What is the Form of Riemann Tensor in 3D?

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The Riemann tensor is defined independently of the coordinate system and the manifold's dimension. In 3D, its specific form raises questions, particularly regarding its relationship with the Christoffel symbols and their derivatives. A reference to Nakahara's "Geometry, Topology and Physics" highlights the general relation applicable across dimensions. A linked forum thread provides a proof of the 3D Riemann tensor, but confusion arises due to a negative sign difference in the Ricci scalar term presented by another user. Understanding this discrepancy is crucial for grasping the Riemann tensor's behavior in three dimensions.
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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