Skip to main content
SpringerLink
Log in

Isometric immersions of higher codimension into the product S k × H n+p−k

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we obtain a sufficient and necessary condition for a simply connected Riemannian manifold (M n, g) to be isometrically immersed, as a submanifold with codimension p ≥ 1, into the product S k × H n+p−k of sphere and hyperboloid.

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. Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and spin geometry. Math. Z., 249, 545–580 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonnet, O.: Mémorie sur la théorie des surfaces applicables sur une surface donnée. École Polytech., 42 (1867)

  3. Chen, Q., Cui, Q.: Isometric immersions into S m × R and H m × R with high codimensions. Results Math., 57, 319–333 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Daniel, B.: Isometric immersions into S n × R or H n × R and applications to minimal surfaces. Trans. Amer. Math. Soc., 361, 6255–6282 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dillen, F.: Equivalence theorems in affine differential geometry. Geom. Dedicata., 32, 81–92 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dillen, F.: Conjugate connections and Radon’s theorem in affine differential geometry. Monatsh. Math., 109, 221–235 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kobayashi, S., Nomizu, K.: Foudations of Differential Geometry, Vol. II. Interscience Publishers, New York, 1969

    Google Scholar 

  8. Kowaltzyk, D.: Hypersurfaces (M n, g) in S k × H n−k+1. ArXiv: math/0903.3510vl

  9. Lira, J. H., Tojeiro, R., Vitório, F.: A Bonnet theorem for isometric immersions into products of space forms. Arch. Math. (Basel), 95, 469–479 (2010)

    Google Scholar 

  10. Spivak, M.: A Comprehensive Introduction to Differential Geometry, Vol. 4, 2nd Edition, Publish or Perish, USA, 1979

    Google Scholar 

  11. Tenenblat, K.: On isometric immersions of Riemannian manifolds. Bol. Soc. Brasil. Mat., 2(2), 23–36 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Tian Qun Zhang

    Present address: Luoyang No. 43 Middle School, Luoyang, 471012, He’nan, P. R. China

Authors and Affiliations

  1. Department of Mathematics, He’nan Normal University, Xinxiang, 453007, P. R. China

    Xing Xiao Li & Tian Qun Zhang

Authors
  1. Xing Xiao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tian Qun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Xiao Li.

Additional information

Supported by NSFC (Grant Nos. 11171091, 11371018) and partially supported by NSF of He’nan Province (Grant No. 132300410141)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X.X., Zhang, T.Q. Isometric immersions of higher codimension into the product S k × H n+p−k . Acta. Math. Sin.-English Ser. 30, 2146–2160 (2014). https://doi.org/10.1007/s10114-014-1184-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-014-1184-1

Keywords

MR(2010) Subject Classification

Search

Navigation