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In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1). Although this definition ''includes'' hyperkähler manifolds, these are often excluded from the definition of a quaternion-Kähler manifold by imposing the condition that the scalar curvature is nonzero, or that the holonomy group is equal to Sp(''n'')·Sp(1). The definition introduced by Edmond Bonan〔.〕 in 1965, uses a 3-dimensional subbundle ''H'' of End(T''M'') of endomorphisms of the tangent bundle to a Riemannian ''M'', that in 1976 Stefano Marchiafava and Giuliano Romani called ''I fibrato di Bonan'' . For ''M'' to be quaternion-Kähler, ''H'' should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions which act on T''M'' preserving the metric. Simultaneously, in 1965, Edmond Bonan and Vivian Yoh Kraines〔.〕 constructed the parallel 4-form. It was not until 1982 that Edmond Bonan proved an outstanding result : the analogue of hard Lefschetz theorem 〔 . * * * * *. *. *. * Edmond Bonan, ''Isomorphismes sur une variété presque hermitienne quaternionique'', Proc. of the Meeting on Quaternionique Structures in Math.and Physics SISSA , Trieste, (1994), 1-6. 〕 for compact Sp(''n'')·Sp(1)-manifold. == Ricci curvature == Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies as the only manifolds of special holonomy with non-zero Ricci curvature. In fact, these manifolds are Einstein.If an Einstein constant of a quaternion-Kähler manifold is zero, it is hyperkähler. This case is often excluded from the definition. That is, quaternion-Kähler is defined as one with holonomy reduced to Sp(''n'')·Sp(1) and with non-zero Ricci curvature (which is constant). Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quaternion-Kähler manifold」の詳細全文を読む スポンサード リンク
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