Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 38D / i

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- Solution i

Solution

The coordinate expression for the Lie bracket of vector fields is

[V, W]^{α} = V^{μ} \partial_{μ} W^{α} - W^{μ} \partial_{μ} V^{α} .

For the Levi-Civita connection,

V^{μ} \nabla_{μ} W^{α} - W^{μ} \nabla_{μ} V^{α} = V^{μ} \partial_{μ} W^{α} - W^{μ} \partial_{μ} V^{α} + Γ_{μν}^{α} (V^{μ} W^{ν} - W^{μ} V^{ν}) .

The final term vanishes because the Christoffel symbols are symmetric in

μ, ν

while the vector product in parentheses is antisymmetric. Hence

[V, W]^{α} = V^{μ} \nabla_{μ} W^{α} - W^{μ} \nabla_{μ} V^{α} .

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