On closed hypersurfaces of constant scalar curvatures and mean curvatures in Sⁿ⁺¹

Shaoping Chang

Vol. 165 (1994), No. 1, 67–76

DOI: 10.2140/pjm.1994.165.67

Abstract

We consider in this note the following question: given a closed Riemann n-manifold of constant scalar curvature, how can it be minimally immersed in the round (n + 1)-sphere? Our main result states that the immersion has to be isoparametric if the number of its distinct principal curvatures is three identically. This provides another piece of supporting evidence to a conjecture of Chern.

Mathematical Subject Classification 2000

Primary: 53C40

Secondary: 53C42

Milestones

Received: 2 June 1992

Published: 1 September 1994