Cohomogeneity one hypersurfaces of Euclidean Spaces

Abstract

Abstract. We study isometric immersions f : M n −→ R n+1 into Euclidean space of dimension n + 1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either n ≥ 3 and M n is compact or if n ≥ 5 and the connected components of the flat part of M n are bounded. We also provide several sufficient conditions for f to be a hypersurface of revolution. Mathematics Subject Classification (2000). 53A07, 53C42.

Polar actions on compact Euclidean hypersurfaces

“…Corollary 4 generalizes similar results in [15], [1], and [11] for compact Euclidean G-hypersurfaces of cohomogeneity one. In part (v), weakening the assumption to non-negativity of the sectional curvatures of some principal G-orbit implies f to be a multirotational hypersurface in the sense of [7]:…”

“…Remark 20 Theorems 1 and 3 yield as a special case the main theorem in [11]: any compact hypersurface of R n+1 with cohomogeneity one under the action of a closed connected subgroup of its isometry group is given as G(γ ), where G ⊂ SO(n + 1) acts on R n+1 with cohomogeneity two (hence polarly) and γ is a smooth curve in a (two-dimensional) section which is invariant under the Weyl group W of the G-action. We take the opportunity to point out that starting with a smooth curve β in a Weyl chamber σ of (which is identified with the orbit space of the G-action) which is orthogonal to the boundary ∂σ of σ is not enough to ensure smoothness of γ = W (β), or equivalently, of G(β), as claimed in [11].…”

“…We take the opportunity to point out that starting with a smooth curve β in a Weyl chamber σ of (which is identified with the orbit space of the G-action) which is orthogonal to the boundary ∂σ of σ is not enough to ensure smoothness of γ = W (β), or equivalently, of G(β), as claimed in [11]. One should require, in addition, that after expressing β locally as a graph z = f (x) over its tangent line at a point of intersection with ∂σ , the function f be even, that is, all of its derivatives of odd order vanish and not only the first one.…”

“…Proposition 8 implies, for instance, that if a closed connected subgroup of I so(M n ) acts on M n with cohomogeneity one then either f is a rotation hypersurface over a plane curve or it is free of totally geodesic points, and in particular it is rigid (see [11], Theorem 1). As another consequence we have:…”

“…In fact, the crucial step in the proof of Kobayashi's theorem is to show that the isometry group of the hypersurface can be realized as a closed subgroup of O(n + 1). The idea of the proof of Theorem 1 actually appears already in [11], where Euclidean G-hypersurfaces of cohomogeneity one, i.e., with principal orbits of codimension one, are considered.…”