What is the Form of Riemann Tensor in 3D?

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SUMMARY

The Riemann tensor is defined independently of the coordinate system and the manifold's dimension, as established in "Geometry, Topology and Physics" by Nakahara, specifically in Chapter 7. In the context of 3D manifolds, the Riemann tensor exhibits a specific form that can be derived from the Christoffel symbols and their derivatives. A discussion on Physics Forums highlights a proof of the 3D Riemann tensor, where a notable difference arises from the treatment of the Ricci scalar term, particularly the inclusion of a negative sign in the final expression as noted by user samalkhaiat.

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sadegh4137
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hi

Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it.

see for example Geometry,Topology and physics By Nakahara Ch.7
In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel symbol and it's derivative. ( that relation is independent of dimension of manifold)
I don't know why Riemann tensor in 3D has the following form

https://www.physicsforums.com/attachment.php?attachmentid=7457&d=1154966304

In this page
https://www.physicsforums.com/showthread.php?t=128275
you can see a prove for 3D Riemann tensor but i can't understand why it has different form
 
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Post #7 in that thread has the exact same form of the equation in your attachment. The only difference is samalkhaiat pulled out a negative sign in the Ricci scalar term in the final expression.
 
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