The beauty of such curvature tensor lies in the fact that it has the flavour of
Riemann curvature tensor R if the scalar triple (a, b, c) [equivalent to] (0, 0, 0), Conformal curvature tensor C [13] if (a, b, c) [equivalent to] (- [1/[2n-1]], - [1/[2n-1]], 1), Conharmonic curvature tensor [??] [16] if (a, b, c) [equivalent to] (- [1/[2n-1]], - [1/[2n-1]], 0), Concircular curvature tensor E ([2], p.
In theory, the geometry of space-time, that is, the components of the
Riemann curvature tensor, can be reconstructed by taking measurements of the deviation of two adjacent light paths (geodesics) [3,4].
The parity-violating Chern-Simons term is defined as a contraction of the
Riemann curvature tensor with its dual and the Chern-Simons scalar field [1].
The curvature of the space-time is represented by the Riemann tensor [R.sub.abcd], where [R.sub.ab] = [R.sup.c.sub.acb] is the contraction of
Riemann curvature tensor, R = [R.sup.a.sub.a], the Ricci scalar, [kappa] = 8[pi]G/[c.sup.4] and [LAMBDA], the cosmological constant.
Following, we calculate the
Riemann curvature tensor which is given by
The
Riemann curvature tensor (also known as Riemann-Christoffel tensor) is a (1,3) tensor field whose coordinate components are given in terms of the coordinate components of the connection as follows:
where R denotes the
Riemann curvature tensor field corresponding to the solution g.
Almost Hermitian manifolds with J-invariant
Riemann curvature tensor. Rendiconti del Seminario Mathematico della Universita e Politecnico di Torino, 1975-76, 34, 487-498.
They define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, execute its
Riemann curvature tensor, and show that T(1) is a Kahler-Einstein manifold with negative Ricci and sectional curvatures.
where a semicolon indicates a covariant derivative (with respect to the coordinates of [V.sub.4]), a comma an ordinary derivative and [R.sub.abcd] stands for the [V.sub.4]
Riemann curvature tensor [1].
Then one can define a projected Riemann curvature with respect to the vector [l.sup.a] by projecting all the indices of the
Riemann curvature tensor, leading to (by the very definition, the Riemann tensor [R.sub.abcd] has a generic form [[partial derivative].sub.b][[partial derivative].sub.c] [g.sub.ad] in local inertial frame; thus the above projection ensures that [g.sub.ab] has only double spatial derivatives when [l.sub.a] is a timelike vector; all time derivatives appearing in [R.sub.abcd] are single in nature and hence vanish in the local inertial frame all together; this is the prime motivation of introduction of this projected Riemann tensor; the same can be ascertained from the Gauss-Codazzi equation as well; see [1,5])