Almost complex structures in 6D with nondegenerate Nijenhuis tensors and large symmetry groups

TL;DR

This paper classifies almost complex structures in 6D with nondegenerate Nijenhuis tensors, identifying the sub-maximal symmetry group dimension as 10 and characterizing the geometric structures that realize this symmetry.

Contribution

It establishes the sub-maximal automorphism group dimension for such structures and characterizes the specific homogeneous spaces that attain this dimension.

Findings

Maximal automorphism group dimension is 14, corresponding to G_2.

Sub-maximal dimension is 10, realized by three specific spaces.

All structures with automorphism group dimension 9 have semi-simple isotropy.

Abstract

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$, i.e. the largest realizable dimension that is less than 14, is $\dim G=10$. Next we prove that only 3 spaces realize this, and all of them are strictly nearly (pseudo-) K\"ahler and globally homogeneous. Moreover, we show that all examples with $\dim Aut(J)=9$ have semi-simple isotropy.