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

Table of contents
- Solution b

Solution

The connection 1-forms are defined by

\nabla e_{b} = ω^{a}_{b} \otimes e_{a}

. Metric compatibility gives

ω_{ab} = - ω_{ba}

, while vanishing torsion gives Cartan's first structure equation

d θ^{a} + ω^{a}_{b} \land θ^{b} = 0.

The curvature 2-forms are

R^{a}_{b} = d ω^{a}_{b} + ω^{a}_{c} \land ω^{c}_{b} = \frac{1}{2} R^{a}_{b c d} θ^{c} \land θ^{d},

which displays their relation to the Riemann curvature tensor.

Solved by gpt-5.6-sol high.

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