Book contents
- Frontmatter
- Contents
- List of Illustrations
- Preface
- 1 Hyperbolic space and its isometries
- 2 Discrete groups
- 3 Properties of hyperbolic manifolds
- 4 Algebraic and geometric convergence
- 5 Deformation spaces and the ends of manifolds
- 6 Hyperbolization
- 7 Line geometry
- 8 Right hexagons and hyperbolic trigonometry
- Bibliography
- Index
Preface
Published online by Cambridge University Press: 02 December 2009
- Frontmatter
- Contents
- List of Illustrations
- Preface
- 1 Hyperbolic space and its isometries
- 2 Discrete groups
- 3 Properties of hyperbolic manifolds
- 4 Algebraic and geometric convergence
- 5 Deformation spaces and the ends of manifolds
- 6 Hyperbolization
- 7 Line geometry
- 8 Right hexagons and hyperbolic trigonometry
- Bibliography
- Index
Summary
To a topologist a teacup is the same as a bagel, but they are not the same to a geometer. By analogy, it is one thing to know the topology of a 3-manifold, another thing entirely to know its geometry—to find its shortest curves and their lengths, to make constructions with polyhedra, etc. In a word, we want to do geometry in the manifold just like we do geometry in euclidean space.
But do general 3-manifolds have “natural” metrics? For a start we might wonder when they carry one of the standards: the euclidean, spherical or hyperbolic metric. The latter is least known and not often taught; in the stream of mathematics it has always been something of an outlier. However it turns out that it is a big mistake to just ignore it! We now know that the interior of “ most” compact 3-manifolds carry a hyperbolic metric.
It is the purpose of this book to explain the geometry of hyperbolic manifolds. We will examine both the existence theory and the structure theory.
Why embark on such a study? Well after all, we do live in three dimensions; our brains are specifically wired to see well in space. It seems perfectly reasonable if not compelling to respond to the challenge of understanding the range of possibilities. In particular, it is not at all established that our own universe is euclidean space, as many so like to believe.
I will briefly summarize the recent history of our subject.
- Type
- Chapter
- Information
- Outer CirclesAn Introduction to Hyperbolic 3-Manifolds, pp. xiii - xviiiPublisher: Cambridge University PressPrint publication year: 2007