Brendan Guilfoyle

An Indefinite Kähler Metric on the Space of Oriented Lines

The total space of the tangent bundle of a K?hler manifold admits a canonical K?hler structure. Parallel translation identifies the space of oriented affine lines in with the tangent bundle of . Thus the round metric on induces a K?hler structure on which turns out to have a metric of neutral signature. It is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the Euclidean metric on . The geodesics of this metric are either planes or helicoids in . The signature of the metric induced on a surface in is determined by the degree of twisting of the associated line congruence in , and it is shown that, for Lagrangian, the metric is either Lorentz or totally null. For such surfaces it is proved that the Keller-Maslov index counts the number of isolated complex points of inside a closed curve on .

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).

On the three-dimensional Blaschke-Lebesgue problem

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n � 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces). 2000 MSC: 52A40, 52A15

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Abstract Given an oriented Riemannian surface ( Σ , g ) , its tangent bundle T Σ enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure J , a pseudo-metric G with neutral signature and a symplectic structure Ω . We give a local classification of those surfaces of T Σ which are both Lagrangian with respect to Ω and minimal with respect to G . We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R 3 or R 1 3 induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in T S 2 or T H 2 respectively. We relate the area of the congruence to a second-order functional F = ∫ H 2 − K d A on the original surface.

The definitions of three-dimensional landmarks on the human face: an interdisciplinary view

The analysis of shape is a key part of anatomical research and in the large majority of cases landmarks provide a standard starting point. However, while the technology of image capture has developed rapidly and in particular three‐dimensional imaging is widely available, the definitions of anatomical landmarks remain rooted in their two‐dimensional origins. In the important case of the human face, standard definitions often require careful orientation of the subject. This paper considers the definitions of facial landmarks from an interdisciplinary perspective, including biological and clinical motivations, issues associated with imaging and subsequent analysis, and the mathematical definition of surface shape using differential geometry. This last perspective provides a route to definitions of landmarks based on surface curvature, often making use of ridge and valley curves, which is genuinely three‐dimensional and is independent of orientation. Specific definitions based on curvature are proposed. These are evaluated, along with traditional definitions, in a study that uses a hierarchical (random effects) model to estimate the error variation that is present at several different levels within the image capture process. The estimates of variation at these different levels are of interest in their own right but, in addition, evidence is provided that variation is reduced at the observer level when the new landmark definitions are used.

Level sets of functions and symmetry sets of surface sections

We prove that the level sets of a real Cs 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 singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also analyse one of the cases coming from a degenerate critical point, corresponding to an elliptic cusp of Gauss on a surface, where the differentiability is reduced to C[s/4]. However in all our applications we assume C∞ smoothness.

THE CASIMIR EFFECT BETWEEN NON-PARALLEL PLATES BY GEOMETRIC OPTICS

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 complex geometry of the space of oriented affine lines in

A CONVERGING LAGRANGIAN FLOW IN THE SPACE OF ORIENTED LINES

{\Bbb{R}}^3

Marginally trapped surfaces in spaces of oriented geodesics

to describe reflection of rays off a surface. This allows the quantities in the optical approximation to the Casimir effect to be calculated. To illustrate this we determine explicitly and in closed form the geometric optics approximation of the Casimir force between two non-parallel plates. By making one of the plates finite we regularise the divergence that is caused by the intersection of the planes. In the parallel plate limit we prove that our expression reduces to Casimirs original result.

A STRUCTURE THEOREM FOR STATIONARY PERFECT FLUIDS

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is the Steiner point of the initial surface, which remains constant under the flow. To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition.Moreover, this flow converges to a holomorphic Lagrangian section, which forms the set of oriented lines through a point. The coordinates of the Steiner point are projections of the support function into the first non-zero eigenspace of the spherical Laplacian and are given by explicit integrals of initial surface data.