On the classification of $3$-dimensional spherical Sasakian manifolds
- Authors: Sykes D.1, Schmalz G.1, Ezhov V.V.2,3
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Affiliations:
- University of New England
- Flinders University
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- Issue: Vol 85, No 3 (2021)
- Pages: 191-202
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133867
- DOI: https://doi.org/10.4213/im9046
- ID: 133867
Cite item
Abstract
In this article we regard spherical hypersurfaces in $\mathbb{C}^2$ with a fixed Reeb vector field as $3$-dimensional Sasakian manifolds. We establish a correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, those used in Stanton's description of rigid spheres, and those arising from the rigid normal forms. We also describe geometrically the moduli space for rigid spheres and provide a geometric distinction between Stanton hypersurfaces and those found in [1]. Finally, we determine the Sasakian automorphism groups of rigid spheres and detect the homogeneous Sasakian manifolds among them.
About the authors
Daniel Sykes
University of New England
Email: dsykes4@myune.edu.au
Gerd Schmalz
University of New England
Email: schmalz@une.edu.au
PhD
Vladimir Vladimirovich Ezhov
Flinders University; Lomonosov Moscow State University, Faculty of Mechanics and MathematicsReferences
- V. Ezhov, G. Schmalz, “Explicit description of spherical rigid hypersurfaces in $mathbb{C}^2$”, Complex Anal. Synerg., 1:1 (2015), 2, 10 pp.
- N. K. Stanton, “A normal form for rigid hypersurfaces in $mathbf{C}^2$”, Amer. J. Math., 113:5 (1991), 877–910
- A. Isaev, J. Merker, “On the real-analyticity of rigid spherical hypersurfaces in $mathbb{C}^2$”, Proc. Amer. Math. Soc., 147:12 (2019), 5251–5256
- M. S. Baouendi, P. Ebenfelt, L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Math. Ser., 47, Princeton Univ. Press, Princeton, NJ, 1999, xii+404 pp.