Abstract: In this talk, I will discuss invariants of almost complex manifolds, among which the rank of the Nijenhuis tensor. I will focus on parallelizable manifolds, studying the local behaviour of such rank along curves of almost complex structures, and describing how to produce structures whose Nijenhuis tensor has prescribed rank. I will complete the discussion classifying 6-nilmanifolds admitting a left-invariant almost complex structure of given rank, underlining how the topology of the manifold imposes constraints on the values of the rank. The theory will be complemented with several examples. At the end of the talk, I shall briefly recall certain invariants of complex manifolds and present the possibility to define their generalization to almost complex manifolds.
The talk is based on the preprint arxiv.org:2211.08340 and on work in progress joint with Prof. Adriano Tomassini.