What is the Form of Riemann Tensor in 3D?

[sadegh4137]

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

 

[WannabeNewton]

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.