Riemann-Christoffel tensor

Riemann-Christoffel tensor

[′rē‚män ′kris·tə·fəl ‚ten·sər]
(mathematics)
The basic tensor used for the study of curvature of a Riemann space; it is a fourth-rank tensor, formed from Christoffel symbols and their derivatives, and its vanishing is a necessary condition for the space to be flat. Also known as curvature tensor.
References in periodicals archive ?
Lanczos, "A remarkable property of the Riemann-Christoffel tensor in four dimensions," Annals of Mathematics, vol.
Calculating the physically observable components of the Riemann-Christoffel tensor [X.sub.ik] (28) for the de Sitter vacuum bubble, we find
If we use these vectors, the Riemann-Christoffel tensor can be written (see also Section 4.1, Eqs (A1)-(A5)) as
In the difference appearing in the Riemann-Christoffel tensor only four terms remain therefore:
For example: [G.sub.[Mu][Nu]] is obtained from the Riemann-Christoffel tensor, that is, from the quantity which largely determines the structure of space-time, by the most straightforward method, namely by the contraction of two indices.
Moreover, Zelmanov proven that any non-holonomic space has nonzero Riemannian curvature (nonzero Riemann-Christoffel tensor) due to [g.sub.0i] [not equal to] 0.
He followed that procedure by which the Riemann-Christoffel tensor was built: proceeding from the non-commutativity of the second derivatives of an arbitrary vector
Briefly, as one calculates the Ricci tensor [R.sub.[alpha][beta]] = [g.sup.[mu]v][R.sub.[alpha][mu]v[beta]] by the contraction of the Riemann-Christoffel tensor
Our solution of the deviation equations depends on a specific formula for the space metric whereby we calculate the Riemann-Christoffel tensor. Because the sources of gravitational waves (double stars, pulsars, etc.) are far away from us, we expect received gravitational waves to be weak and plane.
He followed that procedure by which the Riemann-Christoffel tensor was built: proceeding from the noncommutativity of the second derivatives of an arbitrary vector
As is well known, the components of the Riemann-Christoffel tensor satisfy the identities
To answer the question let us recall that Zelmanov, following the same procedure by which the Riemann-Christoffel tensor was introduced, after considering non-commutativity of the chr.inv.-second derivatives of a vector *[[nabla].sub.i] *[[nabla].sub.k] [Q.sub.l] - +[[nabla].sub.k]*[[nabla].sub.i][Q.sub.l] = 2[A.sub.ik]/[c.sup.2] *[partial derivative][Q.sub.l]/[partial derivative]t + [H.sup....j.sub.lki][Q.sub.j], had obtained the chr.