Arnold's Seminar March 2, 1999 Boris Kruglikov "Nijenhuis tensors in pseudoholomorphic curves neighborhoods" Abstract: Nijenhuis tensor is the unique obstruction to integrability of almost complex structures. In dimension 4 it has very simple geometric reformulation in terms of distributions ets. This allows to connect to pseudoholomorphic curves in 4-dim almost complex manifolds some simple geometrical objects such as foliations of these curves. In complex case Arnold proved that in nonresonant situations a neighborhood of a holomorphic torus is equivalent to a neighborhood of the zero section of its normal bundle. In the almost complex case this normal bundle is not holomorphic but is only almost holomorphic. We describe the structure of these bundles and discuss the equivalence problem. The answer can be formulated in terms of the distributions of the Nijenhuis tensor. We also describe normal forms of 1-jets of almost complex structures on a curve in a 4-dim almost complex manifold. Small neighborhood of pseudoholomorphic torus almost never contains other pseudoholomorphic tori. But it is possible to find many pseudoholomorphic cylinders, which are analogs of the cylinders {r=const} in Arnold's coordinates. These results are similar to the ones of Moser/Bangert. We also discuss the restrictions imposed on Nijenhuis tensor distributions by the condition of complex monodromy and transports of a pseudoholomorphic foliation of a neighborhood.