Question

How can we prove that in 2D the Ricci scalar is proportional to the Ricci tensor? I started with the second Bianchi identity, set 2 axes equal - and it led me to the fact that in 2D the covariant derivative of the Ricci tensor is zero. However, I cannot prove that the Ricci tensor is proportional to the Ricci scalar from it.

Answer 1

In 2D space, the Ricci tensor is proportional to the Ricci scalar because the Ricci scalar is twice the Gaussian curvature and there's only one independent component. Therefore, the Ricci tensor is R/2 multiplied by the metric tensor.

Step-by-step explanation:

To prove in 2D that the Ricci scalar is proportional to the Ricci tensor, we rely on the fact that the Ricci tensor in two dimensions can be expressed as geometric quantities. In 2D, the Ricci tensor is related to Gaussian curvature K, and it holds that the Ricci scalar R is simply 2K in a 2D manifold.

Consequently, in a 2D space, which has only one independent component due to symmetries of the Riemann curvature tensor, the Ricci tensor Rij can be written as Rij = R/2 · gij, where gij is the metric tensor and R is the Ricci scalar. This shows that the Ricci tensor is proportional to the Ricci scalar, with the proportionality constant being 1/2 in this case.