curvature tensor


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curvature tensor

[′kər·və·chər ‚ten·sər]
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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.