 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-59/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Use metric compatibility and differentiate the projector:

\nabla_{α} P^{ϵ}_{β} = (\nabla_{α} n^{ϵ}) n_{β} + n^{ϵ} \nabla_{α} n_{β} .

Projecting the derivative and lower tensor index kills the first term, by part (a). The remaining term is the extrinsic curvature:

P^{α}_{μ} P^{β}_{ν} \nabla_{α} P^{ϵ}_{β} = n^{ϵ} P^{α}_{μ} P^{β}_{ν} \nabla_{α} n_{β} = K_{μν} n^{ϵ} .

The second projection may be inserted in the definition of

K

, since differentiating

n^{β} n_{β} = - 1

gives

n^{β} \nabla_{α} n_{β} = 0

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