Non-degenerate para-complex structures in 6D with large symmetry groups

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-Kähler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.