Geometric 3-manifolds
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== Construction and examples == | == Construction and examples == | ||
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− | {{beginthm|Theorem}} | + | {{beginthm|Theorem}}There are eight 3-dimensional model geometries: |
- the round sphere: $X=S^3, G=O(4)$ | - the round sphere: $X=S^3, G=O(4)$ | ||
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- the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$ | - the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$ | ||
− | - the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$. | + | - the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.{{endthm}} |
</wikitex> | </wikitex> | ||
Revision as of 15:38, 8 June 2010
Contents |
1 Introduction
Let a group act on a manifold by homeomorphisms.
A -manifold is a manifold with a -atlas, that is, a collection of homeomorphismsIf we change the initial data and , the developing map changes by composition with an element of .
If , analytic continuation along a loop representing gives a chart that is comparable to , since they are both defined at . Let be the element of such that . The mapis a group homomorphism and is called the holonomy of .
If we change the initial data and , the holonomy homomorphisms changes by conjugation with an element of .
A -manifold is complete if the developing map is surjective.
[Thurston1997] Section 3.4
Definition 1.1. A model geometry is a smooth manifold together with a Lie group of diffeomorphisms of , such that:
a) is connected and simply connected;
b) acts transitively on , with compact point stabilizers;
c) is not contained in any larger group of diffeomorphisms of with compact point stabilizers;
d) there exists at least one compact -manifold.
[Thurston1997] Definition 3.8.1
A 3-manifold is said to be a geometric manifold if it is a -manifold for a 3-dimensional model geometry .
2 Construction and examples
Theorem 2.1.There are eight 3-dimensional model geometries:
- the round sphere:
- Euclidean space:
- hyperbolic space:
-
-
- the universal covering of the unit tangent bundle of the hyperbolic plane:
- the Heisenberg group:
- the 3-dimensional solvable Lie group with conjugation .3 Invariants
...
4 Classification/Characterization
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5 Further discussion
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6 References
- [Thurston1997] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997. MR1435975 (97m:57016) Zbl 0873.57001
This page has not been refereed. The information given here might be incomplete or provisional. |