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 equation
$$
d\theta^a+\omega^a{}_b\wedge\theta^b=0.
$$
The <curvature 2-forms> are
$$
\mathcal R^a{}_b=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b
=\frac12R^a{}_{bcd}\theta^c\wedge\theta^d,
$$
which displays their relation to the <Riemann curvature tensor>.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-309/3/b/solution