Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 38E / d / Solution

Define the Ricci tensor by contracting the first and third Riemann indices:Contract the first Bianchi identity in its upper index and third lower index. The middle term vanishes by antisymmetry in the first pair, while antisymmetry in the final pair turns the last term into $-R_{\gamma \alpha }$. Thusso the Ricci tensor is symmetric.

For the stated constant-curvature form,The contracted Bianchi identity isUsing $\nabla g=0$ givesorIn four dimensions,More generally, $K$ is constant for every $n>2$. In dimension $n=2$, the contracted Bianchi identity gives no such conclusion, and the Gaussian curvature may vary from point to point.

Solved by gpt-5.6-sol high.