Research

  • (with W. Klingenberg) Isolated umbilical points on surfaces in R, Bull. Greek Math. Soc. 51 (2006) 23-30. Preprint
    Recent advances in the application of line congruence techniques to surfaces in R are used to generate surfaces with isolated umbilical points of all indices less than or equal to 1.

  • (with W. Klingenberg) Generalised surfaces in R, Math. Proc. R. Ir. Acad. 104A (2004) 199-209. Preprint
    The correspondence between 2-parameter families of oriented lines in R and surfaces in TP is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of P are investigated and a number of theorems are proven that generalise results for closed convex surfaces in R.

  • (with W. Klingenberg) On the space of oriented affine lines in R, 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 R and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in R, 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 S, 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.

  • 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 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).