GEODESICS IN HYPERBOLIC SPACE AND NUMBER THEORY
Abstract. Geometry studies geodesics in various settings, in particular on hyperbolic surfaces. The distribution of geodesics on arithmetic surfaces gives information on the arithmetic of quadratic forms, an important branch of Number Theory. 1. Geodesics and the sphere A geodesic curve is the path...
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| Format: | Text |
| Language: | English |
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| Online Access: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.3782 http://comet.lehman.cuny.edu/petridis/geodaeten.pdf |
| Summary: | Abstract. Geometry studies geodesics in various settings, in particular on hyperbolic surfaces. The distribution of geodesics on arithmetic surfaces gives information on the arithmetic of quadratic forms, an important branch of Number Theory. 1. Geodesics and the sphere A geodesic curve is the path of a point in our space that is moving without friction and without external forces. A geodesic curve minimizes the distance between two points, at least when these are close enough to each other. While on our standard euclidean space the shortest distance between two points is given by the length of the line segment between them, the situation becomes more interesting and more complicated in general. For example the earth is (approximately) a sphere. When we fly from Frankfurt to Los Angeles, the plane goes over Greenland. This route at high latitude corresponds to the fact that the shortest path (geodesic) on the sphere is along a great circle, i.e., a circle centered at the center of the sphere (earth). Another example would be a trip from London (on the 0 o meridian) and |
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