General Relativity: An Introduction for Physicists · Problem 2.9

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · embedded-manifolds-volume-area

Problem

Show that the line element of a 3-sphere of radius

a

embedded in four-dimensional Euclidean space can be written in the form

ds^2 = a^2[d\chi^2 + \sin^2\chi(d\theta^2 + \sin^2\theta\,d\phi^2)].

Hence, in this three-dimensional non-Euclidean space, calculate the area of the 2-sphere defined by

\chi = \chi_0

. Also find the total volume of the three-dimensional space.