Almost Complex Projective Structures and Their Morphisms

We discuss almost complex projective geometry and the relations to a distinguished class of curves. We present the geometry from the viewpoint of the theory of parabolic geometries and we shall specify the classical generalizations of the concept of the planarity of curves to this case. In particular, we show that the natural class of J-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving of this class turns out to be the necessary and sufficient condition on diffeomorphisms to become homomorphisms or anti-homomorphisms of almost complex projective geometries. 1. Almost complex projective structures An almost complex structure [8] is a smooth manifold equipped with a smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice-versa. The sufficient condition is vanishing of the Nijenhuis tensor (1) N(X,Y ) = [X,Y ] + J([JX, Y ] + [X, JY ])− [JX, JY ] . Definition 1.1. Let M be a smooth manifold of dimension 2n. An almost complex structure on M is a smooth trace-free affinor J in Γ(T M ⊗ TM) satisfying J2 = − idTM . For better understanding, we describe an almost complex structure at each tangent space in a fixed basis, i.e. with the help of real matrices: J = ( 0 −1 1 0 ) , J = J 0 . . . 0 .. .. 0 0 . . . J  , where J represents the multiplication by i = √ −1 on each tangent space. We can equivalently define an almost complex structure (M,J) as a reduction of the linear 2000 Mathematics Subject Classification: primary 53B10.