Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature (Q1423798)

Let \(M^n\) be a compact oriented hypersurface of \(S^{n+1}\), \(n\geq 3\). The main results of the paper read as follows. Theorem 1: If \(M\) has nonnegative Ricci curvature and \(\pi_1(M)\) is infinite, then \(M\) is isometric to a product \(H(r)= S^n(r)\times S^1(\sqrt{1-r^2})\). Theorem 2: If \(M\) has two distinct principal curvatures \(\lambda\) and \(\mu\) of multiplicities \(1\) and \(n-1\), and the square \(S\) of the norm of the second fundamental form of \(M\) satisfies the inequality \(S\geq S(H)\), where \(H\) is the mean curvature and \[ S(H) = n + n^3H^2/(2n-2) + n(n-2)|H|\sqrt{n^2H^2 + 4(n-1)}/(2n-2), \] then \(H\) is constant, \(S = S(H)\) and \(M\) is isometric to an \(H(r)\) with \(r\geq{\sqrt{(n-1)/n}}\).