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The Ricci Theorem in Tensor Analysis

The Ricci Theorem in tensor analysis is that the covariant derivative of the metric tensor or its inverse are zero; i.e., all components are zero. Let gij be the metric tensor for some coordinate system (x1,…,xn) for n dimensional space. Then formally,

Ricci's Theorem (First part):
gij, k = 0

where 0 is an n×n×n× array of zeroes.


The second part of Ricci's Theorem is that

gij, k = 0

The second part could have been derived from the first part by noting that

gipgpj = δij

where δij is the Kronecker delta; i.e., δij=0 if i≠j and is equal to 1 if i=j.

Covariant differentiation of the above equation results in

gip, kgpj + gipgpj, k= δij, k = 0

Since gij, k=0 the above equation reduces to

gip, kgpjgpj = 0

This is a system of linear equations in the unknowns gij, k with all the constants in the equations equal to zero. Since the coefficients matrix of the equation is the metric tensor and the metric tensor has an inverse the only solution to the equations is

gij, k = 0

as was previously proven.

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