“Riemann Tensor”的概念、定义、习题解题方法及翻译-数学辞典

单词

Riemann Tensor

释义

Riemann Tensor

A Tensor sometimes known as the Riemann-Christoffel Tensor. Let

(1)

where the quantity inside the

is a Christoffel Symbol of the Second Kind. Then

(2)

Broken down into its simplest decomposition in

-D,


	(3)

Here,

is the Ricci Tensor,

is the Curvature Scalar, and

is theWeyl Tensor. In terms of the Jacobi Tensor

(4)

The Riemann tensor is the only tensor that can be constructed from the Metric Tensor and its first and secondderivatives,

(5)

where

are Connection Coefficients and

are CommutationCoefficients. The number of independent coordinates in

-D is

(6)

and the number of Scalars which can be constructed from

and

(7)

In 1-D,


1	0	0
2	1	1
3	6	3
4	20	14

See also Bianchi Identities, Christoffel Symbol of the Second Kind, Commutation Coefficient,Connection Coefficient, Curvature Scalar, Gaussian Curvature, Jacobi Tensor, PetrovNotation, Ricci Tensor, Weyl Tensor

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