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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-56/2/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 56 / 2 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

Contracting the assumed Riemann curvature tensor gives the Ricci tensor

R_{b d} = g^{a c} R_{ab c d} = \frac{R}{12} (4 g_{b d} - g_{b d}) = \frac{R}{4} g_{b d} .

(1)

The contracted Bianchi identity and metric compatibility now imply

\frac{1}{4} \nabla_{b} R = \nabla^{a} R_{ab} = \frac{1}{2} \nabla_{b} R .

(2)

Thus $\nabla_{b} R = 0$ , so the scalar curvature is constant on each connected component of spacetime. The sectional curvature is consequently the constant

κ = R /12

. This is the four-dimensional case of the Schur theorem in pseudo-Riemannian geometry.

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