Wilhelm Klingenberg

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).

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

PARABOLIC CLASSICAL CURVATURE FLOWS

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 neutral Kähler surface with applications in geometric optics

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.

Geodesic Flow on the Normal Congruence of a Minimal Surface

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we show decouples to highest order. We determine the conditions for this system to be parabolic and investigate the lower order terms. The zeroth order terms are shown to form a Hamiltonian system, which is therefore completely integrable. We find conditions on parabolic curvature flows that ensure boundedness of various geometric quantities and investigate some examples, including powers of mean curvature flow, Gauss curvature flow and mean radius of curvature flow, as well as the non-homogeneous Bloore flow. As a new tool we introduce the Radii of Curvature diagram of a surface and its canonical hyperbolic metric. The relationship between these and the properties of Weingarten surfaces are also discussed.

FLOW OF REAL HYPERSURFACES BY THE TRACE OF THE LEVI FORM

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2+1, in which there is a rich interplay with symplectic geometry.This is a survey about conformal mappings between pseudo-Riemannian manifolds and, in particular, conformal vector fields defined on such. Mathematics Subject Classification (2000). Primary 53C50; Secondary 53A30; 83C20.We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle

Generalised Surfaces in R 3

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