 Solution

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

 Top articles  Latest articles New article in topic

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

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

Using the Leibniz rule for the connection, or first checking decomposable tensors and extending linearly, the commutator acts on each covariant index:

[\nabla_{α}, \nabla_{β}] W_{μν} = - R^{ρ}_{μα β} W_{ρ ν} - R^{ρ}_{να β} W_{μ ρ} .

(1)

Choose

W_{μν} = g_{μν}

. Metric compatibility makes the left side zero. Applying the formula with commutator indices

β, γ

gives

0 = - R^{ρ}_{μ β γ} g_{ρ α} - R^{ρ}_{α β γ} g_{μ ρ} = - R_{αμ β γ} - R_{μα β γ} .

(2)

Hence

R_{μα β γ} = - R_{αμ β γ} .

(3)

 Read the full article

 New to topics? Read the docs here!