**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 *T*1*M* be the tangent bundle of unit-length vectors on the manifold *M*, and let *T*1*H* 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

### Other related articles:

... 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 ...

### Famous quotes containing the words surfaces and/or flow:

“Footnotes are the finer-suckered *surfaces* that allow tentacular paragraphs to hold fast to the wider reality of the library.”

—Nicholson Baker (b. 1957)

“Only the *flow* matters; live and let live, love and let love. There is no point to love.”

—D.H. (David Herbert)