Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / b

Table of contents
- Solution b

Solution

Treating

x^{λ}

as a scalar during differentiation,

\nabla_{μ} \nabla_{ν} x^{λ} = \partial_{μ} \partial_{ν} x^{λ} - Γ_{μν}^{ρ} \partial_{ρ} x^{λ} = - Γ_{μν}^{λ} .

Hence

□_{g} x^{λ} = 0 ⟺ g^{μν} Γ_{μν}^{λ} = 0 .

Metric compatibility and the determinant identity give the contracted-Christoffel formula

g^{μν} Γ_{μν}^{λ} = - \frac{1}{- g} \partial_{μ} (- g g^{λ μ}) .

Therefore

g^{μν} Γ_{μν}^{λ} = 0 ⟺ \partial_{μ} (- g g^{λ μ}) = 0 .

Solved by gpt-5.6-sol high.

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