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A non-Sasakian Lefschetz $ K$-contact manifold of Tievsky type


Authors: Beniamino Cappelletti-Montano, Antonio De Nicola, Juan Carlos Marrero and Ivan Yudin
Journal: Proc. Amer. Math. Soc. 144 (2016), 5341-5350
MSC (2010): Primary 53C25, 53D35
DOI: https://doi.org/10.1090/proc/13187
Published electronically: June 3, 2016
MathSciNet review: 3556276
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Abstract | References | Similar Articles | Additional Information

Abstract: We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of $ K$-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.


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  • [1] L. Auslander, L. Green, and F. Hahn, Flows on homogeneous spaces, Annals of Mathematics Studies No. 53, Princeton University Press, Princeton, N.J. 1963.
  • [2] Chal Benson and Carolyn S. Gordon, Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc. 108 (1990), no. 4, 971-980. MR 993739, https://doi.org/10.2307/2047955
  • [3] Indranil Biswas, Marisa Fernández, Vicente Muñoz, and Aleksy Tralle, On formality of Sasakian manifolds, J. Topol. 9 (2016), no. 1, 161-180. MR 3465845, https://doi.org/10.1112/jtopol/jtv044
  • [4] Charles P. Boyer and Krzysztof Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. MR 2382957
  • [5] Beniamino Cappelletti-Montano, Antonio De Nicola, Juan Carlos Marrero, and Ivan Yudin, Sasakian nilmanifolds, Int. Math. Res. Not. IMRN 15 (2015), 6648-6660. MR 3384493, https://doi.org/10.1093/imrn/rnu144
  • [6] Beniamino Cappelletti-Montano, Antonio De Nicola, Juan Carlos Marrero, and Ivan Yudin, Examples of compact $ K$-contact manifolds with no Sasakian metric, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 9, 1460028, 10. MR 3270291, https://doi.org/10.1142/S0219887814600287
  • [7] Beniamino Cappelletti-Montano, Antonio De Nicola, and Ivan Yudin, Hard Lefschetz theorem for Sasakian manifolds, J. Differential Geom. 101 (2015), no. 1, 47-66. MR 3356069
  • [8] Xiaoyang Chen, On the fundamental groups of compact Sasakian manifolds, Math. Res. Lett. 20 (2013), no. 1, 27-39. MR 3126719, https://doi.org/10.4310/MRL.2013.v20.n1.a3
  • [9] Marisa Fernández and Alfred Gray, Compact symplectic solvmanifolds not admitting complex structures, Geom. Dedicata 34 (1990), no. 3, 295-299. MR 1066580, https://doi.org/10.1007/BF00181691
  • [10] Tamehiro Fujitani, Complex-valued differential forms on normal contact Riemannian manifolds, Tôhoku Math. J. (2) 18 (1966), 349-361. MR 0212730
  • [11] Keizo Hasegawa, Complex and Kähler structures on compact solvmanifolds, J. Symplectic Geom. 3 (2005), no. 4, 749-767. Conference on Symplectic Topology. MR 2235860
  • [12] Akio Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289-331 (1960). MR 0124918
  • [13] Bogusław Hajduk and Aleksy Tralle, On simply connected $ K$-contact non-Sasakian manifolds, J. Fixed Point Theory Appl. 16 (2014), no. 1-2, 229-241. MR 3346752, https://doi.org/10.1007/s11784-015-0210-y
  • [14] H. Kasuya, Cohomologies of Sasakian groups and Sasakian solvmanifolds, Ann. Mat. Pura Appl., first published online: November 7, 2015. doi:10.1007/s10231-015-0543-6.
  • [15] Y. Lin, Lefschetz contact manifolds and odd dimensional symplectic geometry, Preprint arXiv:1311.1431.
  • [16] V. Muñoz, J. A. Rojo, and A. Tralle, Homology Smale-Barden manifolds with K-contact and Sasakian structures, Preprint arXiv:1601.06136.
  • [17] Vicente Muñoz and Aleksy Tralle, Simply connected K-contact and Sasakian manifolds of dimension 7, Math. Z. 281 (2015), no. 1-2, 457-470. MR 3384880, https://doi.org/10.1007/s00209-015-1494-8
  • [18] Agustí Roig and Martintxo Saralegi-Aranguren, Minimal models for non-free circle actions, Illinois J. Math. 44 (2000), no. 4, 784-820. MR 1804319
  • [19] Masahiko Saito, Sur certains groupes de Lie résolubles. II, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 7 (1957), 157-168 (French). MR 0097463
  • [20] A. M. Tievsky, Analogues of Kähler geometry on Sasakian manifolds, Ph.D. Thesis, Massachusetts Institute of Technology, 2008. Available in http://dspace.mit.edu/
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Additional Information

Beniamino Cappelletti-Montano
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Email: b.cappellettimontano@gmail.com

Antonio De Nicola
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Email: antondenicola@gmail.com

Juan Carlos Marrero
Affiliation: Unidad Asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica” Departamento de Matemáticas, Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de La Laguna, La Laguna, Tenerife, Spain
Email: jcmarrer@ull.edu.es

Ivan Yudin
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Email: yudin@mat.uc.pt

DOI: https://doi.org/10.1090/proc/13187
Received by editor(s): October 22, 2015
Received by editor(s) in revised form: February 15, 2016
Published electronically: June 3, 2016
Additional Notes: This work was partially supported by CMUC – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020 (second and fourth author), by MICINN (Spain) grants MTM2012-34478 and MTM2015-64166-C2-2-P (second and third author), by Prin 2010/11 – Varietà reali e complesse: geometria, topologia e analisi armonica – Italy (first author), and by the exploratory research project in the frame of Programa Investigador FCT IF/00016/2013 (fourth author)
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society

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