Date of Award

2009

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Department

Institute of Technology Tralee

First Advisor

Dr. Brendan Guilfoyle

Abstract

In this thesis we construct a Kähler structure (J, Ω, G) on the space L(H3) of oriented geodesics of hyperbolic 3-space H3 and investigate its properties. We prove that (L(H3),J) is biholomorphic to (see thesis pdf), and that the Kähler metric G is of neutral signature, conformally flat and scalar flat.

We establish that the identity component of the isometry group of the metric G on L(H3) is isomorphic to the identity component of the hyperbolic isometry group. We show that the geodesics of G correspond to ruled minimal surfaces in H3, which are totally geodesic iff the geodesics are null.

We then study 2-dimensional submanifolds of the space L(H3) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in H3 orthogonal to the geodesics of Σ.

We prove that the induced metric on a Lagrangian surface in L(H3) has zero Gauss curvature iff the orthogonal surfaces in H3 are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in L(H3) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H3

Comments

Submitted for the Degree of Doctor of Philosophy in Mathematics

Access Level

info:eu-repo/semantics/openAccess

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