Properties of curvature tensor in 3 dimensions?

[im_hammer]

Is there any properties with the curvature tensors in 3 dimensions?
(Maybe between the Ricci tensor and the Ricci scalar, they are proportional to each other? )

I heard about it in a lecture, but I can not remember the details. The 3 dimensional case is not discussed in many reference books.

Thank you

 

[haushofer]

In three dimensions, the number of components of the Ricci tensor equals that of the Riemann tensor. The Riemann tensor can be completely expressed in terms of the Ricci tensor (as an exercise, you can try to write it down based on the symmetries). From that you can derive the relation between the Ricci tensor and the Ricci scalar.

These things are often mentioned in books about (2+1)-GR (like the one of Carlip). The number of DOF's of the gravitational field go like (D-3), so in 2+1 dimensions the gravitational field becomes "trivial" if you don't add any extra terms to the Hilbert action.

 

[bcrowell]

Im_hammer, are you asking about 2+1 dimensions, or 3 Euclidean dimensions?

 

[Bill_K]

In three dimensions the Weyl tensor vanishes.