Tangent and normal bundles in almost complex geometry

In this paper we define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. These are intrinsically related to the Gromov D-operator. As an application we deduce normal forms of 1-jets of almost complex structures along a submanifold. In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the question of non-deformation and persistence of pseudoholomorphic curves.Comment: 25 pages; More detailed relations between normal bundles structures are added. Links with other works on the topic - mostly almost complex bundles structures - are developpe

Involutivity of field equations

We prove involutivity of Einstein, Einstein-Maxwell and other field equations by calculating the Spencer cohomology of these systems. Relation with Cartan method is traced in details. Basic implications through Cartan-Kahler theory are derived.Comment: 13 pages; this version is updated with new field equations (radiation, dust etc) - they are proved involutive, Spencer cohomology calculate

Nijenhuis tensors in pseudoholomorphic curves neighborhoods

Normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve are considered. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudoholomorphic tori we investigate the problem of pseudoholomorphic foliation of the neighborhood. We obtain some results on nonexistence of the tori deformation.Comment: 27 page

Examples of integrable sub-Riemannian geodesic flows

Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples are obtained as "holonomization" of sub-Riemannian ones. A feature of non-holonomic situation is non-compactness of the phase space. We also exhibit a Liouvulle-integrable Hamiltonian system with topological entropy of all integrals positive.Comment: 21 pages; Answer to the self-posed question is added: Is it possible to construct Liouville-integrable Hamiltonian system with positive topological entropies of all integrals? Yes and we present an exampl

Invariants and submanifolds in almost complex geometry

In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of higher-dimensional pseudoholomorphic submanifolds