Tokyo Journal of Mathematics

Real Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensor

Xiaomin CHEN

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.

Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 587-601.

Dates
First available in Project Euclid: 18 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1513566023

Mathematical Reviews number (MathSciNet)
MR3908812

Zentralblatt MATH identifier
07053494

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citation

CHEN, Xiaomin. Real Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensor. Tokyo J. Math. 41 (2018), no. 2, 587--601. https://projecteuclid.org/euclid.tjm/1513566023


Export citation

References

  • J. Berndt and Y. J. Suh, Hypersurfaces in Kaehler manifold, Proc. A.M.S. \bf143 (2015), 2637–2649.
  • J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flows in complex quadrics, Inter. J. Math. \bf24 (2013), 1350050, 18 pp.
  • T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci *-tensor, Tokyo J. Math. 25 (2002), 473–483.
  • T. A. Ivey and P. J. Ryan, The $^{*}$-Ricci tensor for hypersurfaces in $\mathbb{CP}^n$ and $\mathbb{C }\mathrm{H}^n$, Tohoku Math. J. \bf34 (2011), 445–471.
  • G. Kaimakamis and K. Panagiotidou, *-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86 (2014), 408–413.
  • S. Klein, Totally geodesic submanifolds in the complex quadric, Diff. Geom. Appl. \bf26 (2008), 79–96.
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Wiley Classics Library ed., A Wiley-Interscience Publ., 1996.
  • H. Reckziegel, On the geometry of the complex quadric, Geometry and Topology of Submanifolds VIII, Brussels/Nordfjordeid, World Sci. Publ., River Edge, NJ, 1995, 302–315.
  • Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Adv. Math. \bf281 (2015), 886–905.
  • Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Inter. J. Math. 25 (2014), 1450059, 17 pp.
  • Y. J. Suh, Real hypersurfaces in the complex quadric with parallel normal Jacobi operator, Math. Nachr. \bf289 (2016), 1–10.
  • Y. J. Suh, Real hypersurfaces in the complex quadric with harmonic curvature, J. Math. Pure. Appl. \bf106 (2016), 393–410.
  • Y. J. Suh, Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor, J. Geom. Phys. \bf106 (2016), 130–142.
  • Y. J. Suh, Pseudo-anti commuting Ricci tensor and Ricci soliton real hypersurfaces in the complex quadric, J. Math. Pure. Appl. \bf107 (2017), 429–450.
  • S. Tachibana, On almost-analytic vectors in almost-Kählerian manifolds, Tohoku Math. J. \bf11 (1959), 247–265.