A closed hypersurface with constant scalar and mean curvatures in $\mathbb{S}^4$ is isoparametric

Abstract

We call KABCD, its contractions KAC = EB K ABCB and K = Y<A,B K ABAB, respectively, the curvature tensor, the Ricci curvature tensor and the scalar curvature of A/" 4 , respectively.When AT 4 is the unit sphere § 4 , it turns out that KABCD = ^AG^BD-^AD^BC-Next, we restrict all tensors to M 3 .

A new characterization of the Clifford torus via scalar curvature pinching

“…Therefore, M is the Clifford torus In particular, if H = 0, the problem above is still open. In the case where M is a closed hypersurface with constant mean curvature and constant scalar curvature in S 4 , Conjecture A was solved by Almeida-Brito [1] and Chang [5]. It is well known that the possible values of the squared length of the second fundamental forms of all closed isoparametric hypersurfaces with constant mean curvature H in the unit sphere form a discrete set I(⊂ R).…”

Properties of Biharmonic Submanifolds in Spheres

“…The proof uses the fact that a proper biharmonic hypersurface in S 4 has constant mean curvature, and thus constant scalar curvature, and a result in [11].…”

Closed hypersurfaces of S4(1) with constant mean curvature and zero Gauß–Kronecker curvature

“…Defining the functions µ i := λ i − H , we have that i µ i = 0. The following classical formulas for the Laplacians of S and f 3 are well known and can be found in many papers such as [5][6][7][8][9]:…”