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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 2

Contracting the <Riemann curvature tensor> gives the nonzero components of the <Ricci tensor>:
$$
R_{rr}=R^\phi{}_{r\phi r}=-1,
\qquad
R_{\phi\phi}=R^r{}_{\phi r\phi}=-\sinh^2r.
$$
The <Ricci scalar> is the contraction with the <inverse metric>. Therefore
$$
\begin{aligned}
R
&=g^{rr}R_{rr}+g^{\phi\phi}R_{\phi\phi}\\
&=\frac1{a^2}(-1)
+\frac1{a^2\sinh^2r}(-\sinh^2r)\\
&=\boxed{-\frac2{a^2}}.
\end{aligned}
$$
It is independent of both coordinates, so the <hyperbolic plane> has constant negative scalar curvature.

Solution

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

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

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

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