Compact hypersurfaces in a unit sphere with infinite fundamental group

Abstract

It is our purpose to study curvature structures of compact hypersurfaces in the unit sphere S n+1 (1). We proved that the Riemannian product S 1 (√1 - c 2 ) × S n-1 (c) is the only compact hypersurfaces in S n+1 (1) with infinite fundamental group, which satisfy r > n-2/n-1 and S < (n-1)n(r-1)+2/n-2 + n-2/n(r-1)+2, where n(n-1)r is the scalar curvature of hypersurfaces and c 2 = n-2/nr. In particular, we obtained that the Riemannian product S 1 (√1 - c2) × S n-1 (c) is the only compact hypersurfaces with infinite fundamental group in S n+1 (1) if the sectional curvatures are nonnegative.