Date of Award
Doctor of Philosophy
Institute of Technology, Tralee
Dr. Brendan Guilfoyle
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
Georgiou, Nikos, "The Geometry of the Space of Oriented Geodesics of Hyperbolic 3-Space" (2009). Theses [online].
Available at: https://sword.cit.ie/allthe/325