We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familiar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.
It is proved in the paper that near every pseudoholomorphic disk on an almost complex manifold a disk of almost the same size in any close direction passes. As an application the Kobayashi-Royden pseudonorm for almost complex manifolds is defined and studied.
We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy.
We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The result is new even in the complex case.
We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conjecture does not exceed fifteen and give criteria for rigidity of 3-webs.
We construct the simplest chaotic system with a two-point attractor.