Skip to main content
SpringerLink
Log in

An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let \(N^{2n+1}(c)\) be an \((2n+1)\)-dimensional Sasakian space form with Sasakian structure \((\varphi ,\xi , \eta ,g)\) and constant \(\varphi \)-sectional curvature c. An n-dimensional submanifold \(M^n\) of \(N^{2n+1}(c)\) is called integral if the contact form \(\eta \) restricted to \(M^n\) vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H. Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baikoussis, C., Blair, D.E., Koufogiorgos, T.: Integral submanifolds of Sasakian space forms \(M^7(k)\). Results Math. 27, 207–226 (1995)

    Article  MathSciNet  Google Scholar 

  2. Berndt, J., Console, S., Olmos, C.: Submanifolds and Holonomy, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL (2016)

    Book  Google Scholar 

  3. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhäuser, Boston (2010)

    Book  Google Scholar 

  4. Blair, D.E., Carriazo, A.: The contact Whitney sphere. Note Mat. 20(2000/01), 125–133 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Borrelli, V., Chen, B.-Y., Morvan, J.M.: Une caractérisation géométrique de la sphère de Whitney. C. R. Acad. Sci. Paris Sér. I Math. 321, 1485–1490 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Castro, I., Urbano, F.: Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes. Ann. Glob. Anal. Geom. 13, 59–67 (1995)

    Article  MathSciNet  Google Scholar 

  7. Castro, I., Montealegre, C.R., Urbano, F.: Closed conformal vector fields and Lagrangian submanifolds in complex space forms. Pac. J. Math. 199, 269–302 (2001)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.-Y.: Jacobi’s elliptic functions and Lagrangian immersions. Proc. R. Soc. Edinb. Sect. A 126, 687–704 (1996)

    Article  MathSciNet  Google Scholar 

  9. Chen, B.-Y., Vrancken, L.: Lagrangian submanifolds satisfying a basic equality. Math. Proc. Camb. Philos. Soc. 120, 257–266 (1996)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y., Houh, C.-S., Lue, H.-S.: Totally real submanifolds. J. Differ. Geom. 12, 473–480 (1977)

    Article  MathSciNet  Google Scholar 

  11. Cheng, X., Hu, Z.: An optimal inequality on locally strongly convex centroaffine hypersurfaces. J. Geom. Anal. 28, 643–655 (2018)

    Article  MathSciNet  Google Scholar 

  12. Dillen, F., Vrancken, L.: \(C\)-totally real submanifolds of \(S^7(1)\) with nonnegative sectional curvature. Math. J. Okayama Univ. 31, 227–242 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Dillen, F., Vrancken, L.: \(C\)-totally real submanifolds of Sasakian space forms. J. Math. Pures Appl. 69, 85–93 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Dillen, F., Li, H., Vrancken, L., Wang, X.: Lagrangian submanifolds in complex projective space with parallel second fundamental form. Pac. J. Math. 255, 79–115 (2012)

    Article  MathSciNet  Google Scholar 

  15. Ferus, D.: Symmetric submanifolds of euclidean space. Math. Ann. 247, 81–93 (1980)

    Article  MathSciNet  Google Scholar 

  16. Kobayashi, S.: Isometric imbeddings of compact symmetric spaces. Tohoku Math. J. 20, 21–25 (1968)

    Article  MathSciNet  Google Scholar 

  17. Li, H., Vrancken, L.: A basic inequality and new characterization of Whitney spheres in a complex space form. Israel J. Math. 146, 223–242 (2005)

    Article  MathSciNet  Google Scholar 

  18. Li, H., Wang, X.: Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space. Results Math. 59, 453–470 (2011)

    Article  MathSciNet  Google Scholar 

  19. Luo, Y.: On Willmore Legendrian surfaces in \(S^5\) and the contact stationary Legendrian Willmore surfaces. Calc. Var. Partial Differ. Equ. 56(86), 19 (2017)

    MATH  Google Scholar 

  20. Luo, Y.: Contact stationary Legendrian surfaces in \(S^5\). Pac. J. Math. 293, 101–120 (2018)

    Article  Google Scholar 

  21. Montiel, S., Urbano, F.: Isotropic totally real submanifolds. Math. Z. 199, 55–60 (1988)

    Article  MathSciNet  Google Scholar 

  22. Naitoh, H.: Totally real parallel submanifolds in \(P^n(c)\). Tokyo J. Math. 4, 279–306 (1981)

    Article  MathSciNet  Google Scholar 

  23. Naitoh, H.: Parallel submanifolds of complex space forms, I. Nagoya Math. J. 90, 85–117 (1983)

    Article  MathSciNet  Google Scholar 

  24. Naitoh, H.: Parallel submanifolds of complex space forms, II. Nagoya Math. J. 91, 119–149 (1983)

    Article  MathSciNet  Google Scholar 

  25. Naitoh, H., Takeuchi, M.: Totally real submanifolds and symmetric bounded domain. Osaka J. Math. 19, 717–731 (1982)

    MathSciNet  MATH  Google Scholar 

  26. Ogiue, K.: On fiberings of almost contact manifolds. Kodai Math. Sem. Rep. 7, 53–62 (1965)

    Article  MathSciNet  Google Scholar 

  27. O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)

    Article  MathSciNet  Google Scholar 

  28. Ros, A., Urbano, F.: Lagrangian submanifolds of \(\mathbb{C}^n\) with conformal Maslov form and the Whitney sphere. J. Math. Soc. Japan 50, 203–226 (1998)

    Article  MathSciNet  Google Scholar 

  29. Reckziegel, H.: Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. In: Ferus, D. (ed.) Global Differential Geometry and Global Analysis (1984), Lecture Notes in Mathematics, vol. 1156, pp. 264–279. Springer, Berlin (1985)

    Chapter  Google Scholar 

  30. Szabó, Z.I.: Structure theorem on Riemannian symmetric space \(R(X, Y)\cdot R=0\). J. Differ. Geom. 17, 531–582 (1982)

    Article  Google Scholar 

  31. Tanno, S.: Sasakian manifolds with constant \(\varphi \)-holomorphic sectional curvature. Tohoku Math. J. 21, 501–507 (1969)

    Article  MathSciNet  Google Scholar 

  32. Vrancken, L.: Locally symmetric C-totally real submanifolds of \(S^7(1)\). Kyungpook Math. J. 29, 167–186 (1989)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Zejun Hu & Jiabin Yin

Authors
  1. Zejun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiabin Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zejun Hu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors were supported by NSF of China, Grant Number 11771404.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, Z., Yin, J. An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms. J Geom Anal 30, 3373–3397 (2020). https://doi.org/10.1007/s12220-019-00200-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-019-00200-8

Keywords

Mathematics Subject Classification

Search

Navigation