On the Geometry of Spaces of Oriented Geodesics (2009)
Alekseevsky, Dmitri V., Guilfoyle, Brendan, Klingenberg, Wilhelm
Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space (other than the octonion hyperbolic plane), and consider the space L(M) of...
On the three-dimensional Blaschke-Lebesgue problem (2009)
Anciaux, Henri, Guilfoyle, Brendan
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex...
Totally Null Surfaces in Neutral Kaehler 4-Manifolds (2008)
Georgiou, Nikos, Guilfoyle, Brendan, Klingenberg, Wilhelm
We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($\alpha$-planes) or anti-self-dual...
Guilfoyle, Brendan, Klingenberg, Wilhelm
We prove that the index of an isolated umbilic point on a $C^3$-smooth surface in Euclidean 3-space ${\mathbb E}^3$ is less than or equal to one. As a corollary, we establish the Caratheodory...
Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface (2008)
Anciaux, Henri, Guilfoyle, Brendan, Romon, Pascal
Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric...
Lagrangian curves on spectral curves of monopoles (2007)
Guilfoyle, Brendan, Khalid, Madeeha, Ramón-Marí, José J.
We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$ equipped with a natural neutral K\"ahler structure, and prove that they must form real curves. By virtue of the...
A characterization of Weingarten surfaces in hyperbolic 3-space (2007)
Georgiou, Nikos, Guilfoyle, Brendan
We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. Such a surface is...
On C$^2$-smooth Surfaces of Constant Width (2007)
Guilfoyle, Brendan, Klingenberg, Wilhelm
A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established....
On the space of oriented geodesics of hyperbolic 3-space (2007)
Georgiou, Nikos, Guilfoyle, Brendan
We construct a K\"ahler structure (${\mathbb{J}},\Omega,{\mathbb{G}}$) on the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space ${\mathbb{H}}^3$ and investigate its...
On area-stationary surfaces in certain neutral Kaehler 4-manifolds (2006)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary,...
A Neutral Kaehler Metric on Space of Time-like Lines in Lorentzian 3-space (2006)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We study the neutral K\"ahler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all...
Geodesic Flow on the Normal Congruence of a Minimal Surface (2006)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We study the geodesic flow on the normal line congruence of a minimal surface in ${\Bbb{R}}^3$ induced by the neutral K\"ahler metric on the space of oriented lines. The metric is lorentz with...
Geodesic Flow on Global Holomorphic Sections of TS^2 (2006)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{TS}^2\to {S}^2$ induced by the neutral K\"ahler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we...
On Hamilton's Characteristic Functions for Reflection (2006)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We review the complex differential geometry of the space of oriented affine lines in ${\Bbb{R}}^3$ and give a description of Hamilton's characteristic functions for reflection in an oriented C$^1$...
Reflection in a Translation Invariant Surface (2005)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal...
Level sets of functions and symmetry sets of smooth surface sections (2005)
Diatta, Andre, Giblin, Peter, Guilfoyle, Brendan, Klingenberg, Wilhelm
We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the...
The Geometry of Focal Sets (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm
The space ${\Bbb{L}}$ of oriented lines, or rays, in ${\Bbb{R}}^3$ is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral K\"ahler metric which...
A Structure Theorem for Stationary Perfect Fluids (2004)
It is proven that, under mild physical assumptions, an isolated stationary relativistic perfect fluid consists of a finite number of cells fibred by invariant annuli or invariant tori. For axially...
An Indefinite Kaehler Metric on the Space of Oriented Lines (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm
The total space of the tangent bundle of a K\"ahler manifold admits a canonical K\"ahler structure. Parallel translation identifies the space ${\Bbb{T}}$ of oriented affine lines in ${\Bbb{R}}^3$...
The Casimir Effect Between Non-Parallel Plates by Geometric Optics (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm, Sen, Siddhartha
Recent work by Jaffe and Scardicchio has expressed the optical approximation to the Casimir effect as a sum over geometric quantities. The first two authors have developed a technique which uses the...
Reflection of a wave off a surface (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm
Recent advances in twistor theory are applied to geometric optics in ${\Bbb{R}}^3$. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit...
Generalised Surfaces in ${\Bbb{R}}^3$ (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex...
On the Space of Oriented Affine Lines in $R^3$ (2004)
Guilfoyle, Brendan, Klingenberg, Wilhelm
We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean ${\Bbb{R}}^3$ and the tangent bundle to the 2-sphere. These can be utilised...