Compact hypersurfaces in a locally symmetric manifold

Let M be an n-dimensional compact hypersurface in a locally symmetric manifold Nn+1. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M. Let vertical bar Phi vertical bar(2) be the nonnegative C-2-function on M defined by vertical bar Phi vertical bar(2) = S - nH(2). In this paper, we prove that if M is oriented and has constant mean curvature and vertical bar Phi vertical bar satisfies P-n,P-H,P-delta (vertical bar Phi vertical bar) >= 0, then (1) vertical bar Phi vertical bar(2) = 0, (i) H = 0 and M is totally geodesic in Nn+1, (ii) H 6 not equal 0 and M is totally umbilical in the unit sphere Sn+1 (1); or (2) vertical bar Phi vertical bar(2) = B-H if and only if (i) H = 0 and M is a Clifford torus, (ii) H not equal 0, n >= 3, and M is an H (r)-torus with r(2) < (n-1)/n, (iii) H not equal 0, n = 2, and M is an H (r)-torus with 0 < r < 1, r(2) not equal 1/2. If M has constant normalized scalar curvature R, <(R)over bar> = R - 1 >= 0, (R) over tilde = R - delta and S satisfies phi(n), ((R) over bar, (R) over tilde) (delta), (S) >= 0, then (1) M is totally umbilical in Sn+1 (1); or (2) M is a product S-1 (root 1-r(2)) x Sn-1 (r), r = root n-2/n(R+1), where P-n,P-H,P-delta (x) and phi(n,(R) over bar, (R) over tilde) (delta) (x) are defined by (1.7) and (1.10).