Abstract
Let M be an n-dimensional orientable compact hypersurface in an (n + 1)dimensional real space form (M) over bar (c), n >= 2. If the lengths parallel to R parallel to, parallel to A parallel to and parallel to del alpha parallel to of the curvature tenser field R, the shape operator A, the gradient del alpha of the mean curvature a and the scalar curvature S of the hypersurface M satisfy the inequality
1/2 parallel to R parallel to(2) <= cS + delta parallel to A parallel to(2) - n(n - 1) parallel to del alpha parallel to(2)
where delta = min Ric = (p is an element of M) min(vTpM parallel to v parallel to=1) Ric(p)(v), Ric is Ricci curvature of the hypersurface, then it is shown that M is an extrinsic sphere in (M) over bar (c). In particular we deduce that the condition 1/2 parallel to R parallel to(2) <= delta parallel to A parallel to(2) - n(n - 1) parallel to del alpha parallel to(2) characterizes spheres in the Euclidean space Rn+1 among the compact orientable hypersurfaces whose Ricci curvatures are bounded below by a constant delta > 0.