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...

In this article, we consider the almost Hermitian structure on TM induced by a pair of a metric and an affine connection on M . We find the conditions under which TM admits almost Kähler structures, Kähler structures and Einstein metrics, respectively. Moreover, we give two examples of Kähler-Einstein structures on TM . 2000 Mathematics Subject Classification: 53C55, 53C15, 53C25.

This paper studies groups of symplectomorphisms of ruled surfaces M for symplectic forms with varying cohomology class. This cohomology class is characterised by the ratio μ of the size of the base to that of the fiber. By considering appropriate spaces of almost complex structures, we investigate how the topological type of these groups changes as μ increases. If the base is a sphere, this cha...

Gray & Hervella gave a classification of almost Hermitian structures (g, I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g, I, J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyperKäher or locally conformal hyper...

We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spinc quantization. We prove the analog of Kodaira vanishing for the Spinc Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Lap...

We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the ChernMoser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense “maximally CR”. One of the main difference...

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.