Compact hypersurfaces in a unit sphere with infinite fundamental group

Qing-Ming Cheng

Vol. 212 (2003), No. 1, 49–56

DOI: 10.2140/pjm.2003.212.49

Abstract

It is our purpose to study curvature structures of compact hypersurfaces in the unit sphere Sⁿ⁺¹(1). We proved that the Riemannian product S¹() ×Sⁿ⁻¹(c) is the only compact hypersurfaces in Sⁿ⁺¹(1) with infinite fundamental group, which satisfy r ≥ and S ≤ (n − 1) + , where n(n − 1)r is the scalar curvature of hypersurfaces and c² = . In particular, we obtained that the Riemannian product S¹() × Sⁿ⁻¹(c) is the only compact hypersurfaces with infinite fundamental group in Sⁿ⁺¹(1) if the sectional curvatures are nonnegative.

Milestones

Received: 2 August 2002

Published: 1 November 2003