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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-1/35d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 35D / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

Apply the product rule for the Lie derivative to

\tilde{g} = e^{2 ϕ} g

L_{ξ} \tilde{g} = e^{2 ϕ} [2 ξ (ϕ) g + L_{ξ} g] = e^{2 ϕ} (2 ξ^{c} ϕ_{, c} + ψ) g .

(1)

The prescribed transport equation makes this zero, so $ξ$ is a Killing vector field for $\tilde{g}$ . This is a conformal rescaling that absorbs the original infinitesimal dilation.

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