S -curvature of Doubly Warped Product of Finsler Manifolds
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
S-curvature of Doubly Warped Product of Finsler Manifolds Zhao YANG College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China E-mail : [email protected]
Yong HE1) Department of Mathematics, Xinjiang Normal University, Urumqi 830017, P. R. China E-mail : [email protected]
Xiao Ling ZHANG College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China E-mail : [email protected] Abstract Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature. Keywords Doubly warped product of Finsler manifold, weak isotropic S-curvature, isotropic mean Berwald curvature, weakly Berwald manifolds MR(2010) Subject Classification
1
53B40, 53C60
Introduction
The notion of warped product generalizes that of a surface of revolution. It was introduced mathematically by O’Neill and Bishop for studying Riemannian manifolds of negative curvature [6]. The recent studies show that warped product is useful in theoretical physics, particularly in general relativity [4, 12]. O’Neill obtained the curvature formulae of a warped product M ×f N of Riemann manifolds in terms of curvatures of M and N . Dobarro and Lami established the relationship between the scalar curvature of a warped product Riemann manifold M ×f N and that of M and N [8]. In [5], Bertola and Gouthier classified certain warped products with constant sectional curvature or satisfying the Einstein condition. The notion of warped product was later extended to the case of Finsler manifolds by the work of Asanov [1, 2], who gave the generalization of the Schwarzschild metric in the Finslerian setting and obtained some models of relativity theory described through the warped product of Finsler metrics. In [13], Kozma–Peter–Varga defined their warped product for Finsler metrics. They related Cartan connection ∇ of a warped product Finsler M ×f N to the Cartan connections Received October 4, 2019, revised January 22, 2020, accepted June 5, 2020 Supported by Natural Science Foundation of Xinjiang Uygur Autonomous Region, China (Grant No. 2015211C277) 1) Corresponding author
S-curvature of Doubly Warped Product of Finsler Manifolds
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of M and N , and concluded that completeness of doubly warped product can be related to completeness of its components. Baagherzadeh and Rezaii [3] obtained necessary and sufficient conditions for a warped product Finsler space to be of constant curvature or of scalar curvature. More recently, He and Zhong extended the notion of warped product to complex Finsler manifold [10]. And warped product complex Finsler manifold was studied by many authors [9, 11]. The S-curvature is one of the most important non-Riemannian quantities in Finsler geometry which was originally introduced for the volume comparis
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