Helga Baum

Humboldt-Universität zu Berlin | HU Berlin · Department of Mathematics

This review is based on lectures given by the authors during the Summer School Geometric Flows and the Geometry of Space-Time at the University of Hamburg, September 19–23, 2016. In the first part we describe the algebraic classification of connected Lorentzian holonomy groups. In particular, we specify the holonomy groups of locally indecomposable...

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzi...

Zu den grundlegenden Objekten, die in der Eichfeldtheorie auftreten, gehören Gruppen mit differenzierbarer Struktur. Im ersten Kapitel werden wir einige grundlegende Eigenschaften und Aussagen über solche Gruppen behandeln, die wir später benötigen werden.

In this survey we review the state of art in Lorentzian holonomy theory. We explain the recently completed classification of connected Lorentzian holonomy groups, we describe local and global metrics with special Lorentzian holonomy and some topological properties, and we discuss the holonomy groups of Lorentzian manifolds with parallel spinors as...

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinuous g...

In this section we give a short introduction to Cartan connections and define their holonomy groups. In particular, we explain the relation to holonomy groups of principal fibre bundle connections and to holonomy groups of covariant derivatives in associated vector bundles. Details can be found in [KN63], [Sh97] and [Ba09].

In diesem Abschnitt definieren wir die grundlegenden Objekte, mit denen wir uns in diesem Buch beschäftigen wollen, die lokal-trivialen Faserbündel.

In diesem Kapitel beschreiben wir Lie-Gruppen, die durch Parallelverschiebungen in G-Hauptfaserbündeln mit gegebenem Zusammenhang entlang geschlossener Wege entstehen. Es wird sich zeigen, dass dadurch die ‘kleinste’ Gruppe entsteht, auf die man die Strukturgruppe des Hauptfaserbündels reduzieren kann, ohne die durch den Zusammenhang gegebene Diffe...

Charakteristische Klassen sind Kohomologieklassen, die man Hauptfaser- und Vektorbündeln zuordnet. Man kann sie benutzen, um nachzuweisen, dass zwei Bündel nicht isomorph sind. Insbesondere messen sie den Grad der Nichttrivialität eines Bündels. Charakteristische Klassen trifft man an vielen Stellen in der Geometrie, Topologie und Analysis. Sie bes...

In diesem Abschnitt wollen wir das sogenannte Yang-Mills-Funktional auf dem Raum der Zusammenhangsformen C(P) eines G-Hauptfaserbündels P über einer semi-Riemannschen Mannigfaltigkeit (M,g) näher studieren. Dieses Funktional ist durch das Integral über die Länge der Krümmungsform definiert: L : A Î C(P) ® òM || FA ||2 dMg .L : A \in \cal{C}(P) \lo...

In diesem Kapitel werden wir die notwendigen Begriffe und Konzepte für die Differentialrechnung auf Hauptfaserbündeln und ihren assoziierten Vektorbündeln bereitstellen. Der zentrale Begriff dafür ist der eines Zusammenhanges im Hauptfaserbündel, der es uns ermöglicht, horizontale Richtungen im Totalraum des Bündels auszuzeichnen und dadurch Schnit...

Wir wollen nun die allgemeine Holonomietheorie, die wir im Kapitel 4 behandelt haben, auf den speziellen Fall des Levi-Civita-Zusammenhangs Riemannscher und pseudo-Riemannscher Mannigfaltigkeiten anwenden. Im Anhang findet der Leser eine Zusammenstellung der wichtigsten Begriffe aus der Riemannschen Geometrie, die wir in diesem Abschnitt voraussetz...

This paper is a survey of recent results about conformal Killing spinors in Lorentzian geometry based on a lecture given during the Summer Program Symmetries and Overdetermined Systems of Partial Differential Equations at IMA, Minnesota, 17.07.06 – 04.08.06. In particular, we will focus on a special class of geometries admitting conformai Killing s...

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface for any weakly irreducible holonomy representation with parallel spinors, t.m. with a holonomy group \(G < imes \mathbb{R}^{n-2} \subset SO(1,n-1)\) ,...

This survey intends to introduce the reader to holonomy theory of Cartan con-nections. Special attention is given to the normal conformal Cartan connection, uniquely defined for a class of conformally equivalent metrics, and to its holonomy group -the 'conformal holonomy group'. We explain the relation between confor-mal holonomy group and existenc...

Es wird die Twistorgleichung auf Lorentz-Spin-Mannigfaltigkeiten untersucht. Bekanntermaßen existieren Lösungen der Twistorgleichung auf den pp-Mannigfaltigkeiten, den Lorentz-Einstein-Sasaki Mannigfaltigkeiten und den Fefferman-Räumen. Es wird gezeigt, dass in den kleinen Dimensionen 3,4 und 5 Twistor-Spinoren ohne 'Singularitäten' nur für diese g...

In the present paper we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariant metric of Lorentzian signature (1,n−1), of signature (2,n−2) and of signature (p,q), where...

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Killing fields. Furthermore, we derive all geometries with singularity free twistor spinors that occur up to dimension 6.

This paper is a survey on special geometric structures that admit conformal Killing spinors based on lectures, given at the "Workshop on Special Geometric Structures in String Theory", Bonn, September 2001 and at ESI, Wien, November 2001. We discuss the case of Lorentzian signature and explain which geometries occur up to dimension 6.

We calculate the zeta-invariant of Dirac operators D 2 A coupled to instantons A. 1 Introduction Let (M n ; g) be a closed Riemannian spin manifold and denote by S the spinor bundle of (M n ; g). Furthermore, let P be a G-principal bundle on M , % : G ! GL(V ) a faithful unitary representation and E := P Theta % V the associated vector bundle. Each...

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there ar...

: We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties. Keywords: Twistor equation, Twistor spinors, Lorentzian manifolds, CR-geometry, Fefferman spaces. MS classification: 58G30, 53C50, 53A50. 1 Introduction In the pre...

: The paper deals with twistor spinors on Lorentzian manifolds. In particular, we explain a relation between a certain class of Lorentzian twistor spinors and the Fefferman spaces of strictly pseudoconvex spin manifolds which appear in CR-geometry. Keywords: Twistor equation, Twistor spinors, Lorentzian manifolds, CRgeometry, Fefferman spaces. MS c...

Let M n be an n-dimensional oriented Riemannian manifold with a fixed spin structure. We understand the spin structure as a reduction P of the SO(n)-principal bundle of M n to the universal covering Spin(n) SO(n) (n ≥ 3) of the special orthogonal group.

We prove a sharp lower bound for the first positive eigenvalue of Dirac operators coupled to instantons and discuss the limit case. 1 Introduction Let (M n ; g) be a closed, connected Riemannian spin manifold and let S denote the spinor bundle of (M; g). Let G be a compact Lie group, P a G-principal bundle on M n and % : G ! GL(V ) an unitary repre...

It is proved that the eigenvalues of the Dirac operator on even dimensional space-- and time--oriented pseudo--Riemannian spin manifolds lie symmetric to the real and to the imaginary axes, and that the Dirac operator does not have residual spectrum if the associated Riemannian metric is complete. 1 Introduction Let (M n;k ; g) be a space-- and tim...

We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties. Contents 1 Introduction 1 2 Algebraic prelimeries 3 3 Lorentzian twistor spinors 5 4 Pseudo-hermitian geometry 11 5 Fefferman spaces 18 6 Spinor calculus for S 1 -bu...

We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors. 1 Introduction In [Wa 89] McKenzie Y. Wang described the possible holonomy groups of complete simply connected irreducible non-flat Riemannian spin manifolds M n which admit parallel spinors....

: In this paper we give a review on normally hyperbolic operators of Huygens type. The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Gunther. Key words: Hyperbolic operators, Huygens' principle, conformal geometry MSC 1991: 58G 16 Contents 1. Introduction : : : : : : : : : : : : : : : : : :...

We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.

We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.

We consider the Dirac operator D of a Lorentzian spin manifold of even dimension n ≥ 4. We prove that the square D2 of the Dirac operator on plane wave manifolds and the shifted operator D2 − K on Lorentzian space forms of constant sectional curvature K are of Huygens type. Furthermore, we study the Huygens property for coupled Dirac operators on f...

Without Abstract

Let (P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let DA be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A(u) of A with tangent vector ω and denote Tω:= (DA(u)−f) (u=0. The coefficients of the asymptotic ex...

This paper is a survey of recent results concerning twistor and Killing spinors on Lorentzian manifolds based on lectures given at CIRM, Luminy, in June 1999, and at ESI, Wien, in October 1999. After some basic facts about twistor spinors we explain a relation between Lorentzian twistor spinors with lightlike Dirac current and the Fefferman spaces...