San José State University

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 The Riemann-Christoffel Tensor

The Riemann-Christoffel tensor arises as the difference of cross covariant derivatives. Let Ai be any covariant tensor of rank one. Then

### Ai, jk − Ai, kj = RijkpAp

Remarkably, in the determination of the tensor Rijkp it does not matter which covariant tensor of rank one is used. The tensor Rijkp is called the Riemann-Christoffel tensor of the second kind. As the notation indicates it is a mixed tensor, covariant of rank 3 and contravariant of rank 1. This has to be proven.

The determination of the nature of Rijkp goes as follows.

• The general formula for the covariant derivative of a covariant tensor of rank one, Ai, is

#### Ai, j = ∂Ai/∂xj − {ij,p}Ap

• For a covariant tensor of rank two, Bij, the formula is:

#### Bij, k = ∂Bij/∂xk − {ik,p}Bpj − {kj,p}Bip

• Ai, j is such a tensor so the above formula applies to it as well. Therefore

#### Ai, jk = ∂Ai, j/∂xk − {ik,p}Ap, j − {kj,p}Ai, p where (Ai, j), k has been written as Ai, jk.

• Replacing Ai, j by ∂Ai/∂xj − {ij,p}Ap and carrying out the indicated differentiation yields

#### Ai, jk = (∂²Ai/∂xkxj) − (∂{ij,p}/∂xk)Ap − {ij,p}(∂Ap/∂xk) − {ik,p}(∂Ap/∂xj) + {ik,p}{pj,q}Aq − {kj,p}(∂Ai/∂xp) + {kj,p}{ip,r}Ar

• If the indices j and k are interchanged the result is Ai, kj; i.e.,

#### Ai, kj = (∂²Ai/∂xjxk) − (∂{ik,p}/∂xj)Ap − {ik,p}(∂Ap/∂xj) − {ij,p}(∂Ap/∂xk) + {ij,p}{pk,q}Aq − {jk,p}(∂Ai/∂xp) + {jk,p}{ip,r}Ar

• The cross partial derivatives (∂²Ai/∂xjxk) and (∂²Ai/∂xkxj) are equal. The partial derivatives of Ap with respect to xj and xk appear in both expressions although in different positions. Thus subtracting the expression for Ai, kj from the one for Ai, jk yields

#### Ai, jk − Ai, kj = − (∂{ij,p}/∂xk)Ap + {ik,p}{pj,q}Aq + (∂{ik,p}/∂xj)Ap − {ij,p}{pk,q}Aq

• The summation indices p and q in the two terms involving a double summation can be interchanged without affecting the result. This allows the above result to be expressed as

#### Ai, jk − Ai, kj = − (∂{ij,p}/∂xk)Ap + {ik,q}{qj,p}Ap (∂{ik,p}/∂xj)Ap − {ij,q}{qk,p}Ap

• The above result can further simplified as

#### Ai, jk − Ai, kj = [∂{ik,p}/∂xj) − ∂{ij,p}/∂xk + {ik,q}{qj,p} − {ij,q}{qk,p}]Ap

• Let the expression within the brackets be denoted as Rijkp so the above is represented as

#### Ai, jk − Ai, kj = RijkpAp

• The expression on the left-hand side of the above equation is the difference of two tensors of covariant rank 3. Therefore it is a tensor of covariant rank 3. The term Rijkp on the right-hand side when multiplied times the components of an arbitrary covariant tensor of rank 1 and summed yields a covariant tensor of rank 3. Therefore by the Tensor Quotient Theorem Rijkp is a mixed tensor of covariant rank 3 and contravariant rank 1. Thus the notation is justified.
• Thus the condition for the cross covariant derivatives to be equal is that the Riemann-Christoffel tensor of the second kind be identically equal to zero; i.e.,

#### Ai, jk = Ai, kj if and only if  Rijkp = 0 for all i, j, k and p.

(To be continued.)