Harmonic Maps and Torse-Forming Vector Fields

Year 2020, Volume: 13 Issue: 1, 87 - 93, 30.01.2020

Ahmed Mohammed Cherif Mustapha Djaa

https://doi.org/10.36890/iejg.555344

Abstract

In this paper, we prove that any harmonic map from a compact orientable Riemannian manifold
without boundary (or from complete Riemannian manifold) (M, g) to Riemannian manifold (N, h)
is necessarily constant, with (N, h) admitting a torse-forming vector field satisfying some condition.

Keywords

Harmonic maps, bi-harmonic maps, torse-forming vector fields

References

• [1] Adati, T., Miyazawa, T.: On P-Sasakian manifolds satisfying certain conditions. Tensor, N.S. 33, 173–178 (1979).
• [2] Deshmukh S., Falleh, R. A.: Conformal vector fields and conformal transformations on a Riemannian manifold. Balkan Journal of Geometry and Its Applications. 17(1), 9–16 (2012).
• [3] Djaa M., Elhendi M., Ouakkas, S.: On the Biharmonic Vector Fields. Turkish Journal of Mathematics. 36, 463–474 (2012).
• [4] Baird P., Wood J. C.: Harmonic morphisms between Riemannain manifolds. Clarendon Press, Oxford (2003).
• [5] Chen, B. Y.: Rectifying submanifolds of Riemannian manifolds and torqued vector fields. Kragujevac J. Math. 41(1), 93-103 (2017).
• [6] Chen, B. Y.: Classification of torqued vector fields and its applications to Ricci solitons. Kragujevac J. Math. 41(2), 239–250 (2017).
• [7] De, U. C., De, B. K.: Some properties of a semi-symmetric metric connection on a Riemannian manifold. Istanbul Univ. Fen Fak. Mat. Der. 54, 111-117 (1995).
• [8] Eells, J., Lemaire L.: A report on harmonic maps. Bull. London Math. Soc. 16, 1–68 (1978).
• [9] Eells, J., Lemaire L.: Another report on harmonic maps. Bull. London Math. Soc. 20, 385–524 (1988).
• [10] Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86, 109–160 (1964).
• [11] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tˆhoku Math. J. 24, 93–103 (1972).
• [12] Kowolik, J.: On some Riemannian manifolds admitting torse-forming vector fields. Dem. Math. 18. 3, 885-891 (1985).
• [13] Cherif, A. M.: Some results on harmonic and bi-harmonic maps. International Journal of Geometric Methods in Modern Physics. 14 (7), (2017).
• [14] Schouten, J. A.: Ricci-Calculus. 2nd ed. Springer-Verlag, Berlin (1954).
• [15] Vanderwinden, A.J.: Exemples d’applcations harmoniques. PHD Thesis. Universite Libre de Bruxelles (1992).
• [16] Wang, Z. P., Ou, Y. L., Yang, H.C.: Biharmonic maps from tori into a 2-sphere, Chin. Ann. Math. Ser. B. 39(5), 861–878 (2018).
• [17] Xin, Y.: Geometry of harmonic maps. Fudan University, (1996).
• [18] Yau, S. T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975).
• [19] Yano, K.: Concircular geometry. I. Concircular transformations. Proc. Imp. Acad. Tokyo. 16, 195–200 (1940).
• [20] Yano, K.: On the torse-forming directions in Riemannian spaces. Proc. Imp. Acad. Tokyo. 20, 340–345 (1944).