Riemann Curvature Tensor

The Riemann Curvature Tensor and its associated tensor are rank four tensors, that describe the curvature of a space by taking the sum of the changes in the covariant derivatives over a closed loop. It comes in handy when ascertaining the curvature of things, and hence is useful in general relativity. The Riemann tensor is entirely covariant, while the associated tensor has its first index raised. The Riemann tensor is also known as the Riemann Cristoffel tensor.

Riemann tensors

covariant tensor R_ijkl; contravariant tensor Rⁱ_jkl

Rⁱ_jkl=g^ibR_bjkl

(As usual the Einstein Summation Principle holds. Whenever two indices are repeated on one side of an equivalence relation, and one is contravariant, and the other covariant, then they are summed through. Also g^ab is the contravariant metric tensor whose covariant "opposite" is g_ab where g^abg_bc=δ^a_c)

The Riemann curvature tensor is a tensor because it transforms according to the tensor transformation rules. That is from the coordinate system xⁱ(The tensor is a function of these coordinates) to the coordinate system y^j:

R^a_bcd=(∂y^a/∂xⁱ)(∂x^j/∂y^b)(∂x^k/∂y^c)(∂x^l/∂y^d)Rⁱ_jkl

R_abcd=(∂xⁱ/∂y^a)(∂x^j/∂y^b)(∂x^k/∂y^c)(∂x^l/∂y^d)R_ijkl

The Riemann Curvature tensor, is a method to find the curvature of a specific set of coordinates via a parallel transport. When multiplied with three vectors, it tells the direction of the resultant vector of parallel transporting one of the vectors with the two others. This, however, is a somewhat sketchy interpretation. Regardless there are several useful identities for this tensor:

Rⁱ_jkl=∂Γⁱ_jl/∂x^k-∂Γⁱ_jk/∂x^l+Γ^a_jlΓⁱ_ak-Γ^a_jkΓⁱ_al

Here Gamma is the Cristoffel symbol and is not a tensor even though it has indexical components.

A_iRⁱ_jkl=A_j;kl-A_j;lk

(here the semi colon denotes a covariant derivative in the indices which follow it)

R_ijkl=-R_jikl

R_ijkl=-R_ijlk

R_ijkl=R_klij

when contracted along two indices one gets the Ricci tensor:

R_ij=g^abR_aibj

When this is contracted one gets the Ricci Scalar.

R=g^abR_ab

The Ricci tensor and the ricci scalar are then useful in general relativity as they are used for the einstein tensor:

G_ab=R_ab-R g_ab/2

The Gaussian curvature or total curvature of a surface whose coordinates define the Riemann tensor can be found by:

K=ε^ijε^klR_ijkl/4

Here ε^ij is the alternating tensor or the Levi-Civita tensor. That's all for now

Godamn! math is a pain in the ass to do in HTML. Anywho most of my information is from Introduction to Tensor Calculus and Continuum Mechanics by J.H. Heinbockel, and Cosmology and Particle AstroPhysics by Lars Bergström and Ariel Goobar. The former is available free on the internet at http://www.math.odu.edu/~jhh/counter2.html. The latter cost me a shitload of money and isn't really all that great. Also in E2 abstract index notation may be useful reading.

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