TAILIEUCHUNG - Generalized Berwald metrics

In this paper, we consider a class of Finsler metrics called generalized Berwald metrics which contains the class of Berwald metrics as a special case. We prove that every generalized Berwald metrics with non-zero scalar flag curvature or isotropic Berwald curvature is a Randers metric. | Turk J Math 36 (2012) , 475 – 484. ¨ ITAK ˙ c TUB doi: Generalized Berwald metrics Esmaeil Peyghan and Akbar Tayebi Abstract In this paper, we consider a class of Finsler metrics called generalized Berwald metrics which contains the class of Berwald metrics as a special case. We prove that every generalized Berwald metrics with non-zero scalar flag curvature or isotropic Berwald curvature is a Randers metric. Then we prove that on generalized Berwald metrics, the notions of generalized Landsberg and Landsberg curvatures are equivalent. Key Words: Berwald metric, landsberg metric, randers metric 1. Introduction For a Finsler metric F = F (x, y), its geodesics curves are given by the system of differential equations c¨i + 2Gi (c) ˙ = 0 , where the local functions Gi = Gi (x, y) are called the spray coefficients. A Finsler metric is called a Berwald metric if Gi are quadratic in y ∈ Tx M for any x ∈ M . The Berwald spaces can be viewed as Finsler spaces modeled on a single Minkowski space [6]. On the other hand, various interesting special forms of Cartan, Landsberg and Berwald tensors have been obtained by some Finslerians. The Finsler spaces having such special forms have been called C-reducible, isotropic Berwald curvature and isotropic Landsberg curvature, etc. [4][5][7][9][11][12][13]. In [8], Matsumoto introduced the notion of C-reducible metrics and proved that any Randers metric is C-reducible. Later on, Matsumoto-H¯ oj¯ o proved that the converse is true, too [10]. A Randers metric F = α + β is just a Riemannian metric α perturbated by a one-form β which has important applications both in mathematics and physics [14]. In [4], Shen-Chen by using the structure of Funk metric, introduce the notion of isotropic Berwald metrics. This motivates us to study special forms of Berwald curvature for other important special Finsler metrics. We call a Finsler metric F to be generalized Berwald metric if its Berwald curvature satisfies the relation B

• Toán học
• Generalized Berwald metrics
• Berwald metric
• Landsberg metric
• Randers metric
• Non zero scalar flag curvature