On the mean curvature of submanifolds with nullity

Abstract

In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical n-dimensional minimal submanifolds with index of relative nullity \(n-2\), fully described by Dajczer and Florit (Ill J Math 45:735–755, 2001) in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal n-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank \(n-2\ge 2,\) which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension \(n=3\) turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.

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

References

  1. 1.

    Abe, K.: On a class of hypersurfaces of \({\mathbb{R}}^{2n+1}\). Duke Math. J. 41, 865–874 (1974)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alias, L.J., Melendez, J.: Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space. Geom. Dedicata 199, 273–280 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Canevari, S., Machado De Freitas, G., Guimarães, F., Manfio, F., Dos Santos, J.P.: Complete submanifolds with relative nullity in space forms. arXiv:1910.040550

  4. 4.

    Chern, S.S., Kuiper, N.: Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space. Ann. of Math. (2) 56, 422–430 (1952)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dajczer, M., Florit, L.: A class of austere submanifolds. Illinois J. Math. 45, 735–755 (2001)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dajczer, M., Gromoll, D.: Gauss parametrizations and rigidity aspects of submanifolds. J. Differential Geom. 22, 1–12 (1985)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dajczer, M., Gromoll, D.: Rigidity of complete Euclidean hypersurfaces. J. Differential Geom. 31, 401–416 (1990)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dajczer, M., Kasioumis, Th., Savas-Halilaj, A., Vlachos, Th: Complete minimal submanifolds with nullity in Euclidean space. Math. Z. 287, 481–491 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dajczer, M., Rodríguez, L.: Complete real Kähler minimal submanifolds. J. Reine Angew. Math. 419, 1–8 (1991)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dajczer, M., Tojeiro, R.: On compositions of isometric immersions. J. Differential Geom. 36, 1–18 (1992)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dajczer, M., Tojeiro, R.: Submanifolds of constant sectional curvature with parallel or constant mean curvature. Tohoku Math. J. (2) 45, 43–49 (1993)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dajczer, M., Tojeiro, R.: Submanifold Theory. Beyond an Introduction. Universitext. Springer, New York (2019)

    Google Scholar 

  13. 13.

    Ejiri, N.: Equivariant minimal immersions of \({\mathbb{S}}^2\) into \({\mathbb{S}}^{2m}(1)\). Trans. Amer. Math. Soc. 297, 105–124 (1986)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Enomoto, K.: Umbilical points on surfaces in \({\mathbb{R}}^N\). Nagoya Math. J. 100, 135–143 (1985)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Florit, L.: On submanifolds with nonpositive extrinsic curvature. Math. Ann. 298, 187–192 (1994)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Florit, L., Zheng, F.: On nonpositively curved Euclidean submanifolds: splitting results. Comment. Math. Helv. 74, 53–62 (1999)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Florit, L., Zheng, F.: On nonpositively curved Euclidean submanifolds: splitting results II. J. Reine Angew. Math. 508, 1–15 (1999)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Florit, L., Zheng, F.: A local and global splitting result for real Kähler Euclidean submanifolds. Arch. Math. (Basel) 84, 88–95 (2005)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Furuhata, H.: A cylinder theorem for isometric pluriharmonic immersions. Geom. Dedicata 66, 303–311 (1997)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Guimarães, F., Machado De Freitas, G.: Cylindricity of complete Euclidean submanifolds with relative nullity. Ann. Global Anal. Geom. 49, 253–257 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hartman, P.: On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures. II. Trans. Amer. Math. Soc. 147, 529–540 (1970)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hartman, P.: On Complete hypersurfaces of nonnegative sectional curvatures and constant \(m\)th mean curvature. Trans. Amer. Math. Soc. 245, 363–374 (1978)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Hasanis, Th, Savas-Halilaj, A., Vlachos, Th.: Minimal hypersurfaces with zero Gauss-Kronecker curvature. Illinois J. Math. 49, 523–529 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Klotz, T., Osserman, R.: Complete surfaces in \(E^3\) with constant mean curvature. Comment. Math. Helv. 41, 313–318 (1966/1967)

  25. 25.

    Lawson, H.B.: Some intrinsic characterizations of minimal surfaces. J. Analyse Math. 24, 151–161 (1971)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Maltz, R.: Cylindricity of isometric immersions into Euclidean space. Proc. Amer. Math. Soc. 53, 428–432 (1975)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Moore, J.D.: Submanifolds of constant positive curvature I. Duke Math. J. 44, 449–484 (1977)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nomizu, K., Smyth, B.: A formula of Simons’ type and hypersurfaces with constant mean curvature. J. Differential Geom. 3, 367–377 (1969)

    MathSciNet  Article  Google Scholar 

  29. 29.

    O’Neill, B.: Umbilics of constant curvature immersions. Duke Math. J. 32, 149–159 (1965)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Vlachos, Th.: Congruence of minimal surfaces and higher fundamental forms. Manuscripta Math. 110, 77–91 (2003)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The first named author would like to acknowledge financial support by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI), Grant No: 133, Action: Support for Postdoctoral Researchers.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Th. Vlachos.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kanellopoulou, A.E., Vlachos, T. On the mean curvature of submanifolds with nullity. Ann Glob Anal Geom 58, 79–108 (2020). https://doi.org/10.1007/s10455-020-09717-6

Download citation

Keywords

  • Index of relative nullity
  • Relative nullity distribution
  • Mean curvature
  • Cylinder
  • Elliptic submanifolds
  • Minimal surfaces
  • Curvature ellipse

Mathematics Subject Classification

  • Primary 53C42; Secondary 53C40
  • 53B25