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HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

Part of: Classical differential geometry Local differential geometry Complex spaces with a group of automorphisms Automorphic functions

Published online by Cambridge University Press:  08 November 2019

D. ALEKSEEVSKY Open the ORCID record for D. ALEKSEEVSKY [Opens in a new window] ,
K. HASEGAWA Open the ORCID record for K. HASEGAWA [Opens in a new window]  and
Y. KAMISHIMA Open the ORCID record for Y. KAMISHIMA [Opens in a new window]
D. ALEKSEEVSKY
Affiliation:
Institute for Information Transmission Problems, Bolshoi Karetnyi per., 19, 127994Moscow, Russia email dalekseevsky@iitp.ru
K. HASEGAWA
Affiliation:
Department of Mathematics, Faculty of Education, Niigata University, 8050 Ikarashi-Nino-cho, Nishi-ku, 950-2181Niigata, Japan email hasegawa@ed.niigata-u.ac.jp
Y. KAMISHIMA
Affiliation:
Department of Mathematics, Josai University, Keyaki-dai 1-1, Sakado, 350-0295Saitama, Japan email kami@josai.ac.jp
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Abstract

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

MSC classification

Primary: 32M10: Homogeneous complex manifolds 53A30: Conformal differential geometry
Secondary: 53B35: Hermitian and Kählerian structures
Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 83 - 96
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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References

Alekseevsky, D., Cortés, V., Hasegawa, K. and Kamishima, Y., Homogeneous locally conformally Kähler and Sasaki manifolds, Internat. J. Math. 26(06) (2015), 1–29.Google Scholar
Boothby, W. M. and Wang, H. C., On contact manifolds, Ann. of Math. (2) 68(3) (1958), 721–734.10.2307/1970165Google Scholar
Borel, A., Kählerian coset spaces of semi-simple Lie groups, Proc. Natl Acad. Sci. USA 40 (1954), 721–734.10.1073/pnas.40.12.1147Google Scholar
Boyer, C. P. and Galicki, K., Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.Google Scholar
Dorfmeister, J. and Nakajima, K., The fundamental conjecture for homogeneous Kähler manifolds, Acta. Math. 161 (1988), 23–70.10.1007/BF02392294Google Scholar
Grauert, H., Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263–273.10.1007/BF01351803Google Scholar
Hano, J., On Kählerian homogeneous spaces of unimodular Lie groups, Amer. J. Math. 79 (1957), 885–900.Google Scholar
Hasegawa, K. and Kamishima, Y., Locally conformally Kähler structures on homogeneous spaces, Progr. Math. 308 (2015), 353–372.10.1007/978-3-319-11523-8_13Google Scholar
Hasegawa, K. and Kamishima, Y., Compact homogeneous locally conformally Kähler manifolds, Osaka J. Math. 53 (2016), 683–703.Google Scholar
Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry II, Wiley Classics Library, John Wiley & Sons, New York, 1996.Google Scholar
Koszul, J. K., Sur la forme hermitienne canonique des espaces homogènes complexes, Canad. J. Math. 7 (1955), 562–576.Google Scholar
Nakajima, K., Homogeneous Kähler manifolds of non-degenerate Ricci curvature, J. Math. Soc. Japan 42 (1990), 475–494.10.2969/jmsj/04230475Google Scholar
Sawai, H., Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures, Geom. Dedicata 125 (2007), 93–101.10.1007/s10711-007-9140-1Google Scholar
Sawai, H., Structure theorem for Vaisman completely solvable solvmanifolds, J. Geom. Phys. 114 (2017), 581–586.10.1016/j.geomphys.2017.01.002Google Scholar
Vinberg, E. B., Gindikin, S. G. and Piatetskii-Shapiro, I. I., Classification and canonical realization of complex homogeneous domains, Trans. Moscow Math. Soc. 12 (1963), 404–437.Google Scholar