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The local structure of trans-Sasakian manifolds. (English) Zbl 0772.53036

A trans-Sasakian structure is, in some sense, an analogue of a locally conformal Kähler structure on an almost Hermitian manifold. Two remarkable subclasses of trans-Sasakian structures are those called \({\mathcal C}_ 5\)- and \({\mathcal C}_ 6\)-structures, which contain the Kenmotsu and Sasakian structures respectively. In this paper the author has completely characterized the local nature of trans-Sasakian structures on differentiable manifolds of dimension \(\geq 5\). This has been done through two stages: 1) characterizing the local nature of \({\mathcal C}_ 5\)- and \({\mathcal C}_ 6\)-structures; 2) showing that a trans- Sasakian structure is either of class \({\mathcal C}_ 5\) or of class \({\mathcal C}_ 6\). The author has finally obtained some examples of 3-dimensional trans-Sasakian manifolds which are neither of class \({\mathcal C}_ 5\) nor of class \({\mathcal C}_ 6\).
Reviewer: N.L.Youssef (Giza)

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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