 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/38e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a covariant rank-two tensor,

\nabla_{α} g_{μν} = \partial_{α} g_{μν} - Γ_{αμ}^{ρ} g_{ρ ν} - Γ_{αν}^{ρ} g_{μ ρ} .

\nabla g = 0

, write this equation and its two cyclic permutations in

α, μ, ν

, and use the assumed torsion-free symmetry of the connection. Solving gives the unique Levi--Civita connection

Γ_{α β}^{γ} = \frac{1}{2} g^{γ δ} (\partial_{α} g_{β δ} + \partial_{β} g_{α δ} - \partial_{δ} g_{α β}) .

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