Hans-Bert Rademacher

Conformal vector fields on pseudo-Riemannian spaces

Abstract We study conformal vector fields on pseudo-Riemannian manifolds, in particular on Einstein spaces and on spaces of constant scalar curvature. A global classification theorem for conformal vector fields is obtained which are locally gradient fields. This includes the case of a positive metric as well as the case of an indefinite metric.

Existence of closed geodesics on positively curved Finsler manifolds

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

The second closed geodesic on Finsler spheres of dimension n > 2

We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n > 2. 2000 MSC classification: 53C22; 53C60; 58E10

Twistor spinors and gravitational instantons

We construct a complete Riemannian metric on the four-dimensional vector space ℝ4 which carries a two-dimensional space of twistor spinor with common zero point. This metric is half-conformally flat but not conformally flat. The construction uses a conformal completion at infinity of theEguchi-Hanson metric on the exterior of a closed ball in ℝ4.

On a generic property of geodesic flows

In this paper we are concerned with a sufficient condition for a Riemannian metric on a compact simply–connected manifold to have infinitely many geometrically distinct closed geodesics. 1969 Gromoll–Meyer proved in [GM] that for every Riemannian metric on a compact simply–connected manifold M there are infinitely many geometrically distinct closed geodesics if the sequence (bi(ΛM ;F ))i of Betti numbers of the free loop space ΛM is unbounded for some field F . Using the theory of minimal models Vigue-Poirrier/Sullivan proved that the rational cohomology algebra H∗(M ; I Q) of M is generated by a single element if and only if the sequence (bi(ΛM ; I Q))i of Betti numbers of the free loop space ΛM of M is bounded. It is a conjecture that the same statement holds for all fields of prime characteristic, partial results are due to McCleary–Ziller [MZ] and Halperin/Vigue-Poirrier [HV]. Now we turn to the manifolds for which the hypothesis of Gromoll– Meyer’s theorem does not hold, for example spheres and projective spaces. Then stability properties of the closed geodesics become important. A closed geodesic is hyperbolic if the linearized Poincare map has no eigenvalue of norm 1. From the bumpy metrics theorem due to Abraham [Ab] and Anosov [An2] and from a pertubation result due to Klingenberg–Takens [KT] one can conclude: A C4–generic metric on a compact manifold has either a non–hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic. In the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by Moser [Mo]. In the second case there are infinitely many closed geodesics if M is simply–connected due to results by Hingston [Hi] and the author [Ra1]. Hence it follows from these results that a C4–generic metric on a compact simply–connected manifold has infinitely many geometrically distinct closed geodesics. We remark that it is an open question whether there is a Riemannian metric on a simply–connected compact manifold all of whose closed geodesics are hyperbolic. In [Ra2] we show that the examples of metrics on the 2–

Conformal transformations of pseudo-Riemannian manifolds

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

Solitons of Discrete Curve Shortening

E

Asymptotically Euclidean Ends of Ricci Flat Manifolds, and Conformal Inversions

, endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised

AN EQUIVARIANT CW -COMPLEX FOR THE FREE LOOP SPACE OF A FINSLER MANIFOLD

G

On the average indices of closed geodesics

-structure and characterised by an