It addresses graduate students and researchers and serves as a reference book for experts in the field.
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Du kanske gillar. Lifespan David Sinclair Inbunden. Spara som favorit. Skickas inom vardagar. Time-evolution in low-dimensional topological spaces is a subject of puzzling vitality. This book is a state-of-the-art account, covering classical and new results.
A -manifold is complete if the developing map is surjective. Definition 1. A model geometry is a smooth manifold together with a Lie group of diffeomorphisms of , such that:.
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A 3-manifold is said to be a geometric manifold if it is a -manifold for a 3-dimensional model geometry. Theorem 2.
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There are eight 3-dimensional model geometries:. Let be the connected component of the identity of , and let be the stabiliser of.
Case Then has constant sectional curvature. The Cartan Theorem implies that up to rescaling is isometric to one of. Let be the -invariant vector field such that, for each , the direction of is the rotation axis of.
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In our setting this implies that the flow of acts by isometries. Hence the flowlines define a 1-dimensional foliation with embedded leaves. The quotient is a 2-dimensional manifold, which inherits a Riemannian metric such that acts transitively by isometries. Thus has constant curvature and is up to rescaling isometric to one of.
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Case 2a: is a foliation. Thus is a flat bundle over. Case 2b: is a contact structure.
For one would obtain for the group of isometries of that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case.
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For one obtains. Namely, is the subgroup of the group of automorphisms of the standard contact structure on consisting of those automorphisms which are lifts of isometries of the x-y-plane. Then is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is.