Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME RESULTS ON THE GEOMETRY OF A NON-CONFORMAL DEFORMATION OF A METRIC

  • Received : 2021.06.03
  • Accepted : 2021.11.09
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let (Mm, g) be an m-dimensional Riemannian manifold. In this paper, we introduce a new class of metric on (Mm, g), obtained by a non-conformal deformation of the metric g. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. In the last section we characterizes some class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when (Mm, g) is an Euclidean space.

Keywords

Acknowledgement

The authors would like to thank the reviewer for his insightful comments and suggestions that helped us to improve the paper.

References

  1. P. Baird, A. Fardoun, and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Global Anal. Geom. 34 (2008), no. 4, 403-414. https://doi.org/10.1007/s10455-008-9118-8
  2. P. Baird and D. Kamissoko, On constructing biharmonic maps and metrics, Ann. Global Anal. Geom. 23 (2003), no. 1, 65-75. https://doi.org/10.1023/A:1021213930520
  3. A. Balmus, Biharmonic properties and conformal changes, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 50 (2004), no. 2, 361-372.
  4. A. Balmus, S. Montaldo, and C. Oniciuc, Biharmonic maps between warped product manifolds, J. Geom. Phys. 57 (2007), no. 2, 449-466. https://doi.org/10.1016/j.geomphys.2006.03.012
  5. R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of S3, Internat. J. Math. 12 (2001), no. 8, 867-876. https://doi.org/10.1142/S0129167X01001027
  6. N. E. Djaa and F. Latti, Geometry of generalized F-harmonic maps, Bull. Transilv. Univ. Brasov Ser. III 13(62) (2020), no. 1, 101-114. https://doi.org/10.31926/but.mif
  7. J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385-524. https://doi.org/10.1112/blms/20.5.385
  8. W. J. Lu, Geometry of warped product manifolds and its applications, Ph. D. thesis, Zhejiang University, 2013.
  9. S. Ouakkas, R. Nasri, and M. Djaa, On the f-harmonic and f-biharmonic maps, JP J. Geom. Topol. 10 (2010), no. 1, 11-27.