Abstract
A class of hypersurfaces in a real space form satisfying n(2)\\ grad alpha\\(2) less than or equal to \\del A\\(2), where a is the mean curvature and del A is the covariant derivative of the Weingarten map A, includes the hypersurfaces of constant mean curvature as well as the hypersurfaces with parallel second fundamental form. In this paper it is shown that positively curved compact hypersurfaces in this class are totally umbilical which generalizes the results of Lawson [3] and Nomizu-Smyth [4]. An example of a hypersurface of a real space form which is in this class and has nonconstant mean curvature and whose second fundamental form is not parallel is also discussed.