Contact geometric theory of Anosov flows in dimension three and related topics
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This thesis consists of the author's work on the contact and symplectic geometric theory of Anosov flows in low dimensions, as well as the related topics from Riemannian geometry. This includes the study of the interplay between various geometric, topological and dynamical features of such flows. After reviewing some basic elements from the theory of contact and symplectic structures in low dimensions, we discuss a characterization of Anosov flows on three dimensional manifolds, purely in terms of those geometric structure. This is based on the previous observations of Mitsumatsu and Eliashberg-Thurston in the mid 90s, and in the context of a larger class of dynamics, namely projectively Anosov flows. Our improvement of those observations, which have been left unexplored to a great extent in our view, facilitates employing new geometric tools to the study of questions about (projectively) Anosov flows and vice versa. We then discuss another characterization of Anosov three flows, in terms of the associated underlying Reeb dynamics. Beside the contact topological consequences of this result, it sheds light on contact geometric interpretation of the existence of an invariant volume form for these flows, a condition which is well known to have deep consequences in the dynamics of the flow from the viewpoint of the long term behavior of the flow (transitivity) and measure theory (ergodicity). The implications of these results on various related theories, namely, Liouville geometry, the theory of contact hyperbolas and bi-contact surgery, are discussed as well. As contact Anosov flows are an important and well studied special case of volume preserving Anosov flows, we also make new observation regarding these flows, utilizing the associated Conley-Zehnder indices of their periodic orbits, a classical tool from the field of contact dynamics. We finally discuss some Riemannian geometric motivations in the study of contact Anosov flows in dimension three. In particular, this bridges our study to the curvature properties of Riemannian structures, which are compatible with a given contact manifold. Our study of the curvature in this context goes beyond the study of Anosov dynamics, although has implications on the topic. In particular, we investigate a natural curvature realization for compatible Riemannian structures, namely Ricci-Reeb realization problem. The majority of the results in this manuscript, with the exception of some parts of Chapter 5, can be found in the author's previous papers.