Compact Hypersurfaces in a Unit Sphere with Infinite Fundamental Group

It is our purpose to study curvature structures of compact hypersurfaces in the unit sphere S(1). We proved that the Riemannian product S( √ 1 − c2) ×Sn−1(c) is the only compact hypersurfaces in S(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 = n−2 nr . In particular, we obtained that the Riemannian product S( √ 1 − c2) × Sn−1(c) is the only compact hypersurfaces with infinite fundamental group in S(1) if the sectional curvatures are nonnegative.