Polar actions on compact Euclidean hypersurfaces

Abstract

Given an isometric immersion f : M n → 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 I so 0 (M n ) of the isometry group I so(M n ) of M n admits an orthogonal representation :If G is a closed connected subgroup of I so(M n ) acting polarly on M n , we prove that (G) acts polarly on R n+1 , and we obtain that f (M n ) is given as (G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the (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.

Euclidean hypersurfaces with a totally geodesic foliation of codimension one

“…The case in which M n is assumed to be compact in Theorem 4 was already considered in [7]. Isometric immersions f :…”

“…The case in which M n is assumed to be compact in Theorem 4 was already considered in [7]. Isometric immersions f : M n → R n+1 , n ≥ 3, of a warped product connected Riemannian manifold free of flat points M n = N n−k × ρ L k , with k ≥ 2, were shown in [3] to be, even locally, either rotation hypersurfaces, products with R k of hypersurfaces of R n−k+1 or products with R k−1 of cones over hypersurfaces of S n−k+1 ⊂ R n−k+2 .…”