Abstract

Extract The 2-sphere The 2-sphere is the two-dimensional manifold of constant positive curvature; it can be realized as the surface of a sphere in three-dimensional Euclidean space. Here are three versions of the metric of the 2-sphere, corresponding to the three coordinate choices shown in Fig. B.1a: where K=1/R2=4k2⁠. Note that R is here being used for the radius of our sphere; it is not the Ricci scalar! Consider the length of a curve from (θ,ϕ)=(0,0) to (θ0, 0) at fixed ϕ. By symmetry, this curve is geodesic. Using metric 1 and eqn (8.27) the length of the curve is We can observe from this that all the points at θ=θ0 are at the same geodesic distance a≡Rθ0 from the origin, therefore these points form a circle of radius a. The circumference of this circle is where to compute the length we used...

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