A 'three-sphere' is the three-dimensional surface in four-dimensional Euclidean space (coordinates x, y, z, w), given by the equation x2 + y2 + z2 + w2 = r2, where r is the radius of the sphere.
(a) Define new coordinates (r, θ, φ, χ) by the equations w = r cos χ, z = r sin χ cos θ, x = r sin χ sin θ cos φ, y = r sin χ sin θ sin φ. Show that (θ, φ, χ) are coordinates for the sphere. These generalize the familiar polar coordinates.
(b) Show that the metric of the three-sphere of radius r has components in these coordinates gχχ = r2, gθθ = r2 sin2 χ, gφφ = r2 sin2 χ sin2 θ, all other components vanishing. (Use the same method as in Exer. 28.)