 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/37d/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Lower the free index of

K

. The Killing equation makes the last two indices of

\nabla_{ν} \nabla_{μ} K_{α}

antisymmetric after the appropriate derivative is moved. Combining the three resulting Ricci identities cyclically gives

\nabla_{ν} \nabla_{μ} K_{α} = \frac{1}{2} ([\nabla_{ν}, \nabla_{μ}] K_{α} - [\nabla_{μ}, \nabla_{α}] K_{ν} + [\nabla_{α}, \nabla_{ν}] K_{μ}) .

Applying the covector form of the Ricci identity and then the first Bianchi identity reduces this to

\nabla_{ν} \nabla_{μ} K_{α} = R_{αμν β} K^{β} .

Raising

α

\nabla_{ν} \nabla_{μ} K^{α} = R^{α}_{μν β} K^{β} .

Contracting

μ

and

ν

and using the definition and symmetries of the Ricci tensor gives

\nabla_{μ} \nabla^{μ} K^{α} = - R^{α}_{β} K^{β} .

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