[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).
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.
[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).
Mohammed Cherif, A., & Djaa, M. (2020). Harmonic Maps and Torse-Forming Vector Fields. International Electronic Journal of Geometry, 13(1), 87-93. https://doi.org/10.36890/iejg.555344
AMA
Mohammed Cherif A, Djaa M. Harmonic Maps and Torse-Forming Vector Fields. Int. Electron. J. Geom. January 2020;13(1):87-93. doi:10.36890/iejg.555344
Chicago
Mohammed Cherif, Ahmed, and Mustapha Djaa. “Harmonic Maps and Torse-Forming Vector Fields”. International Electronic Journal of Geometry 13, no. 1 (January 2020): 87-93. https://doi.org/10.36890/iejg.555344.
EndNote
Mohammed Cherif A, Djaa M (January 1, 2020) Harmonic Maps and Torse-Forming Vector Fields. International Electronic Journal of Geometry 13 1 87–93.
IEEE
A. Mohammed Cherif and M. Djaa, “Harmonic Maps and Torse-Forming Vector Fields”, Int. Electron. J. Geom., vol. 13, no. 1, pp. 87–93, 2020, doi: 10.36890/iejg.555344.
ISNAD
Mohammed Cherif, Ahmed - Djaa, Mustapha. “Harmonic Maps and Torse-Forming Vector Fields”. International Electronic Journal of Geometry 13/1 (January 2020), 87-93. https://doi.org/10.36890/iejg.555344.
JAMA
Mohammed Cherif A, Djaa M. Harmonic Maps and Torse-Forming Vector Fields. Int. Electron. J. Geom. 2020;13:87–93.
MLA
Mohammed Cherif, Ahmed and Mustapha Djaa. “Harmonic Maps and Torse-Forming Vector Fields”. International Electronic Journal of Geometry, vol. 13, no. 1, 2020, pp. 87-93, doi:10.36890/iejg.555344.
Vancouver
Mohammed Cherif A, Djaa M. Harmonic Maps and Torse-Forming Vector Fields. Int. Electron. J. Geom. 2020;13(1):87-93.