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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / c / Solution

Using the stated convention for the Riemann curvature tensor, the first independent component is

R^{r}_{ϕ r ϕ} = \partial_{r} Γ_{ϕϕ}^{r} - Γ_{ϕ r}^{ϕ} Γ_{ϕϕ}^{r} = - (cosh^{2} r + sinh^{2} r) + cosh^{2} r = - sinh^{2} r .

The other is

R^{ϕ}_{r ϕ r} = - \partial_{r} Γ_{r ϕ}^{ϕ} - Γ_{r ϕ}^{ϕ} Γ_{ϕ r}^{ϕ} = csch^{2} r - coth^{2} r = - 1,

using the hyperbolic Pythagorean identity

coth^{2} r - csch^{2} r = 1

. Antisymmetry in the final two indices supplies the components with those indices reversed. These are the curvature of the two-dimensional hyperbolic polar metric components.

Solved by gpt-5.6-sol high.

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