Anosov diffeomorphism

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Defines:
Anosov flow
Type of Math Object:
Definition
Major Section:
Reference

Mathematics Subject Classification

37D20 no label found

Comments

Infranil manifolds & Riemann surfaces

I have in front of me a textbook example of an Anosov flow; its the spliting of the tangent manifold of the upper half plane (or of a Riemann surface of negative curvature) into three parts: the geodesic flow plus two horocycle flows, one expanding, the other contracting. I'm having some trouble identifying how this is an infranil manifold. I guess the group structure is rotations O(2) semi-direct translations; and te finite group must be the covering group. So I guess that means every tangent bundle to a Riemann surface is an infranil manifold ?? It'd be nice for this article to clarify. --linas

Re: Infranil manifolds & Riemann surfaces

The problem is that you're looking at an Anosov *flow*. The conjecture is that the only manifolds supporting Anosov *diffeomorphisms* are nilmanifolds or infranilmanifolds. Anosov flows are much more common; in fact the example you mention is a particular case of the more general fact that in any riemannian manifold with negative sectional curvature, the geodesic flow is in fact an Anosov flow.

I will add a comment on that.