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E Algebraic Groups 111 E.1 Algebraic Group Actions Holonomy Groups of Flat Pseudo-Riemannian Homogeneous Manifolds Zur Erlangung des wissenschaftlichen Grades eines Doktors der Naturwissenschaften von der Fakultät für Mathematik des Karlsruher Instituts für Technologie genehmigte Dissertation von Dipl.Inform.Wolfgang Globke Tag der mündlichen Prüfung: 29. Juni 2011 Referent: HDoz.Dr.Oliver Baues Korreferent: Prof.Dr.Enrico Leuzinger Version of June 11, 2011. iii Acknowledgments This thesis grew during my time as a research assistant at the workgroup for metric geometry under the supervision of HDoz.Dr.Oliver Baues. During the last year of intense work on this thesis he took great care in inspecting and discussing my results (literally to the last minute). While doing so, he unrelentingly pressed for rigour and clarity in my work, and this certainly helped me improve my thesis both in style and in substance. When discussing mathematics, Oliver has a striking ability to ask the right questions (and give many right answers), and his questions repeatedly inspired me to investigate new aspects of my subject. On a personal level, we had a very friendly working relationship and the time spent working at the institute was quite enjoyable. Without a doubt I can say that I am happy to have Oliver as my Doktorvater. Prof.Dr.Enrico Leuzinger, my thesis’ second referee, chairs the workgroup for metric geometry. In this position he grants his staff the freedom to pursue their scientific interests. Whenever necessary, he takes some time to discuss questions and offer helpful advice. Prof.Joseph A. Wolf kindly provided a first draft of the revised chapter on flat pseudo-Riemannian homogeneous spaces from the 6th edition of his book on spaces of constant curvature. This draft inspired me to search for a proof that the linear holonomy of a Wolf group should always be abelian, which eventually lead to the discovery of examples with non-abelian linear holonomy, from which the thesis developed on. Jessica Hoffmann has been a dear and supportive friend for many years and also provided a non-mathematician’s view on my thesis’ introduction. That my time working at the metric geometry workgoup was most enjoyable is not in small part owed to my former and current colleagues, Slavyana Geninska,Gabi Link, Andreas Weber,Hannes Riesterer and Sebastian Grensing. They also kindly proofread an earlier version of my thesis and gave valuable remarks to improve the exposition. Before my thesis got off the ground, it was Klaus Spitzmuller¨ who repeatedly reminded me to stay focused on my work and not let me get distracted by my teaching duties. Our workgroup secretary Anne-Marie Vacchi- ani’s friendly and helpful manner greatly eased the burden of the uninevitable administrative duties to be performed. Finally, Sarah Vanessa Schlafmutze¨ “provided” food and shelter throughout the work on this thesis. My thanks to all of you! Karlsruhe, June 2011 Wolfgang Globke iv v Notation and Conventions Throughout this text, we use the following notation: The pseudo-Euclidean space Rn endowed with an indefinite inner • product of signature r; s (where n = r + s) is denoted by Rr;s. In n;0 0;n n 1;1 1;n 1 particular, R R is the Euclidean space, and R − R − is the Minkowski space. Unless stated otherwise, the vectors e1;:::; en denote the canonical • basis of Rn. Elements of Rn are represented by column vectors. To save space, • these columns will sometimes also be written as n-tuples without further remark. The kernel and the image of a linear map A are denoted by ker A and • im A, respectively. Groups will be denoted by boldface letters, G, and Lie algebras will • be denoted by German letters, g. The Lie algebra of a Lie group G is denoted by Lie(G). The neutral element of an abstract group G is denoted by 1G or 1. For • matrix groups, we also write I or I for the n n-identity matrix. n × The action of a group element g on elements x of some set is denoted • by g:x. The one-dimensional additive and multiplicative groups are denoted • by G+ and G×, respectively. By a mild abuse of language, when we speak of the Zariski closure of • a group G Aff(Rn), we shall always mean the real Zariski closure, that is, the ⊂R-points of its complex Zariski closure. The differential of a smooth map f is denoted by f or d f . • ∗ Different parts of a proposition or a theorem are labeled by (a), (b), • etc., and the parts of the proof referring to these are labeled the same way. Different steps in the proof of one statemend are labeled by small Roman numerals (i), (ii), etc. See also the appendices for some standard notations. vi Contents vii Contents Notation v Introduction xi I Flat Pseudo-Riemannian Homogeneous Spaces 1 1 Isometries of Flat Pseudo-Riemannian Homogeneous Spaces 1 1.1 Flat Pseudo-Riemannian Manifolds . .1 1.2 Killing Fields and the Development Representation . .3 1.3 Wolf Groups . .4 2 Representations of Wolf Groups 9 2.1 Some Bookkeeping . .9 2.2 The Matrix Representation . 12 2.3 Translation Parts . 15 2.4 Criteria for Fixed Points . 16 2.5 Dimension Bounds . 22 3 The Centraliser 25 3.1 Algebraic Properties . 25 3.2 Matrix Representation of the Centraliser . 28 3.3 Remarks on Translationally Isotropic Domains . 29 4 Compact Flat Homogeneous Spaces 33 4.1 Compact Flat Pseudo-Riemannian Homogeneous Spaces . 33 4.2 Lie Algebras with Bi-Invariant Metric . 35 5 Orbits of Wolf Groups 37 5.1 The Orbits . 37 5.2 An Algebraic Principal Bundle . 37 5.3 The Affine and Metric Structure on the Orbits . 40 viii Contents 5.4 Pseudo-Riemannian Submersions . 46 6 The Lorentz Case and Low Dimensions 49 6.1 Riemann and Lorentz Metrics . 49 6.2 Generalities on Abelian Wolf Groups . 50 6.3 Signature (n 2; 2) ........................ 51 − 6.4 Dimension 5........................... 55 ≤ 6.5 Dimension 6 . 56 II Main Examples 63 7 Miscellanea 63 7.1 On Open Orbits . 63 7.2 Formulae . 64 7.3 A Criterion for Properness . 65 8 Abelian Holonomy (3,5) 69 9 Compact (3,3) 71 9.1 A Nilpotent Lie Group with Flat Bi-Invariant Metric . 71 9.2 A Lattice in G ........................... 72 9.3 The Development Representation of G ............. 72 9.4 The Linear Holonomy Group . 74 10 Non-Abelian Holonomy, Incomplete (4,4) 77 10.1 The Group Generators . 77 10.2 The Centraliser . 78 10.3 The Open Orbit of the Centraliser . 80 10.4 The Complement of the Open Orbit . 82 11 Non-Abelian Holonomy, Complete (7,7) 87 11.1 The Group Generators . 87 11.2 The Centraliser . 88 Contents ix 11.3 The Orbits of Γ ........................... 89 11.4 A Global Slice . 91 12 Non-Degenerate Orbits, Complete (7,7) 93 12.1 The Group Generators . 93 12.2 The Centraliser . 94 12.3 The Tensor S ............................ 94 III Appendix 97 A Pseudo-Euclidean Spaces and their Isometries 97 A.1 Isotropic Subspaces . 97 A.2 Pseudo-Euclidean Isometries . 99 BAffine Manifolds 103 B.1 Affine Transformations . 103 B.2 Affine Vector Fields . 103 C Pseudo-Riemannian Manifolds 105 C.1 Pseudo-Riemannian Isometries . 105 C.2 Killing Fields . 105 C.3 Bi-Invariant Metrics . 106 D Discrete Groups and Proper Actions 109 D.1 Proper Definition of Proper Action . 109 E Algebraic Groups 111 E.1 Algebraic Group Actions . 111 E.2 Algebraic Homogeneous Spaces . 114 F Cechˇ Cohomology and Fibre Bundles 115 F.1 Cechˇ Cohomology . 115 F.2 Fibre Bundles and Principal Bundles . 116 x Contents G Unipotent Groups 119 G.1 Unipotent Groups as Lie Groups . 119 G.2 Unipotent Groups as Algebraic Groups . 119 G.3 Malcev Coordinates . 120 G.4 Lattices in Unipotent Groups . 122 G.5 Homogeneous Spaces of Unipotent Groups . 123 G.6 Algebraic Principal Bundles for Unipotent Groups . 124 References 128 Index 132 xi Introduction In flatness is the preservation of the world. So seek the Wolf in thyself! –Metallica Pseudo-Riemannian manifolds are among the most important objects in geometry, and they are of particular importance for modern mathematical physics. When developing structure theories for certain classes of pseudo- Riemannian manifolds, one studies their symmetries. By this we mean the isometry group of a manifold, and its fundamental group in particular. In this thesis, we study the structure of flat pseudo-Riemannian homogeneous spaces via their affine holonomy groups. For complete manifolds they coincide with the fundamental group. Incomplete manifolds are harder to understand, and here the affine holonomy group can be a more tangible homomorphic image of the fundamental group. Setting the Stage Non-degenerate symmetric bilinear forms appear naturally in many bran- ches of mathematics and physics. In geometry, the positive definite forms define Euclidean geometry in Rn. The indefinite forms define more general geometries, of which Lorentz geometry for signature (n 1; 1) is the most prominent. In physics, Lorentz geometry appears as− the geometry of special relativity. Going from Rn to smooth manifolds, one studies pseudo-Riemannian manifolds M endowed with a pseudo-Riemanian metric ; , a field of h· ·i non-degenerate symmetric bilinear forms ; p on the tangent spaces TpM. Unsurprisingly, the Riemannian case, whereh· ·i the metric is positive definite, is the most studied and best understood of these. In theoretical physics, Riemannian geometry is the language of classical mechanics. Modern theories like general relativity, gravitation and cosmology are built on pseudo-Riemannian geometry. There are significant differences between positive definite and indefinite metrics: The famous theorem by Hopf and Rinow states that for a Rie- mannian manifold M, geodesic completeness is equivalent to metric com- pleteness (meaning any Cauchy sequence converges).
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