A new class of quaternionic manifolds
Let M be a quaternionic manifold with 4-form \(\Omega\) and almost complex structures I, J, K. Define a differential 1-form on M by \[ L_ 1(\nabla \Omega)=\sum^{4n}_{r,s}\nabla_{E_ r}(\Omega)(IE_ r,JE_ s,KE_ s,X). \] A new class of quaternionic manifolds called \(L_ 1KQ\) is introduced by the condition \(L_ 1(\nabla \Omega)=0\). The structure group of an \(L_ 1KQ\)-manifold is Sp(n); conformal but non-homothetic changes of the metric do not preserve this class. Let AKQ denote the class of almost Kähler quaternionic manifolds; the inclusion \(AKQ\subseteq L_ 1KQ\) is shown. The example \(M^{4r+4}=H\times R^{4r+1},\) where H is the Heisenberg group, constructed in the final part of the paper, shows that the inclusion is strict. Moreover it is proved that \(M^{4r+4}\) has no Kählerian quaternionic structure.
scientific article; zbMATH DE number 4080305
scientific article