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...
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...
REAL HYPERSURFACES IN K AHLER MANIFOLDS (2007)
0. Introduction. The present work is concerned with the geometry of embedded real hypersurfaces in a Kahler manifold, where isomorphisms are both holomorphic and isometric in the underlying...
We introduce a curvature invariant for embedded real two-surfaces in almostcomplex four-manifolds and give a Weingarten-type identity which serves to study the geometry of such surfaces near complex...
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,...
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...
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...
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...
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 ${\Bbb{R}}^3$...
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...
Königsberg, Med. F., Diss., 1941 (Nicht f. d. Austausch).