Almost Quaternionic Structures on Quaternionic Kaehler Manifolds

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• 作者：F. ?zdemir
• 关键词：Subbundle ; Almost complex structure ; Almost quaternionic structure ; Torsion tensor ; 53C55 ; 53C15
• 刊名：Bulletin of the Malaysian Mathematical Sciences Society
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：38
• 期：1
• 页码：1-13
• 全文大小：218 KB
• 参考文献：1.Yano, K., Ako, M.: Integrability conditions for almost quaternion structure. Hokaido Math J. 1, 63-6 (1972)CrossRef MATH MathSciNet
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  7.?zdemir, F., Crasmareanu, M.: Geometrical objects associated to a substructure. Turkish Jour. of Math. 35, 717-28 (2011)
  8.Ata, E., Yayl?, Y.: A global condition for the triviality of an almost split quaternionic structure on split complex manifolds. Int. J. Math. Sci. (WASET) 2(1), 47-1 (2008)MATH MathSciNet
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  11.Stong, R.E.: The Rank of an \(f\) -Structure. Kodai Math. Sem. Rep. 29, 207-09 (1977)CrossRef MATH MathSciNet
  
• 作者单位：F. ?zdemir (1)
  
  1. Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469, Maslak-Istanbul, Turkey
  
• 刊物类别：Mathematics, general; Applications of Mathematics;
• 刊物主题：Mathematics, general; Applications of Mathematics;
• 出版者：Springer Singapore
• ISSN：2180-4206

文摘

In this work, we consider a Riemannian manifold \(M\) with an almost quaternionic structure \(V\) defined by a three-dimensional subbundle of \((1,1)\) tensors \(F\), \(G\), and \(H\) such that \(\{F,G,H\}\) is chosen to be a local basis for \(V\). For such a manifold there exits a subbundle \(\mathcal{{H}} (M)\) of the bundle of orthonormal frames \(\mathcal{{O}}(M)\). If \(M\) admits a torsion-free connection reducible to a connection in \(\mathcal{{H}}(M)\), then we give a condition such that the torsion tensor of the bundle vanishes. We also prove that if \(M\) admits a torsion-free connection reducible to a connection in \(\mathcal{{H}}(M)\), then the tensors \(\widetilde{F}^2\), \(\widetilde{G}^2\), and \(\widetilde{H}^2\) are torsion-free, that is, they are integrable. Here \(\widetilde{F}\), \(\widetilde{G}\), \(\widetilde{H}\) are the extended tensors of \(F\), \(G\), and \(H\) defined on \(M\). Finally, we show that if the torsions of \(\widetilde{F}^2\), \(\widetilde{G}^2\) and \(\widetilde{H}^2\) vanish, then \(M\) admits a connection with torsion which is reducible to \(\mathcal{{H}}(M)\), and this means that \(\widetilde{F}^2\), \(\widetilde{G}^2\), and \(\widetilde{H}^2\) are integrable. Keywords Subbundle Almost complex structure Almost quaternionic structure Torsion tensor