Additional Topics in Complex Geometry

Abstract

Before turning to our main topics we first discuss partially hyperbolic diffeomorphisms and holomorphic Anosov flows as introduced by Etienne Ghys [1995]. In Section 13.2 we discuss the geometry of the projectivized holomorphic tangent and cotangent bundles. The study of the projectivized holomorphic tangent bundle naturally raises the question of a complex geodesic flow, which we discuss in Section 13.3. In Section 13.4 we return to the projectivized holomorphic tangent bundle and develop its complex almost contact metric structure. In Section 13.5 we first discuss special directions on complex contact manifolds analogous to our treatment in the real case in Chapter 11 and then discuss complex contact structures on the Lie group SL(2, \( C\!\!\!\!C \)) in detail.