tailieunhanh - On Finsler metrics with vanishing S-curvature

In this paper, we consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We these metrics with vanishing S-curvature. We find some conditions under which such a Finsler metric is Berwaldian or locally Minkowskian. | Turkish Journal of Mathematics Turk J Math (2014) 38: 154 – 165 ¨ ITAK ˙ c TUB ⃝ doi: Research Article On Finsler metrics with vanishing S-curvature Akbar TAYABI1,∗, Hassan SADEGHI1 , Esmaeil PEYGHAN2 Department of Mathematics, Faculty of Science University of Qom, Qom, Iran 2 Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran 1 Received: • Accepted: • Published Online: • Printed: Abstract: In this paper, we consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We study these metrics with vanishing S-curvature. We find some conditions under which such a Finsler metric is Berwaldian or locally Minkowskian. Key words: (α, β) -metric, Berwald metric, S-curvature. 1. Introduction In Finsler geometry, there are several important non-Riemannian quantities: the Cartan torsion C, the Berwald curvature B, the S-curvature S, the new non-Riemannian curvature H, etc. They all vanish for Riemannian metrics; hence they are said to be non-Riemannian [6, 7, 9]. ∂ ∂ i Let (M, F ) be a Finsler manifold. The Finsler metric F on M induced a spray G = y i ∂x i −2G (x, y) ∂y i , which determines the geodesics, where Gi = Gi (x, y) are called the spray coefficients of G . A Finsler metric F is called a Berwald metric if Gi = 1 i j k 2 Γjk (x)y y are quadratic in y ∈ Tx M for any x ∈ M . The Berwald curvature B of Finsler metrics is an important non-Riemannian quantity constructed by L. Berwald. The S-curvature is constructed by Shen for given comparison theorems on Finsler manifolds [10]. A natural problem is to study and characterize Finsler metrics of vanishing S-curvature. It is known that some Randers metrics are of vanishing S-curvature [8, 13]. This is one of our motivations to consider Finsler metrics with vanishing S-curvature. Shen proved that every Berwald metric satisfies S = 0 [10]. In [2], Bao .