Publications

(with W. Klingenberg) On C˛-smooth surfaces of constant width, Tbilisi Math. J. 2 (2009) 1-17. Preprint
An integral inequality for constant width surfaces in Euclidean 3-space is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function and construct explicit examples.
(with W. Klingenberg) On area-stationary surfaces in certain neutral Kaehler 4-manifolds, Beitraege Algebra Geom. 49 (2008) 481-490. Preprint
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. When (N,g) is the round 2-sphere, TN can be identified with the space of oriented affine lines in Rł, and we exhibit a two parameter family of area-stationary tori that are neither holomorphic nor Lagrangian.
(with W. Klingenberg) A neutral Kaehler surface with applications in geometric optics, in Recent developments in pseudo-Riemannian Geometry, European Mathematical Society Publishing House, Zurich (2008) 149-178. Preprint
The space L of oriented lines, or rays, in Euclidean 3-space is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kaehler metric which is closely related to the Euclidean metric on Rł. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in Rł) and the geometry induced on the associated surface in L. The physical context of such sets is geometric optics in a homogeneous isotropic medium, and so, to illustrate the method, we compute the focal set of the k-th reflection of a point source off the inside of a cylinder. The focal sets, which we explicitly parameterize, exhibit unexpected symmetries, and are found to fit well with observable phenomena.
(with W. Klingenberg) Geodesic flow on the normal congruence of a minimal surface, Progr. Math. 265 (2007) 427-436. Preprint
We study the geodesic flow on the normal line congruence of a minimal surface in Rł induced by the neutral Kaehler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description of minimal surfaces in Rł and relate it to the classical Weierstrass representation.
(with W. Klingenberg) Geodesic flow on global holomorphic sections of TS˛, Bull. Belg. Math. Soc. 13 (2006) 1-9. Preprint
We study the geodesic flow on the global holomorphic sections of the bundle TS˛--> S˛ induced by the neutral Kaehler metric on the space of oriented lines of Rł, which we identify with TS˛. This flow is shown to be completely integrable when the sections are symplectic, and the behaviour of the geodesics is described.
(with W. Klingenberg) Reflection in a translation invariant surface, Math. Phys. Anal. Geom. 9 (2006) 225-231. Preprint
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 to the symmetry direction containing the source, while the focal surface is translation invariant. In addition, we show that the focal curve is not physically visible.
(with W. Klingenberg), On Hamilton's characteristic functions for reflection, Irish Math. Soc. Bulletin 57 (2006) 29-40. Preprint
We review the complex differential geometry of the space of oriented affine lines in Rł and give a description of Hamilton's characteristic functions for reflection in an oriented Cš surface in terms of this geometry.
(with W. Klingenberg) Reflection of a wave off a surface, J. Geom. 84 (2006) 55-72. Preprint
Recent investigations of the space of oriented lines in Rł are applied to geometric optics. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming wave is spherical. In each case particular examples are computed exactly and the results plotted to illustrate the outgoing wavefront.
(with W. Klingenberg) Isolated umbilical points on surfaces in Rł, Bull. Greek Math. Soc. 51 (2006) 23-30. Preprint
Recent advances in the application of line congruence techniques to surfaces in Rł are used to generate surfaces with isolated umbilical points of all indices less than or equal to 1.
(with A. Diatta, P. Giblin and W. Klingenberg) Level sets of functions and symmetry sets of surface sections, in Mathematics of Surfaces. Lecture Notes in Computer Science 3604 (2005). Preprint
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 singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.
A structure theorem for stationary perfect fluids, Class. Quantum Grav. 22 (2005) 1599-1606. Preprint
The purpose of this paper is to compare and contrast certain aspects of the relativistic theory of stationary perfect fluids and the non-relativistic theory of Euler flows. Under certain assumptions, we show that the fluid region is divided into a finite number of cells of two types: if the cell does not intersect the boundary it is fibred by tori invariant under the flow, if the cell intersects the boundary it is fibred by annuli invariant under the flow. The flow lines on a torus are either all closed or all dense, while the flow lines on an annulus are all closed.
(with W. Klingenberg and S. Sen) The Casimir effect between non-parallel plates by geometric optics, Reviews in Math. Phys. 17 (2005) 859-880. Preprint
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 Rł 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 Casimir's original result.
(with W. Klingenberg) An indefinite Kaehler metric on the space of oriented lines, J. London Math. Soc. 72 (2005) 497-509. Preprint
The total space of the tangent bundle of a Kaehler manifold admits a canonical Kaehler structure. Parallel translation identifies the space T of oriented affine lines in Rł with the tangent bundle of S˛. Thus, the round metric on S˛ induces a Kaehler structure on T 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 Rł. The geodesics of this metric are either planes or helicoids in Rł. The signature of the metric induced on a surface in T is determined by the degree of twisting of the associated line congruence in Rł, and we show that, for a Lagrangian surface, the metric is either Lorentz or totally null. For such surfaces it is proven that the Keller-Maslov index counts the number of isolated complex points of J inside a closed curve on the surface.
(with W. Klingenberg) Generalised surfaces in Rł, Math. Proc. R. Ir. Acad. 104A (2004) 199-209. Preprint
The correspondence between 2-parameter families of oriented lines in Rł and surfaces in TPš is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of Pš are investigated and a number of theorems are proven that generalise results for closed convex surfaces in Rł.
(with W. Klingenberg) On the space of oriented affine lines in Rł, Archiv der Math. 82 (2004) 81-84. Preprint
We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean Rł and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in Rł, as we illustrate with a number of explicit examples.
The local moduli of Sasakian 3-manifolds, Int. J. Math. Sci. 32 (2002) 117-127. Preprint
The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the scalar curvature is constant is completely solved locally. The resulting Sasakian manifolds include Sł, Nil and SL2R, as well as the Berger spheres. It is also shown that a conformally flat Sasakian 3-manifold is Einstein of positive scalar curvature.
Weyl-type fields with geodesic lines of force, J. Math. Phys. 40 (2000) 2032-2045. Preprint
The static electrogravitational equations are studied and it is shown that an aligned type D metric which has a Weyl-type relationship between the gravitational and electric potential has shearfree geodesic lines of force. All such fields are then found and turn out to be the fields of a charged sphere, charged infinite rod and charged infinite plate. A further solution is also found with shearing geodesic lines of force. This new solution can have m>|e| or m<|e|, but cannot be in the Majumdar-Papapetrou class (in which m = |e|). It is algebraically general and has flat equipotential surfaces.
Interior Weyl-type solutions to the Einstein-Maxwell field equations, Gen. Rel. and Grav. 31 (1999) 1645-1674. Preprint
Static solutions of the electro-gravitational field equations exhibiting a functional relationship between the electric and gravitational potentials are studied. General results for these metrics are presented which extend previous work of Majumdar. In particular, it is shown that for any solution of the field equations exhibiting such a Weyl-type relationship, there exists a relationship between the matter density, the electric field density and the charge density. It is also found that the Majumdar condition can hold for a bounded perfect fluid only if the matter pressure vanishes (that is, charged dust). By restricting to spherically symmetric distributions of charged matter, a number of exact solutions are presented in closed form which generalise the Schwarzschild interior solution. Some of these solutions exhibit functional relations between the electric and gravitational potentials different to the quadratic one of Weyl. All the non-dust solutions are well-behaved and, by matching them to the Reissner-Nordstrom solution, all of the constants of integration are identified in terms of the total mass, total charge and radius of the source. This is done in detail for a number of specific examples. These are also shown to satisfy the weak and strong energy conditions and many other regularity and energy conditions that may be required of any physically reasonable matter distribution.
(with B. Nolan) Yang's gravitational theory, Gen. Rel. and Grav. 30 (1998) 473-495. Preprint
Yang's pure space equations (C.N. Yang, Phys. Rev. Lett. v.33, p.445 (1974)) generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.

Papers to appear

(with N. Georgiou) On the space of oriented geodesics of hyperbolic 3-space, Rocky Mountain J. Math. Preprint
We construct a Kaehler structure (J,W,G) on the space L(Hł) of oriented geodesics of hyperbolic 3-space and investigate its properties. We prove that (L(Hł),J) is biholomorphic to Pšx Pšminus the reflected diagonal, and that the Kaehler 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(Hł) is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesics of G correspond to ruled minimal surfaces in Hł, which are totally geodesic iff the geodesics are null.
(with W. Klingenberg) On Weingarten surfaces in Euclidean and Lorentzian 3-space, Differential Geom. Appl. Preprint
We study the neutral Kaehler metric on the space of time-like lines in Lorentzian, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric. In addition, we give a new characterisation of Weingarten surfaces in Euclidean 3-space and Lorentzian 3-space as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our construction to the classical Weierstrass representation of minimal and maximal surfaces.
(with H. Anciaux) On the three-dimensional Blaschke-Lebesgue problem, Proc. Amer. Math. Soc. Preprint
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 sets of fixed constant width and is still open in dimension greater than 2. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension 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).

Preprints

Einstein metrics adapted to contact structures on 3-manifolds, (2000) Preprint
The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian 3-manifolds. In particular it is shown that a contact 3-manifold admits an adapted riemannian metric if and only if it admits a metric with a divergence-free, constantly twisting, geodesic congruence. The shear of this congruence is identified with the torsion of the associated pseudohermitian structure, while the Tanaka-Webster curvature is identified with certain derivatives of the spin coefficients. The particular case where the associated riemannian metric is Einstein is studied in detail. It is found that the torsion is constant and the field equations are completely solved locally. Hyperbolic space forms are shown not to have adapted contact structures, even locally, while contact structures adapted to a flat or elliptic space form are contact isometric to the standard one.
(with N. Georgiou) A characterization of Weingarten surfaces in hyperbolic 3-space, (2007) Preprint
We study 2-dimensional submanifolds of the space L(Hł) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kaehler structure. Such a surface is Lagrangian iff there exists a surface in Hł orthogonal to the geodesics of S. We prove that the induced metric on a Lagrangian surface in L(Hł) has zero Gauss curvature iff the orthogonal surfaces in Hł are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in L(Hł) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in Hł.
(with M. Khalid and J. J. Ramon Mari) Lagrangian curves on spectral curves of monopoles, (2007) Preprint
We study Lagrangian points on smooth holomorphic curves in TPš equipped with a natural neutral Kaehler structure, and prove that they must form real curves. By virtue of the identification of TPš with the space L(Rł) of oriented affine lines in Euclidean 3-space, these Lagrangian curves give rise to ruled surfaces in Rł, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in Rł, called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in Rł where the number of oriented lines in the complex curve C that pass through the point is less than the degree of C. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.
(with H. Anciaux and P. Romon) Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface, (2008) Preprint
Given an oriented Riemannian surface (S,g), its tangent bundle TS enjoys a natural pseudo-Kaehler structure, that is the combination of a complex structure J, a pseudo-metric G with neutral signature and a symplectic structure W. We give a local classification of those surfaces of TS which are both Lagrangian with respect to W 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 Rł induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS˛ or TH˛ respectively. We relate the area of the congruence to a second-order functional on the original surface.
(with W. Klingenberg) Proof of the Caratheodory conjecture by mean curvature flow in the space of oriented affine lines, (2008) Preprint
We prove that the index of an isolated umbilic point on a Cł-smooth surface in Euclidean 3-space is less than or equal to one. As a corollary, we establish the Caratheodory conjecture, that the number of umbilic points on a closed convex surface in Rł must be greater than one. We do this by first reformulating the problem in terms of the index of an isolated complex point on a Lagrangian surface in TS˛, viewed as the space of oriented geodesics in Rł. The main step in the proof is to establish the existence of stable holomorphic discs with boundary contained on the Lagrangian surface enclosing the complex point. We first show that the existence of such discs implies that the Keller-Maslov index must be greater than or equal to one, which for topological reasons, places a bound on the index of the isolated complex point on the Lagrangian surface. To construct the holomorphic disc we utilize mean curvature flow with respect to the canonical neutral Kaehler metric on TS˛. We prove long-time existence of this flow by a priori estimates and show that, for small enough initial disc, the flowing disc is asymptotically holomorphic. Convergence to a bubbled holomorphic disc is then proven by a version of compactness for J-holomorphic discs with boundary contained in a totally real surface. Continuity up to the boundary assures that the Keller-Maslov index is retained in the limit and this establishes our main result.
(with N. Georgiou and W. Klingenberg) Totally null surfaces in neutral Kaehler 4-manifolds, (2008) Preprint
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 (beta-planes) and so we consider alpha-surfaces and beta-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the alpha-planes are integrable and alpha-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The beta-surfaces are less known and our interest is mainly in their description. In particular, we classify the beta-surfaces of the neutral Kaehler metric on TN, the tangent bundle to a Riemannian 2-manifold N. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the beta-surfaces are affine tangent bundles to curves of constant geodesic curvature on S˛ and H˛, respectively. In addition, we construct the beta-surfaces of the space of oriented geodesics of hyperbolic 3-space.
(with Dmitri V. Alekseevsky and Wilhelm Klingenberg) On the Geometry of Spaces of Oriented Geodesics, (2009) Preprint
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 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)Kaehler structure on L(M).