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

For a covariant rank-two tensor,Impose metric compatibility $\nabla g=0$, write this equation and its two cyclic permutations in $\alpha ,\mu ,\nu$, and use the assumed torsion-free symmetry of the connection. Solving gives the unique Levi--Civita connection

Solved by gpt-5.6-sol high.