A new class of quaternionic manifolds. (English) Zbl 0661.53019
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.
Reviewer: L.Ornea
MSC:
| 53B35 | Local differential geometry of Hermitian and Kählerian structures |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
Keywords:
nilmanifold; quaternionic manifold; \(L_ 1KQ\)-manifold; almost Kähler quaternionic manifolds; Heisenberg group; no Kählerian quaternionic structureReferences:
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