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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 g_{ij} be the metric tensor for some coordinate system (x^{1},…,x^{n}) for n dimensional space. Then formally,
where 0 is an n×n×n× array of zeroes.
Proof:
where the symbol {ij,k} is the Christoffel 3index symbol of the second kind.
From this definition it is obvious that [ij,k]=[ji,k]. From the definition it is easily established that
But from this relation it then follows that the symbols of the first kind can be obtained from those of the second kind by this relation
Therefore the equation previously derived for ∂g_{ij}/∂x^{k} can be expressed as
But from the previous equation the righthand side of this equation is identically zero. Thus
This is the first part of Ricci's Theorem.
The second part of Ricci's Theorem is that
From the definitions of the 3index symbols of the first and second kind and from the inverse relation of g_{ij} and g^{ij} the following formula can be derived
From the previously derived equation the righthand side of the above equation is identically zero. Thus
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g^{ij}_{, k} = 0
This is the second part of Ricci's Theorem
The second part could have been derived from the first part by noting that
where δ^{i}_{j} is the Kronecker delta; i.e., δ^{i}_{j}=0 if i≠j and is equal to 1 if i=j.
Covariant differentiation of the above equation results in
Since g_{ij, k}=0 the above equation reduces to
This is a system of linear equations in the unknowns g^{ij}_{, 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
as was previously proven.
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