On the classification of $3$-dimensional spherical Sasakian manifolds

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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 Mathematics

Email: vladimir.ejov@flinders.edu.au

References

  1. V. Ezhov, G. Schmalz, “Explicit description of spherical rigid hypersurfaces in $mathbb{C}^2$”, Complex Anal. Synerg., 1:1 (2015), 2, 10 pp.
  2. N. K. Stanton, “A normal form for rigid hypersurfaces in $mathbf{C}^2$”, Amer. J. Math., 113:5 (1991), 877–910
  3. 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
  4. 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.

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Copyright (c) 2021 Sykes D., Schmalz G., Ezhov V.V.

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