The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive the second Plebański equation and its geometry. We do not use Kähler or other ...

We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless ...

If a closed manifold M possesses two Riemannian metrics which have the same unparameterized geodesics and are not strictly proportional at each point, then the topological entropy of both geodesic flows is zero. This is the main result of the paper and it has many dynamical and topological corollaries. In particular, such a manifoldM should be finitely covered by the product of a rationally ...

We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions.

We define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. As an application we deduce normal forms of 1-jets of almost complex structures along a pseudoholomorphic submanifold. In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the ...