On Finsler warped product metrics with vanishing E-curvature

Document Type : Original Article

Authors

1 Banaras Hindu University, Varanasi

2 DST-Centre For Interdisciplinary Mathematical Sciences BANARAS HINDU UNIVERSITY, Varanasi

Abstract

In this paper, we study Finsler warped product metric recently introduced by P. Marcal and Z. Shen and find characteristics differential equations for this metric to have vanishing E-curvature. We also prove that if this warped product Finsler metric is projectively flat, then it becomes a Riemannian metric.

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Main Subjects


[1] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49.
[2] J. Nash, C 1 -isometric imbeddings, Ann. of Math. 60(3) (1954), 383-396.
[3] J. Nash,The imbedding problem for Riemannian manifolds, Ann. of Math. 63(1) (1956), 20-63.
[4] B. Chen, Z. Shen and L. Zhao ,Constructions of Einstein Finsler Metrics by Warped Product, Internat. J. Math. 29(11) (2018), 1850081.
[5] L. Kozma, R. Peter and C. Varga, Warped product of Finsler manifolds, Ann Univ Sci Budapest. 44 (2001) 157-170.
[6] H. Liu and X. Mo, Finsler warped product metrics of Douglas type, Canad. Math. Bull. 62(1) (2019), 119-130.
[7] R. Gangopadhyay, A. Shriwastawa and B. Tiwari, On equivalence of two non-Riemannian curvatures in warped product Finsler metrics, arXiv:2011.12045 [math.DG].
[8] B. Tiwari, R. Gangopadhyay and G. K. Prajapati, A class of Finsler spaces with general (α, β)-metrics, Int. J. Geom. Methods Mod. Phys. 16(2) (2019), 1950102.
[9] B. Tiwari, R. Gangopadhyay and G. K. Prajapati, On general (α, β)-metrics with some curvature properties, Khayyam J. Math. 5(2) (2019), 30-39.
[10] R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28(2) (2002), 221-262.
[11] C. Yu and H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl. 29 (2011), 244-254.
[12] X. Mo, N. M. Solorzano and K. Tenenblat, On spherically symmetric Finsler metrics with vanishing Douglas curvature, Differential Geom. Appl. 31 (2013), 746-758.
[13] Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128(2) (1997), 306-328.
[14] Z. Shen, Finsler metrics with K = 0 and S = 0, Canadian J. Math. 55 (2003), 112-132.
[15] Z. Shen, Finsler manifolds of constant positive curvature, Contemp. Math. 196 (1995), 83-93.
[16] H. Zhu, On general (α, β)-metrics with isotropic S-curvature, J. Math. Anal. Appl. 464 (2018), 1127-1142.
[17] X. Cheng and Z. Shen, Randers metric with special curvature properties, Osaka J. Math. 40 (2003), 87-101.
[18] S. S. Chern and Z. Shen, Riemannian-Finsler geometry, World Scientific Publisher, Singapore, 2005.
[19] E. Peyghana and A. Tayebi, On doubly warped product Finsler manifolds, Nonlinear Anal. Real World Appl. 13 (2012), 1703-1720.
[20] M. M. Rezaii and Y. Alipour-Fakhri, On projectively related warped product Finsler manifolds, Int. J. Geom. Methods Mod. Phys. 8(5), (2012) 953-967.
[21] P. Marcal and Z.Shen, Ricci flat Finsler metrics by warped product, arXiv:2012.05699v1 [math.DG] 10 Dec 2020.
[22] G. S. Asanov, Finslerian extensions of Schwarzschild metric, Fortschr. Phys. 40 (1992), 667-693.
[23] G. S. Asanov, Finslerian metric functions over the product and their potential applications, Fortschr. Phys. 40 (1992), 667-693.