Anosov Diffeomorphism - Anosov Flow On (tangent Bundles Of) Riemann Surfaces

Anosov Flow On (tangent Bundles Of) Riemann Surfaces

As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M=HΓ be a Riemann surface of negative curvature, and let T1M be the tangent bundle of unit-length vectors on the manifold M, and let T1H be the tangent bundle of unit-length vectors on H. Note that a bundle of unit-length vectors on a surface is the principal bundle of a complex line bundle.

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Anosov Diffeomorphism - Anosov Flow On (tangent Bundles Of) Riemann Surfaces - Geometric Interpretation of The Anosov Flow
... When acting on the point z=i of the upper half-plane, corresponds to a geodesic on the upper half plane, passing through the point z=i ... The action is the standard Möbius transformation action of SL(2,R) on the upper half-plane, so that A general geodesic is given by with a, b, c and d real, with ad-bc=1 ...

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