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=H\Γ 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.
Read more about this topic: Anosov Diffeomorphism
Famous quotes containing the words flow and/or surfaces:
“Only the flow matters; live and let live, love and let love. There is no point to love.”
—D.H. (David Herbert)
“But ice-crunching and loud gum-chewing, together with drumming on tables, and whistling the same tune seventy times in succession, because they indicate an indifference on the part of the perpetrator to the rest of the world in general, are not only registered on the delicate surfaces of the brain but eat little holes in it until it finally collapses or blows up.”
—Robert Benchley (1889–1945)