Abstract
In the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.
References
[1] H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North-Holland Math. Libr. 68, Elsevier, Amsterdam, 2006. Search in Google Scholar
[2] S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776–797. 10.1090/S0002-9904-1946-08647-4Search in Google Scholar
[3] M. Crasmareanu, Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel, Anal. Math. Phys. 5 (2015), no. 4, 373–389. 10.1007/s13324-015-0102-8Search in Google Scholar
[4] S. Dragomir and D. Perrone, Harmonic Vector Fields: Variational Principles and Differential Geometry, Elsevier, Amsterdam, 2012. 10.1016/B978-0-12-415826-9.00002-XSearch in Google Scholar
[5] Q. He and Y.-B. Shen, Some results on harmonic maps for Finsler manifolds, Int. J. Math. 16 (2005), no. 9, 1017–1031. 10.1142/S0129167X05003211Search in Google Scholar
[6] X. Mo, Harmonic maps from Finsler manifolds, Illinois J. Math. 45 (2001), no. 4, 1331–1345. 10.1142/9789812773715_0009Search in Google Scholar
[7] X. Mo, An Introduction to Finsler Geometry, World Scientific, Hackensack, 2006. 10.1142/6095Search in Google Scholar
[8] X. Mo and Y. Yang, The existence of harmonic maps from Finsler manifolds to Riemannian manifolds, Sci. China Ser. A 48 (2005), no. 1, 115–130. 10.1360/03ys0338Search in Google Scholar
[9] S. Ohta and K. T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), no. 10, 1386–1433. 10.1002/cpa.20273Search in Google Scholar
[10] A. Shahi and B. Bidabad, Harmonic vector fields on Landsberg manifolds, C. R. Math. Acad. Sci. Paris 352 (2014), no. 9, 737–741. 10.1016/j.crma.2014.08.002Search in Google Scholar
[11] A. Shahi and B. Bidabad, Harmonic vector fields on Finsler manifolds, C. R. Math. Acad. Sci. Paris 354 (2016), no. 1, 101–106. 10.1016/j.crma.2015.10.006Search in Google Scholar
[12] Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapor, 2001. 10.1142/4619Search in Google Scholar
[13] Y. Shen and Y. Zhang, Second variation of harmonic maps between Finsler manifolds, Sci. China, Ser. A 47 (2004), no. 1, 39–51. 10.1360/03ys0040Search in Google Scholar
[14] F. Zhang and Q. Xia, Some Liouville-type theorems for harmonic functions on Finsler manifolds, J. Math. Anal. Appl. 417 (2014), no. 2, 979–995. 10.1016/j.jmaa.2014.03.078Search in Google Scholar
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