Nijenhuis tensors and Lie algebras (Q1362540)

One of the most intriguing problems in complex geometry is to investigate how to deform almost complex structures in order to obtain integrable ones. The failure of integrability is fully measured by the Nijenhuis tensor, so it would be interesting to characterize the class of such tensors. In [Riv. Mat. Univ. Parma, V. Ser. 3, 109-121 (1994; Zbl 0852.53032)], \textit{P. de Bartolomeis} defines a totally non-integrable almost complex structure. If \(J\) is an almost complex structure on a manifold \(M\) and \(N_J\) is its Nijenhuis tensor, then \(J\) is totally non-integrable if the tangent space \(T_pM\) at a point \(p\in M\) is spanned by \(\{N_J [p](X,Y);\;X,Y\in T_pM\}\). 2- and 4-dimensional almost complex manifolds are never totally non-integrable. In this paper, the authors study some related questions in the six-dimensional case. By constructing Nijenhuis tensors and special Lie algebras, the authors produce examples of 6-dimensional totally non-integrable manifolds.