Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 37D / b

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- Solution b

Solution

Expanding the two covariant derivatives of a vector, the second partial derivatives cancel in the commutator. The remaining derivatives and products of Christoffel symbols combine into the coordinate definition of the Riemann curvature tensor, giving the Ricci identity

2 \nabla_{[α} \nabla_{β]} V^{γ} = [\nabla_{α}, \nabla_{β}] V^{γ} = R^{γ}_{ρ α β} V^{ρ} .

For a type-

(2, 0)

tensor, the connection acts on both upper indices, so the curvature commutator on a contravariant tensor is

2 \nabla_{[α} \nabla_{β]} T^{μν} = R^{μ}_{ρ α β} T^{ρ ν} + R^{ν}_{ρ α β} T^{μ ρ} .

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