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Year 2021, Volume: 14 Issue: 2, 305 - 312, 29.10.2021
https://doi.org/10.36890/iejg.936026

Abstract

References

  • [1] Cecil, T. E., Ryan, P. J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York (2015).
  • [2] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60, 568-578 (1993).
  • [3] Chen, B. Y.: A general inequality for submanifolds in complex space forms and its applications. Arch. Math. (Basel) 67, 519–528 (1996).
  • [4] Chen, B. Y.: Pseudo Riemannian Geometry, δ-invariants and Applications. World Scientific, Hackensack, New Jersey (2011).
  • [5] Ivey, T. A., Ryan, P. J.: Hypersurfaces in CP2 and CH2 with two distinct principal curvatures. Glasgow Math. J. 58, 137-152 (2016).
  • [6] Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurface in Pn(C). Math. Ann. 276, 487–497 (1987).
  • [7] Sasahara, T.: Real hypersurfaces in the complex projective plane attaining equality in a basic inequality. Houston J. Math. 43, 89-94 (2017).

Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$

Year 2021, Volume: 14 Issue: 2, 305 - 312, 29.10.2021
https://doi.org/10.36890/iejg.936026

Abstract

It was proved in Chen's paper [3] that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$\delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.

References

  • [1] Cecil, T. E., Ryan, P. J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York (2015).
  • [2] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60, 568-578 (1993).
  • [3] Chen, B. Y.: A general inequality for submanifolds in complex space forms and its applications. Arch. Math. (Basel) 67, 519–528 (1996).
  • [4] Chen, B. Y.: Pseudo Riemannian Geometry, δ-invariants and Applications. World Scientific, Hackensack, New Jersey (2011).
  • [5] Ivey, T. A., Ryan, P. J.: Hypersurfaces in CP2 and CH2 with two distinct principal curvatures. Glasgow Math. J. 58, 137-152 (2016).
  • [6] Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurface in Pn(C). Math. Ann. 276, 487–497 (1987).
  • [7] Sasahara, T.: Real hypersurfaces in the complex projective plane attaining equality in a basic inequality. Houston J. Math. 43, 89-94 (2017).
There are 7 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

T. Sasahara 0000-0003-2853-0268

Publication Date October 29, 2021
Acceptance Date October 8, 2021
Published in Issue Year 2021 Volume: 14 Issue: 2

Cite

APA Sasahara, T. (2021). Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$. International Electronic Journal of Geometry, 14(2), 305-312. https://doi.org/10.36890/iejg.936026
AMA Sasahara T. Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$. Int. Electron. J. Geom. October 2021;14(2):305-312. doi:10.36890/iejg.936026
Chicago Sasahara, T. “Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$”. International Electronic Journal of Geometry 14, no. 2 (October 2021): 305-12. https://doi.org/10.36890/iejg.936026.
EndNote Sasahara T (October 1, 2021) Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$. International Electronic Journal of Geometry 14 2 305–312.
IEEE T. Sasahara, “Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$”, Int. Electron. J. Geom., vol. 14, no. 2, pp. 305–312, 2021, doi: 10.36890/iejg.936026.
ISNAD Sasahara, T. “Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$”. International Electronic Journal of Geometry 14/2 (October 2021), 305-312. https://doi.org/10.36890/iejg.936026.
JAMA Sasahara T. Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$. Int. Electron. J. Geom. 2021;14:305–312.
MLA Sasahara, T. “Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$”. International Electronic Journal of Geometry, vol. 14, no. 2, 2021, pp. 305-12, doi:10.36890/iejg.936026.
Vancouver Sasahara T. Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$. Int. Electron. J. Geom. 2021;14(2):305-12.