Nonholonomic Lorentzian Geometry on Some ℍ-Type Groups

TLDR: In this article, the authors considered ℍ-type Carnot groups whose noncommutative multiplication law gives rise to a smooth 2-step bracket generating distribution of the tangent bundle.

Abstract: We consider examples of ℍ-type Carnot groups whose noncommutative multiplication law gives rise to a smooth 2-step bracket generating distribution of the tangent bundle. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1, instead of a positive definite quadratic form. The causal character is defined. We study the reachable set by timelike future directed curves. The parametric equations of geodesics are obtained.

Figures

Figure 2.4: The domain reachable from the origin by timelike geodesics Rt

Figure 2.3: The graphs of functions μ̃(τ) and μ(τ)

Figure 2.2: The graph of projection of timelike geodesic on (x, y)-plane

Figure 2.6: Graphics of logarithmic spirals for α+ iβ and α− iβ respectively.

Figure 2.5: The graph of function μ̄(τ)

Figure 2.1: The graph of timelike geodesic in R3 passing through the origin

Full text

Sub-semi-Riemannian geometry
on Heisenberg-type groups
ANNA KOROLKO
Dissertation for the degree of Philosophiae Doctor (Ph.D.)
Department of Mathematics
University of Bergen
October 2010

Preface
The structure of the thesis is as follows:
Chapter 1. We give an abbreviated review covering the history and applications of sub-
Riemannian geometry and semi-Riemannian geometry, since those are the areas we make a start
from. We introduce also H(eisenberg)-type groups with a natural left-invariant Riemannian
metric. We provide basic definitions and facts from designated research areas and give a short
overview of our contribution to the subject.
Chapter 2. With the above tools in hands, we are prepared to turn to the main results of
the thesis. We introduce principal notions in sub-semi-Riemannian geometry based on ideas of
sub-Riemannian and semi-Riemannian geometries, emphasizing the difference between them.
We construct various examples of H-type groups with left-invariant semi-Riemannian metrics
and describe their geodesics, our main object of interest. We start from the Heisenberg group
with sub-Lorentzian metric and then, in order to exhibit more features of sub-semi-Riemannian
geometry, we pass to higher dimensional examples. Further, we present general H-type groups
with nondegenerate metric of an arbitrary index and the detailed description of geodesics. We
finish the survey with a summary and a list of open questions related to sub-semi-Riemannian
geometry.
Chapter 3. We include five papers, two of which are published, one accepted, one submit-
ted and one is in preparation. The last article is somewhat independent of the rest of the thesis.
It concerns numerical integration on the discrete nonholonomic systems on sub-Riemannian
Heisenberg-type groups.

Acknowledgements
First of all, I wish to thank my principal supervisor Professor Irina Markina for her invaluable
scientific and non-scientific guidance, enormous patience and enthusiasm, and for always sparing time
and energy for discussions with me. At the same time I am very grateful to my co-supervisor Professor
Alexander Vasiliev for all the care and warm support I received from him.
I want to thank my colleagues: Mauricio Godoy, Georgy Ivanov and Pavel Gumenyuk for their
responsive attention and for the lavishly friendly atmosphere in the office. In particular, I am indebted
to Mauricio Godoy who provided me valuable assistance during this project. I thank also Erlend Grong
for his occasional illuminations. The help and friendship of Martin Stolz is also very much appreciated.
Many thanks go to all members of the analysis group in University of Bergen for interesting collaboration
and nice environment.
I would also like to thank Professor Robert McLachlan for the warm hospitality during my 2-months
visit to his research group at Massey University in New Zealand.
I am grateful to the University of Bergen for the financial support. Special thanks to my former
supervisor Professor Sergey Vodopyanov for his positive encouragement to apply for this Ph.D. position.
Thanks to helpful administrative staff of the mathematical institute and to everyone else who in some
way participated in my work under the thesis.
Finally, I express my deep gratitude to my family for their kind support and constant interest in
the process even from the long distance, to my dear Andreas Sandvin for his love and making my life
very happy, and to his warm-hearted family who have all been very supportive during these years.
Anna Korolko
Bergen, October 2010

Contents
1 Introduction 7
1.1 History and motivation ............................... 7
1.1.1 Sub-Riemannian geometry and its applications .............. 7
1.1.2 Semi-Riemannian Geometry ......................... 8
1.1.3 Heisenberg-type groups ........................... 8
1.1.4 Motivation .................................. 8
1.2 Overview of the thesis ................................ 9
1.3 Necessary background ................................ 10
2 Presentation of main results 13
2.1 Semi-Riemannian geometry with constraint .................... 13
2.2 Sub-Lorentzian Heisenberg group .......................... 15
2.3 Sub-Lorentzian Quaternion group and its physical interpretation ........ 18
2.4 Sub-semi-Riemannian Quaternion group with the metric of index 2 ....... 20
2.5 General sub-semi-Riemannian Heisenberg-type group ............... 21
2.6 Summary and open problems ............................ 24
3 Papers A-E 29
3.1 PaperA........................................ 31
3.2 PaperB ........................................ 59
3.3 PaperC........................................ 89
3.1 PaperD........................................ 121
3.1 PaperE ........................................ 147