Page 121 - Intro to Tensor Calculus

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Riemann Christoffel Tensor

Utilizing the equation (1.4.32), it is left as an exercise to show that

A i,jk − A i,kj = A σ R σ ijk

where
∂ σ ∂ σ m σ m σ
σ
R ijk = − + − (1.4.33)
∂x j ik ∂x k ij ik mj ij mk
is called the Riemann Christoffel tensor. The covariant form of this tensor is

i
R hjkl = g ih R jkl . (1.4.34)

It is an easy exercise to show that this covariant form can be expressed in either of the forms

∂ ∂ s s
R injk = [nk, i] − [nj, i]+ [ik, s] − [ij, s]
∂x j ∂x k nj nk
2 2 2 2
1 ∂ g il ∂ g jl ∂ g ik ∂ g jk αβ
or R ijkl = − − + + g ([jk, β][il, α] − [jl, β][ik, α]) .
j
i
j
i
2 ∂x ∂x k ∂x ∂x k ∂x ∂x l ∂x ∂x l
From these forms we find that the Riemann Christoffel tensor is skew symmetric in the first two indices
and the last two indices as well as being symmetric in the interchange of the first pair and last pairs of
indices and consequently
R klij = R ijkl .
R jikl = −R ijkl R ijlk = −R ijkl
In a two dimensional space there are only four components of the Riemann Christoffel tensor to consider.
These four components are either +R 1212 or −R 1212 since they are all related by

R 1212 = −R 2112 = R 2121 = −R 1221 .

In a Cartesian coordinate system R hijk = 0. The Riemann Christoffel tensor is important because it occurs
in differential geometry and relativity which are two areas of interest to be considered later. Additional
properties of this tensor are found in the exercises of section 1.5.