Consider the hyperbolic plane with metric on the upper half-plane

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

Invariance, Ch Curvature

Problem

Consider the hyperbolic plane with metric ds2=(dx2+dy2)/y2\mathrm{d}s^2 = (\mathrm{d}x^2+\mathrm{d}y^2)/y^2 on the upper half-plane y>0y > 0, for which K=−1K=-1. ((a)) Find the hyperbolic distance between the points (0,1)(0,1) and (0,5)(0,5), which lie on the same vertical geodesic (x=0x=0). ((b)) A geodesic triangle in this plane has area (in the same units) A=0.30A = 0.30. Find the sum of its interior angles, in radians.

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