On the Curvature of Metric Contact Pairs

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• 作者：Gianluca Bande (1)
  David E. Blair (2)
  Amine Hadjar (3)
  
• 关键词：Primary 53C25 ; Secondary53B20 ; 53C12 ; 53B35 ; Contact pairs ; metric contact geometry ; foliations ; Vaisman manifolds
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2013
• 出版时间：May 2013
• 年：2013
• 卷：10
• 期：2
• 页码：989-1009
• 全文大小：328KB
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• 作者单位：Gianluca Bande (1)
  David E. Blair (2)
  Amine Hadjar (3)
  
  1. Dipartimento di Matematica e Informatica, Università degli studi di Cagliari, Via Ospedale 72, 09124, Cagliari, Italy
  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824-027, USA
  3. Laboratoire de Mathématiques Informatique et Applications, Université de Haute Alsace -4, Rue des Frères Lumière, 68093, Mulhouse Cedex, France
  
• ISSN：1660-5454

文摘

We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-K?hler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.