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

A note on biharmonic submanifolds of product spaces

  • 149 Accesses

  • 2 Citations


We investigate biharmonic submanifolds of the product of two space forms. We prove a necessary and sufficient condition for biharmonic submanifolds in these product spaces. Then, we obtain mean curvature estimates for proper-biharmonic submanifold of a product of two unit spheres. We also prove a non-existence result in the case of the product of a sphere and a hyperbolic space.

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


  1. 1

    Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic PNMC submanifolds in spheres. Ark. Mat. (2013, in press)

  2. 2

    Caddeo R., Montaldo S., Oniciuc C.: Biharmonic submanifolds in spheres. Isr. J. Math 130, 109–123 (2002)

  3. 3

    Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type. Series in Pure Mathematics, Vol. 1. World Scientific Publishing Co., Singapore (1984)

  4. 4

    Eells J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math 86, 109–160 (1964)

  5. 5

    Fetcu D., Loubeau E., Montaldo S., Oniciuc C.: Biharmonic submanifolds of \({\mathbb{C} P^n}\) . Math. Z. 266, 505–531 (2010)

  6. 6

    Fetcu D., Oniciuc C.: Explicit formulas for biharmonic submanifolds in Sasakian space forms. Pac. J. Math 240(1), 85–107 (2009)

  7. 7

    Fetcu, D., Oniciuc, C., Rosenberg, H.: Biharmonic submanifolds with parallel mean curvature in \({\mathbb{S}^n \times \mathbb{R}}\) , J. Geom. Anal. (2013, in press)

  8. 8

    Jiang G.Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7(4), 389–402 (1986)

Download references

Author information

Correspondence to Julien Roth.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Roth, J. A note on biharmonic submanifolds of product spaces. J. Geom. 104, 375–381 (2013). https://doi.org/10.1007/s00022-013-0168-0

Download citation

Mathematics Subject Classification (2000)

  • 53C42
  • 53C43


  • Biharmonic submanifolds
  • product spaces