Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Biharmonic submanifolds in spheres

  • 209 Accesses

  • 125 Citations

Abstract

We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereS n. In the case of curves inS n we solve explicitly the biharmonic equation.

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

References

  1. [1]

    R. Caddeo, S. Montaldo and C. Oniciuc, biharmonic submanifolds ofS 3, International Journal of Mathematics, to appear.

  2. [2]

    B. Y. Chen,Some open problems and conjectures on submanifolds of finite type, Soochow Journal of Mathematics17 (1991), 169–188.

  3. [3]

    B. Y. Chen and S. Ishikawa,Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu Journal of Mathematics52 (1998), 167–185.

  4. [4]

    B. Y. Chen and K. Yano,Minimal submanifolds of a higher dimensional sphere, Tensor (N.S.)22 (1971), 369–373.

  5. [5]

    I. Dimitric,Submanifolds of E m with harmonic mean curvature vector, Bulletin of the Institute of Mathematics. Academic Sinica20 (1992), 53–65.

  6. [6]

    J. Eells and J.H. Sampson,Harmonic mappings of Riemannian manifolds, American Journal of Mathematics86 (1964), 109–160.

  7. [7]

    H. Gluck,Geodesics in the unit tangent bundle of a round sphere, L'Enseignement Mathématique34 (1988), 233–246.

  8. [8]

    T. Hasanis and T. Vlachos, Hypersurfaces inS 4 with harmonic mean curvature vector field, Mathematische Nachrichten172 (1995), 145–169.

  9. [9]

    G. Y. Jiang,2-harmonic isometric immersions between Riemannian manifolds, Chinese Annals of Mathematics. Series A7 (1986), 130–144.

  10. [10]

    G. Y. Jiang,2-harmonic maps and their first and second variational formulas, Chinese Annals of mathematics. Series A7 (1986), 389–402.

  11. [11]

    H. B. Lawson, Complete minimal surfaces inS 3, Annals of Mathematics (2)92 (1970), 335–374.

  12. [12]

    C. Oniciuc,Biharmonic maps between Riemannian manifolds, Analele Stiintifice ale University Al. I. Cuza Iasi. Mat. (N.S.), to appear.

  13. [13]

    J. Simons,Minimal varieties in Riemannian manifolds, Annals of Mathematics88 (1968), 62–105.

Download references

Author information

Correspondence to R. Caddeo.

Additional information

The first author was supported by G.N.S.A.G.A., Italy.

The second author was supported by “Contratto giovani ricercatori”, University of Cagliari and by G.N.S.A.G.A., Italy.

The third author was supported by a NATO Guest fellowship grant and the grant 6186/25.X.2000, A.N.S.T.I., România.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Caddeo, R., Montaldo, S. & Oniciuc, C. Biharmonic submanifolds in spheres. Isr. J. Math. 130, 109–123 (2002). https://doi.org/10.1007/BF02764073

Download citation

Keywords

  • Riemannian Manifold
  • Sectional Curvature
  • Isometric Immersion
  • Geodesic Curvature
  • Biharmonic Equation