# 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 ?
If we use these vectors, the Riemann-Christoffel tensor can be written (see also Section 4.
In the difference appearing in the Riemann-Christoffel tensor only four terms remain therefore:
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.
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
lki] differs algebraically from the Riemann-Christoffel tensor because of the presence of the space rotation [A.
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.
He followed that procedure by which the Riemann-Christoffel tensor was built: proceeding from the noncommutativity of the second derivatives of an arbitrary vector
projections of the Riemann-Christoffel tensor are [1]: [X.
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.

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