Past exam of the general relativity course of the University of Oxford / 2022 / 1 / a

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Solution

For the Levi-Civita connection, the all-lowered Riemann curvature tensor has the algebraic symmetries

R_{λ μν κ} = - R_{μ λ ν κ} = - R_{λ μ κ ν}, R_{λ μν κ} = R_{ν κλ μ} .

The first Bianchi identity is the cyclic identity

R_{λ μν κ} + R_{λ ν κ μ} + R_{λκ μν} = 0.

These statements concern the Riemann curvature tensor at each point and are unchanged by reversing its overall curvature sign convention.

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