Examples of compact K-contact manifolds with no Sasakian metric

Cappelletti–Montano, Beniamino; Nicola, Antonio De; Marrero, Juan Carlos; Yudin, Ivan

doi:10.1142/s0219887814600287

Cited by 17 publications

(17 citation statements)

References 15 publications

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“…We also remark that the examples of compact non-simply-connected K-contact non-Sasakian manifolds of dimensions 5 and 7 which are nilmanifolds have already been found by Cappelletti-Montano, De Nicola, Marrero, and Yudin in the paper [3].…”

Section: Theoremsupporting

confidence: 56%

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Kim

2015

Int. J. Geom. Methods Mod. Phys.

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The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space CP 2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

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Section: Theoremsupporting

confidence: 56%

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Kim

2015

Int. J. Geom. Methods Mod. Phys.

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“…Then, it is clear that the space of orbits of the Lee vector field of M is N . Thus, using Corollary 6.2 and taking as N the examples of non-Lefschetz compact contact manifolds considered in [4], we obtain examples of compact l.c.s. manifolds of the first kind which satisfy the following conditions:…”

Section: (Basic Lefschetz ⇒ Lefschetz)mentioning

confidence: 99%

Hard Lefschetz theorem for Vaisman manifolds

Cappelletti–Montano

Nicola

Marrero

et al. 2018

Trans. Amer. Math. Soc.

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We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual 1-form coincides with the Lee 1-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics. 2010 Mathematics Subject Classification. Primary 53C25, 53C55, 53D35. (J.C.M.) and by Prin 2010/11 -Varietà reali e complesse: geometria, topologia e analisi armonica -Italy (B.C.M.), and by the exploratory research project in the frame of Programa Investigador FCT IF/00016/2013 (I.Y.). J.C.M. acknowledges the Centre for Mathematics of the University of Coimbra in Portugal for its support and hospitality in a visit where a part of this work was done.

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“…To prove that they use the properties of the cohomology ring which can be drived from the Hard Lefschetz Theorem, cf. [4].…”

Section: Fiω "ωmentioning

confidence: 99%

Sasakian structures a foliated approach

Wolak

2017

Demonstratio Mathematica

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Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The paper presents a foliated approach to Sasakian manifolds on which the author gave several lectures. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and Kählerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds.

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Examples of compact K-contact manifolds with no Sasakian metric

Abstract: Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively

Cited by 17 publications

References 15 publications

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

Hard Lefschetz theorem for Vaisman manifolds

Sasakian structures a foliated approach

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