Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 35D / a / Solution

For $k=1$, put $r=\sin \chi$. The spatial metric tensor becomesThe global closed spatial slice is a 3-sphere with $0\le \chi \le \pi$; the original $r$ chart covers only a hemisphere, so two charts are needed. A circle at fixed $\chi$ in an equatorial angular plane has circumference $\boxed{2\pi a\sin \chi }$. The full spatial volume isIntegrating a single $r\in [0,1]$ chart without its second copy would give only half this volume.