Using the components of the

curvature tensor, we can easily calculate the non-vanishing components of the Ricci tensor S and its covariant derivatives as follows:

The Riemann-Christoffel or

curvature tensor for the gravitational field is then constructed and the Ricci tensor obtained from it as (2.18)-(2.22).

In search of a general theory, it is natural to ask about a basic inequality (corresponding to the inequality (1.1)) involving the Ricci curvature and the squared mean curvature of any submanifold of any Riemannian manifold without assuming any restriction on the Riemann

curvature tensor of the ambient manifold.

[rho] being the vector field associated to the 1-form A and [~.C] is a concircular

curvature tensor given by (2)

where {[R.sub.ijkl]} is the component of the

curvature tensor of [M.sup.n].

and where [G.sub.[mu][upsilon]] is the Einstein

curvature tensor, [T.sub.[mu][upsilon]] is the energy-momentum density tensor, ds is the Schwarzschild line element, and dt and dr are the time and radius differentials.

He proved that in order that a Riemannian manifold admits a semi-symmetric metric connection whose

curvature tensor vanishes, it is necessary and sufficient that the Riemannian manifold be conformally flat.

Let [[GAMMA].sup.p.sub.ij] be the Christoffel symbols and [R.sup.m.sub.ijk] be the components of the

curvature tensor field produced by the pseudo-Riemannian metric [g.sub.ij].

We also define the covariant components of the

curvature tensor (second fundamental form) of the midsurface as

Recently, in tune with Yano and Sawaki [24], the first two authors [19] have introduced and studied generalized quasi-conformal

curvature tensor W in the context of N(k, [mu])-manifold.

where k is the Einstein constant, [T.sub.ab] is the energy-momentum, and [R.sub.ab] is the Ricci

curvature tensor which represents geometry of the spacetime in presence of energy-momentum.

The basic quantity, a spatial dreibein, parameterizes unit quaternions, that is, group elements of SU(2) with nontrivial winding on its group manifold [S.sub.3] which give rise to a connection, in turn, defining the

curvature tensor. Solving the static field equations, this yields solitons whose topological charge can be matched to electric charge after a reduction of the non-Abelian curvature to the 't Hooft tensor is performed.