Presenter: Moses Samuelson-Lynn, College of Science, Mathematics
Authors: Moses Samuelson-Lynn
Faculty Advisor: Aaron Bertram, College of Science, Mathematics
Institution: University of Utah
The upper half plane is often considered with the Euclidean metric (the standard distance function in two dimensions) as a subset, but not a subspace, of R2. However, when given a certain alternative metric, known as the hyperbolic metric, the upper half plane is imbued with an entirely new structure, one which can be directly compared to the structure of hyperbolic space. Hyperbolic space can be thought of as the geometry which would be perceived by someone living on a general three-dimensional manifold with constant negative curvature and two degrees of freedom. It is not at all trivial to create a reasonable embedding of hyperbolic space in three-space, but the fact that it can also be represented using a special distance function on the upper half plane surprisingly reveals the fundamental two-dimensionality of hyperbolic space. In this project, we present this metric. We then explore whether it is a valid distance function by investigating the following statements: (a) it is a mapping of pairs of points in the upper half plane to nonnegative real numbers, (b) given two points A and B in the upper half plane, the distance between them given by the hyperbolic metric is zero if and only if A and B are the same point, and that (c) it satisfies the triangle inequality, that is, given any three points A, B, and C in the upper half plane, the sum of the hyperbolic distance between A and B with the hyperbolic distance between B and C is greater than or equal to the hyperbolic distance between A and C. Finally, we seek to show that it does imbue the upper half plane with hyperbolic structure. Furthermore, we will explore an explicit derivation of this distance function.