Non-degenerate Para-Complex Structures in 6D with Large Symmetry Groups

TL;DR

This paper classifies 6D almost product structures with non-degenerate Nijenhuis tensor, revealing maximal symmetry groups as $G_2^*$ and $sp(4,R)$, and establishing conditions for local homogeneity and transitivity.

Contribution

It determines the possible symmetry group dimensions for non-degenerate para-complex structures in 6D and characterizes the structures with maximal and submaximal symmetries.

Findings

Maximal symmetry group dimension is 14, realized by $G_2^*$.

Submaximal symmetry group dimension is 10, with Lie algebra $sp(4,R)$.

Structures with symmetry dimension ≥9 are locally homogeneous.

Abstract

For an almost product structure $J$ on a manifold $M$ of dimension $6$ with non-degenerate Nijenhuis tensor $N_J$, we show that the automorphism group $G=Aut(M,J)$ has dimension at most 14. In the case of equality $G$ is the exceptional Lie group $G_2^*$. The next possible symmetry dimension is proved to be equal to 10, and $G$ has Lie algebra $sp(4,R)$. Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-K\"ahler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.