for every $$g \in Iso^0(M^n)$$ . If G is a closed connected subgroup of Iso(M n ) acting polarly on M n , we prove that Φ(G) acts polarly on $${\mathbb{R}}^{n+1}$$ , and we obtain that f(M n ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(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 ≥ 3 are characterized by their underlying warped product structure.