Talk:List of formulas in Riemannian geometry

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Concerning Order of Indices of Christoffel symbols and Riemann curvature tensor.: new section

← Previous revision​	Revision as of 03:20, 3 May 2024​
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Following the definition in [[Tensor]], and since <math>{R^{ijk}}_l</math> before was listed as “<math>(3, 1)</math> Riemann curvature tensor”, I think that <math>{R^i}_{jkl}</math> should be listed as “<math>(1, 3)</math> Riemann curvature tensor” instead. --[[UserointedEars|PointedEars]] ([[User talkointedEars|talk]]) 09:42, 1 October 2023 (UTC)	Following the definition in [[Tensor]], and since <math>{R^{ijk}}_l</math> before was listed as “<math>(3, 1)</math> Riemann curvature tensor”, I think that <math>{R^i}_{jkl}</math> should be listed as “<math>(1, 3)</math> Riemann curvature tensor” instead. --[[UserointedEars|PointedEars]] ([[User talkointedEars|talk]]) 09:42, 1 October 2023 (UTC)
== Concerning Order of Indices of Christoffel symbols and Riemann curvature tensor. ==
Hello,
I'm noticing variations in notations and index ordering for definitions of the Riemann curvature tensor and the Christoffel symbols. Using Einstein's "The Meaning of Relativity", 1953, as reference, I find myself wanting to make some comments.
In Einstein's little book on page 71, equation 69 is the Christoffel symbol of the 1st kind. I'd like to propose a new definition, changing it slightly by swapping some indices, which makes no difference for the case of a symmetric metric, but I suggest is helpful when considering an asymmetric metric.
<math display="block"> {
[(g)ij,k]\equiv\frac{1}{2} (g_{ik,j}+g_{kj,i}-g_{ij,k})
} </math>
To differentiate this definition from Chritoffel's, I've adjusted the notation to make it an operator of a rank 2 covariant tensor.
I'm also seeing different representations of the the Riemann curvature tensor, which in Einstein's little book on page 77, equation 77 is:
<math>
R^{\mu}{}_{\sigma\alpha\beta} =
-\partial_{\beta}\Gamma^{\mu}{}_{\sigma\alpha}
+\partial_{\alpha}\Gamma^{\mu}{}_{\sigma\beta}
+\Gamma^{\mu}{}_{\rho\alpha}\Gamma^{\rho}{}_{\sigma\beta}
-\Gamma^{\mu}{}_{\rho\beta}\Gamma^{\rho}{}_{\sigma\alpha}
</math>
which is different from the definition here. In studying the asymmetric metric, I'm finding myself trying to make sense of the order of these indices as defined in Einstein's little book, and now here. [[User:Ric.Peregrino|Ric.Peregrino]] ([[User talk:Ric.Peregrino|talk]]) 03:20, 3 May 2024 (UTC)

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