Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenbergalgebra, we also describe the geometric structure of the manifolds if they are compact.
Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation, curvatures and holonomy. Especially, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.
Contents
1. Lie groups acting isometrically on Lorentzian manifolds
1.1. Definitions and basic properties
1.2. Examples
1.2.1. Product with a compact Riemannian manifold
1.2.2. Two-dimensional affine algebra
1.2.3. Special linear algebra
1.2.4. Heisenberg algebra
1.2.5. Twisted Heisenberg algebras
1.3. Induced bilinear form on the Lie algebra
2. Main theorems
2.1. Algebraic theorems
2.2. Geometric theorems
2.3. Theorems in the homogeneous case
3. Algebraic classification of the Lie algebras
3.1. Symmetric bilinear forms on Lie algebras
3.2. Nilradical
3.3. Radical
3.4. Compact radical: case of the special linear algebra
3.5. Non-compact radical
3.5.1. Form not positive semidefinite: case of the twisted Heisenberg algebra
3.5.2. Form positive semidefinite
3.6. General subgroups of the isometry group
3.6.1. Trivial case
3.6.2. Case of the affine algebra
3.6.3. Case of the Heisenberg algebra
3.6.4. Case of the twisted Heisenberg algebra
3.6.5. Case of the special linear algebra
4. Geometric characterization of the manifolds
4.1. Induced bilinear form is positive semidefinite
4.2. Locally free action
4.3. Induced bilinear form is indefinite
4.3.1. Lorentzian character of orbits
4.3.2. Orthogonal distribution
4.3.3. Structure of the manifold
4.3.4. Lorentzian metrics on the twisted Heisenberg group
5. Compact homogeneous Lorentzian manifolds
5.1. Structure of homogeneous manifolds
5.2. General reductive representation
5.3. Geometry of homogeneous manifolds
5.3.1. Curvature and holonomy of homogeneous semi-Riemannian manifolds
5.3.2. Isometry group contains a cover of the projective special linear group
5.3.3. Isometry group contains a twisted Heisenberg group
5.3.4. Isotropy representation and Ricci-flat manifolds
Objectives and Topics
The primary objective of this thesis is to classify Lie groups that act isometrically and locally effectively on Lorentzian manifolds of finite volume. Furthermore, the research investigates the structure and local geometry of compact homogeneous Lorentzian manifolds where the isometry groups exhibit non-compact connected components, providing a comprehensive analysis of their symmetry and geometric properties.
- Algebraic classification of Lie groups acting on Lorentzian manifolds.
- Investigation of compact homogeneous Lorentzian manifolds with non-compact isometry groups.
- Development of a canonical bilinear form on Lie algebras of isometry groups.
- Geometric characterization of manifolds based on the nature of their isometry groups.
- Analysis of curvature and holonomy in compact homogeneous Lorentzian spaces.
Excerpt from the Book
Introduction
The aim of this thesis is a classification of Lie groups acting isometrically and locally effectively on Lorentzian manifolds of finite volume as well as providing a detailed investigation of compact homogeneous Lorentzian manifolds whose isometry groups have non-compact connected components.
Symmetries play an important role in geometry and other disciplines. In 1872, Klein proposed in his Erlangen program a way how to characterize the classical geometries (for example, Euclidean geometry and Hyperbolic geometry) by using the underlying group of symmetries. In addition, symmetries are fundamental in theoretical physics, where they are related to conserved quantities. Symmetries are also relevant in architecture and biology.
In semi-Riemannian geometry, symmetries correspond to isometries. The study of isometry groups is of great interest, regarding the structure of these groups as well as the way they act on the manifolds. Semi-Riemannian manifolds may allow only one isometry, but they are easier to investigate if the isometry groups are large, in the sense that they are acting transitively on the manifolds. In this case, we speak about homogeneous manifolds. A homogeneous manifold is very symmetric, that is, the manifold looks the same regardless from which point on the manifold it is viewed. Classical examples for homogeneous spaces are the Euclidean and Hyperbolic space.
Summary of Chapters
Lie groups acting isometrically on Lorentzian manifolds: This chapter introduces fundamental definitions, provides Lie algebraic results concerning isometric actions, and defines the crucial induced bilinear form κ.
Main theorems: This section presents the core results regarding the algebraic classification of Lie groups acting on manifolds and the geometric characterization of compact Lorentzian manifolds.
Algebraic classification of the Lie algebras: This chapter provides the detailed proof for the classification of Lie algebras based on the properties of the bilinear form κ and the analysis of radical and nilradical structures.
Geometric characterization of the manifolds: This part focuses on the proofs for the geometric classification, including the analysis of orbits and the structure of the manifolds under investigation.
Compact homogeneous Lorentzian manifolds: This chapter presents an in-depth analysis of compact homogeneous spaces, specifically detailing their geometry, curvature, and holonomy representations.
Keywords
Lorentzian manifolds, Isometry groups, Lie algebras, Homogeneous manifolds, Semi-Riemannian geometry, Twisted Heisenberg groups, Killing vector fields, Compact manifolds, Bilinear form, Ricci-flat, Holonomy, Curvature, Reductive representation, Jordan decomposition.
Frequently Asked Questions
What is the primary scope of this thesis?
The work focuses on classifying Lie groups that act isometrically and locally effectively on Lorentzian manifolds of finite volume, specifically targeting those with non-compact connected isometry groups.
Which mathematical disciplines are utilized?
The thesis heavily relies on differential geometry, Lie group theory, Lie algebra classification, and dynamical systems to characterize the geometric and algebraic properties of the manifolds.
What is the significance of the bilinear form κ?
The bilinear form κ, defined on the Lie algebra of the isometry group, is the central tool used to prove the classification theorems and to establish non-degeneracy conditions for the Lie groups involved.
How is the algebraic classification performed?
The classification is achieved by decomposing the Lie algebra of the isometry group into a direct sum of a compact semisimple part, an abelian part, and a specific "s" part (trivial, affine, Heisenberg, twisted Heisenberg, or sl2-algebra).
What is the role of the twisted Heisenberg algebra in this context?
Twisted Heisenberg algebras emerge as one of the fundamental types of Lie algebras that can act isometrically on compact Lorentzian manifolds, and the work provides a detailed geometric characterization of the manifolds associated with them.
What is the main finding regarding Ricci-flat manifolds?
The research concludes that the isometry group of any Ricci-flat compact homogeneous Lorentzian manifold, which is not flat, must be compact.
How are orbits used to characterize the geometry?
The thesis utilizes the orbits of the action of the subgroups generated by the "s" component of the Lie algebra to establish the Lorentzian character of the manifolds and define foliations for their structural analysis.
What does the "special" homogeneous space imply?
A homogeneous space is defined as "special" in this context when a specific curvature-related map V vanishes, which leads to a simplified metric product representation of the manifold.
- Arbeit zitieren
- Felix Günther (Autor:in), 2011, Isometry groups of Lorentzian manifolds of finite volume and the local geometry of compact homogeneous Lorentz spaces, München, GRIN Verlag, https://www.grin.com/document/179225
