A Study of Hyperbolic Geometry

Abstract

We provide an expansive treatment of the fundamentals of hyperbolic geometry. The explicit computation of hyperbolic isometries enables us to find geodesics, calculate the metric, define the boundary, and compute the sectional curvatures of the canonical hyperbolic space in n dimensions. We also explore the theory of geometric structures on manifolds and state the rigidity theorems for hyperbolic manifolds. Lastly, we see an application of this theory in the triangulation of knot complements and work out a complete hyperbolic structure on the decomposition of the figure-8 knot complement into two ideal tetrahedra.