Flows on 2-dimensional Manifolds: An Overview

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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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  4. 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.

    C3.3 Differentiable Manifolds (2016-2017)

    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.

    What is a manifold?

    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.

    Download Flows On 2 Dimensional Manifolds An Overview

    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.

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