A surface has the metric with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Curvature

Problem

A surface has the metric ds2=dr2+f(r)2 dϕ2\mathrm{d}s^2=\mathrm{d}r^2+f(r)^2\,\mathrm{d}\phi^2 with f>0f>0, where rr is arc length along the curves of constant ϕ\phi and ϕ\phi is an angle. Surfaces of revolution have metrics of this form. ((a)) Find the non-zero Christoffel symbols, and show that Rrϕrϕ=−ff′′R^r{}_{\phi r\phi}=-ff'' and hence that K=−f′′/fK=-f''/f, where a prime means d/dr\mathrm{d}/\mathrm{d}r. ((b)) Evaluate KK for f=asin⁡(r/a)f=a\sin(r/a), for f=rsin⁡αf=r\sin\alpha and for f=bsinh⁡(r/b)f=b\sinh(r/b), and name the surface of this chapter that each describes. ((c)) A map-maker wants a flat map of a small region of each surface that keeps every distance. Using (b), say for which of the three this is possible, and why.

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