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Holonomy Groups of Flat Pseudo-Riemannian Homogeneous Manifolds [Elektronische Ressource] / Wolfgang Globke. Betreuer: O. Baues

156 pages
Holonomy GroupsofFlat Pseudo-RiemannianHomogeneous ManifoldsZur Erlangung des wissenschaftlichen Grades einesDoktorsder Naturwissenschaftenvon der Fakultät für Mathematik desKarlsruher Instituts für TechnologiegenehmigteDissertationvonDipl. Inform. Wolfgang GlobkeTag der mündlichen Prüfung: 29. Juni 2011Referent: HDoz. Dr. Oliver BauesKorreferent: Prof. Dr. Enrico LeuzingerVersion of July 13, 2011.iiiAcknowledgmentsThis thesis grew during my time as a research assistant at the workgroup for metricgeometry under the supervision of HDoz. Dr. Oliver Baues. During the lastyear of intense work on this thesis he took great care in inspecting and discussingmy results (literally to the last minute). While doing so, he unrelentingly pressedfor rigour and clarity in my work, and this certainly helped me improve mythesis both in style and in substance. When discussing mathematics, Oliver has astriking ability to ask the right questions (and give many right answers), and hisquestions repeatedly inspired me to investigate new aspects of my subject. On apersonal level, we had a very friendly working relationship and the time spentworking at the institute was quite enjoyable. Without a doubt I can say that I amhappy to have Oliver as my Doktorvater.Prof. Dr. Enrico Leuzinger, my thesis’ second referee, chairs the workgroup formetric geometry. In this position he grants his sta the freedom to pursue theirscientific interests.
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Holonomy Groupsof
Flat Pseudo-Riemannian
Homogeneous Manifolds
Zur Erlangung des wissenschaftlichen Grades eines
Doktorsder 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 LeuzingerVersion of July 13, 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 sta the freedom to pursue their
scientific interests. Whenever necessary, he takes some time to discuss questions
and oer 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 o 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 Globkeivv
Notation and Conventions
Throughout this text, we use the following notation:
n The pseudo-Euclidean space endowed with an indefinite inner
r;sproduct of signature r; s (where n = r + s) is denoted by . In
n;0 0;n n 1;1 1;n 1particular, is the Euclidean space, and is
the Minkowski space.
Unless stated otherwise, the vectors e ;:::; e denote the canonical1 n
nbasis of .
n Elements of 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 groupG is byLie(G).
The neutral element of an abstract groupG is denoted by 1 or 1. ForG
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
na groupG A( ), we shall always mean the real closure,
that is, the -points of its complex Zariski closure.
The dierential of a smooth map f is denoted by f or d f .
Dierent parts of a proposition or a theorem are labeled by (a), (b),
etc., and the parts of the proof referring to these are the same
way. Dierent steps in the proof of one statemend are labeled by
small Roman numerals (i), (ii), etc.
See also the appendices for some standard notations.
RRRRRRRRRRviContents 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 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 Ane and Metric Structure on the Orbits . . . . . . . . . 40viii 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 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 inG . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.3 The Development Representation ofG . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 88Contents ix
11.3 The Orbits of . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 TensorS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
III Appendix 97
A Pseudo-Euclidean Spaces and their Isometries 97
A.1 Isotropic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Isometries . . . . . . . . . . . . . . . . . . 99
B Ane Manifolds 103
B.1 Ane Transformations . . . . . . . . . . . . . . . . . . . . . . 103
B.2 Ane Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 103
C Pseudo-Riemannian Manifolds 105
C.1 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 Homogeneous Spaces . . . . . . . . . . . . . . . . 114
ˇF Cech Cohomology and Fibre Bundles 115
ˇF.1 Cech . . . . . . . . . . . . . . . . . . . . . . . . 115
F.2 Fibre Bundles and Principal Bundles . . . . . . . . . . . . . . 116x Contents
G Unipotent Groups 119
G.1 Unipotent Groups as Lie Groups . . . . . . . . . . . . . . . . 119
G.2 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 133