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-Nordström 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.

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R 3 with the tangent bundle of S 2 . Thus, the round metric on S 2 induces a Kähler structure on T which turns out to have a metric of neutral signature. It is shown that the isometry group of this metric is isomorphic to the isometry group of the Euclidean metric on R 3 .The geodesics of this metric are either planes or helicoids in R 3 . The signature of the metric induced on a surface Σ in T is determined by the degree of twisting of the associated line congruence in R 3 , and we show that, for Σ Lagrangian, 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 Σ.

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space other than OH 2 , 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).

We study area-stationary, or maximal, surfaces in the space L(H 3 ) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L(H 3 ) is a maximal surface. We then classify Lagrangian maximal surfaces Σ in L(H 3 ) and prove that the family of parallel surfaces in H 3 orthogonal to the geodesics γ ∈ Σ form a family of equidistant tubes around a geodesic.

The correspondence between 2-parameter families of oriented lines in R 3 and surfaces in T P 1 is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of T P 1 are investigated and a number of theorems are proven that generalise results for closed convex surfaces in R 3 . Σ Kdµ = 4π