Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 38D / c / Solution

At any point choose orthonormal coordinates. The antisymmetries and pair symmetry of the Riemann curvature tensor leave only $R_{1212}$ independent. In these coordinates,while $R=2R_{1212}$. Since this is a tensor identity, it follows in every coordinate system that the Ricci tensor in two dimensions satisfiesThe Einstein tensor consequently vanishes identically. Hence the two-dimensional vacuum field equation contains no local gravitational dynamics, while coupling the unmodified equation to matter would require $T_{\alpha \beta }=0$. This is the basic degeneracy of two-dimensional general relativity.

Solved by gpt-5.6-sol high.