- #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
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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