Regular projectively Anosov flows on three-dimensional manifolds

Abstract

We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of T 2 × I-models. We also apply our method to rigidity problems of some group actions.

Nonhomogeneous locally free actions of the affine group

“…As a corollary, if H 1 (M Γ ) is nontrivial, then M Γ admits a C ω nonhomogeneous action. 1 We can also classify actions on nonorientable manifolds by taking an orientable double cover.…”

“…There are two classical examples of locally free GA-actions on closed threedimensional manifolds. Let PSL(2, R) be the group of orientation preserving projective transformations of the real projective line RP 1 and PSL(2, R) its universal covering group. The group PSL(2, R) contains a closed subgroup H which is isomorphic to GA, e.g., the group of elements which fix ∞ ∈ RP 1 = R ∪ {∞}.…”

“…Theorem 1.5 ( [1]). Let M be a closed, connected, oriented, and three-dimensional manifold and ρ a C ∞ locally free action of GA on M .…”

On Anosovity, divergence and bi-contact surgery

“…It is worth mentioning that although projectively Anosov flows have been previously studied in various contexts, such as foliation theory [15,6,33,13], Riemannian geometry [9,10,34,27], hyperbolic dynamics [25,3,35,36] and Reeb dynamics [26], their primary significance for us is that they serve as bridge between Anosov dynamics and contact and symplectic geometry [32] (see Section 2.2), eventually yielding a complete characterization of Anosov flows in terms of such geometries [28]. We also remark that such flows are also called by different names in the literature, including conformally Anosov flows or flows with dominated splitting.…”

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

“…We remark that projectively Anosov flows are previously studied in various contexts, under different names. In the geometry and topology literature, beside projectively Anosov flows, they are referred to as conformally Anosov flows and are studied from the perspectives of foliation theory [20][42] [3], Riemannian geometry of contact structures [9][44] [35] and Reeb dynamics [34]. This is while, in the dynamical systems literature, the term conformally Anosov is preserved for another dynamical concept (for instance see [36][48] [13]) and the dynamical aspects of projectively Anosov flows are studied under the titles flows with dominated splitting (see [11] [46] [45][47] [2]) or eventually relatively pseudo hyperbolic flows [31].…”