Question

Recall that the upper half plane H ((x, y)ly > 0) gives a "model" for hyperbolic space. In this model, distance decreases as one moves up (i.e. the distance between (0, 1) and (1, 1) is 1, the distance between (0,2) and (1,2) is 1/2, and the distance between (0, y) and (1, y) is 1/y. Draw a picture to see that this is strange.) The geodesics on H are given by (1) half circles with center somewhere on the r- axis and (2) vertical lines. If two circles centered on the r-axis (or one circle and one vertical line) intersect (tangentially) on the r-axis, then the corresponding geodesics are asymptotic (get closer and closer to each other but do not intersect) as they approach the -axis 2. Given a geodesic L and a point p not lying on L, show there are an infinite number of geodesics through p which do not intersect L

Answer 1

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