Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 3 / b / Solution

The connection 1-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. Metric compatibility gives ${\omega }_{ab}=-{\omega }_{ba}$, while vanishing torsion gives Cartan's first structure equationThe curvature 2-forms arewhich displays their relation to the Riemann curvature tensor.

Solved by gpt-5.6-sol high.