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

Set

U_{μ} = \partial_{μ} ϕ

. Torsion-freeness and commutation of ordinary partial derivatives imply that its covariant Hessian is symmetric:

\nabla_{α} U_{β} = \nabla_{β} U_{α} .

Take the cyclic sum of the three commutators acting respectively on

U_{γ}, U_{α}, U_{β}

. Their left-hand sides cancel pairwise by this Hessian symmetry, leaving

(R^{ν}_{γ α β} + R^{ν}_{α β γ} + R^{ν}_{β γ α}) U_{ν} = 0.

At a chosen point, a scalar field can be selected whose gradient has any prescribed covector value. The coefficient of

U_{ν}

must therefore vanish. Relabelling indices gives the first Bianchi identity

R^{μ}_{α β γ} + R^{μ}_{β γ α} + R^{μ}_{γ α β} = 0.

Solved by gpt-5.6-sol high.

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