Abstract: Let M
be an orientable hypersurface of an
-dimensional Einstein manifold As for hypersurfaces of a real space
form, one is interested in obtaining conditions under which the hypersurface is
a real space form. In this paper, we are interested in finding conditions under
whichthehypersurfaceMoftheEinsteinmanifoldisanEinsteinmanifold. Let N be the unit normal vector field of the hypersurface M.
We say that the Einstein manifold has constant mixed sectional
curvature with respect to the hypersurface M if the sectional curvatures of of the plane sections containing the
unit normal vector field N are
constant. In this paper, we
show that a compact orientable positively curved hypersurface M of an
-dimensional Einstein manifold of constant mixed sectional
curvature c satisfying the inequality
is an Einstein
manifold, where is the gradient of the mean
curvature a, A
is the shape operator and is the Ricci curvature of the
hypersurface M (cf. Main Theorem).
Keywords and phrases: hypersurfaces of an Einstein manifold, mixed sectional curvature, Ricci curvature, mean curvature.
