R, S, Q & r being Christoffel Riemannian curvature tensor, Ricci tensor
, Ricci operator and scalar curvature respectively.
In Einstein's magical formulation of the Theory of General Relativity he started with equating the Riemann's curvature known as Ricci Tensor
denoted by Ruv with the gravity tensor Tuv and added the metric tensor guv which provides measurement of infinitesimal distances along the curved space.
Other models have been proposed for gravity [1-5] by generalizing a gravity model based on scalar and Ricci tensor
Furthermore, we have included Ricci tensor
components [R.sub.ab], Ricci scalar R, and the stress-energy tensor [T.sub.ab] of space-time (26), (30), and (46) in the relevant section.
[R.sup.[alpha][beta]] = [square root of -g] [R.sup.[alpha].sub.[beta]] (Ricci tensor
A Ricci soliton (see ) is a generalization of the Einstein metric (that is, the Ricci tensor
is a constant multiple of the Riemannian metric g) and is defined on a Riemannian manifold (M, g) by
where [[bar.R].sub.ij] is the Ricci tensor
In a partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere, Bulgarian mathematicians show that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor
of the Biquard connection is zero, and this occurs precisely on 3-Sasakian manifolds.
Ryan: Hypersurfaces with parallel Ricci tensor
, Osaka J.
where S is the Ricci tensor
of type (0, 2) and R is the curvature tensor of type (1 , 3).
The Ricci tensor
[R.sub.ab], obtained by contraction of the Riemann curvature tensor, is given by
A Riemannian manifold ([M.sup.n], g) is called an almost pseudo Ricci symmetric manifold if its Ricci tensor
S of type (0, 2) is not identically zero and satisfies the condition