Let
M be an
n-dimensional
submanifold of an
-dimensional Einstein manifold
As in the case of Real space form
one is interested in obtaining conditions under which the submanifold of a Real
space form is a real space form, in this paper we are interested in finding
conditions underwhich the submanifold
M
of the Einstein manifold is an Einstein manifold. For a unit normal vector field
N on the submanifold
M, we
say that the Einstein manifold
has constant mixed sectional
curvature with respect to the submanifold
M if the sectional curvatures of
of the plane sections spanned by
the unit normal vector field
N and a
unit vector field on the submanifold are constant. In this paper we show that a
totally umbilical submanifold
M of an
Einstein manifold
of constant mixed sectional
curvature
c is an Einstein manifold,
(cf. Theorem 3.2).
We
also obtain condition under which a submanifold of an Einstein manifold is an
intrinsic sphere. Finally, we otain a characterization for a submanifold of a
Real space form to have parallel mean curvature vector field. 
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