Thurston Geometries: Get the Facts on Left-Invariant Metrics

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In summary, Thurston geometries are a type of non-Euclidean geometry that are characterized by their left-invariant metrics. These metrics are defined by the properties they share with the Lie group, a continuous group of symmetries. Thurston geometries have been extensively studied and classified by mathematician William Thurston, and are important in understanding the various types of spaces and their geometric properties.
  • #1
Reverie
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Hi,

Every Thurston Geometry (X,Isom(X)) with the exception of the geometry modeled on S^2xR can be achieved as a 3D Lie group with a left-invariant metric. That is, the space X=G, where G is a 3D Lie group with a left-invariant metric. After picking a left-invariant frame field consisting of three left-invariant vector-fields. The metric is determined by 6 constants.

This is true since all left-invariant metrics on Lie groups are completely determined by the metric at the identity. Since the metrics are in 1-1 correspondence with symmetric matices that don't have 0 as an eigenvalue, this metric is determined by (n+1)n/2 constants. In the 3D case, the metric is completely determined by 6 constants.

This metric must be such that Isom(X) is maximal. That is, Isom(X) achieves its biggest(in the sense of containment) value with the appropriate choice of these 6 constants. Does anyone know the value of these 6 constants for each of the 7 geometries?

I would like to know what the value of these 42 constants(depend on the choice of the chosen left-invariant vectors on each of the seven Lie groups) because they contain the answer to the ultimate question of life, the universe, and everything(Hitchhiker's Guide to the Galaxy) . These left-invariant metrics are the steady state solutions of Ricci flow, and I believe they correspond to Einstein metrics. The Thurston Geometries are used quite frequently in Cosmology and Quantum Theories of Gravity on curved spacetime.

Anyone know the answer?

-Reverie
 
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  • #2
Note to moderators... this is actually a serious question... The reference to Adams' book was for amusement... but the rest is serious... after all... The Answer to The Ultimate Question Of Life, the Universe and Everything...and... I'm really interested in the answer... o:)
 
  • #3
It appears to be essentially worked out in Scott's paper. I'll post a bit more about it when I've worked it out.
 
  • #4
Just noticed this. Your terminology seems somewhat non-standard, but if I understand you correctly, your question is similar to asking for representations of the Lie groups corresponding to the Bianchi classification of three-dimensional Lie algebras as matrix groups, and you are asking for the left and right invariant metrics on each of these matrix groups. If so, I can tell you that if you haven't already worked it out (see Flanders, Differential Forms with Applications to the Physical Sciences for all you need to do this). As you probably know, there is not a perfect overlap between the Bianchi groups and the Thurston geometries, but I can give you left and right invariant metrics for these also.
 

1. What are Thurston geometries?

Thurston geometries are a classification of three-dimensional spaces based on their curvature properties. They were first studied by mathematician William Thurston in the late 1970s and early 1980s.

2. What is a left-invariant metric?

A left-invariant metric is a type of metric (a mathematical concept that measures distance) that remains unchanged when the space is rotated or translated in a specific direction. In the context of Thurston geometries, left-invariant metrics are used to study the curvature of these spaces.

3. How are Thurston geometries useful?

Thurston geometries have proved to be useful in various areas of mathematics, including topology, geometry, and dynamics. They have also been applied in physics and computer science. Understanding the properties of these spaces can lead to insights and applications in these fields.

4. What are the different types of Thurston geometries?

There are eight types of Thurston geometries: Euclidean, hyperbolic, spherical, Nil, Sol, SL(2,R), H^2 x R, and H^3. Each type has its own distinct curvature properties and can be visualized as a specific shape or surface.

5. How are Thurston geometries related to the Poincaré Conjecture?

The Poincaré Conjecture, one of the most famous unsolved problems in mathematics, was proven by Grigori Perelman using techniques and ideas from Thurston geometries. Thurston's work on understanding and classifying three-dimensional spaces played a crucial role in the eventual solution of the conjecture.

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