Afr. Mat. DOI 10.1007/s13370-012-0132-4 On Kropina metrics H. G. Nagaraja Received: 11 March 2012 / Accepted: 20 December 2012 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013 Abstract In this paper we study the curvature properties of Kropina metric. We find expres- sions for Riemann curvature and Ricci curvature of a Kropina metric when the 1-form β is a Killing form of constant length. We give a characterization of projectively flat Kropina metric and Kropina metric with isotropic S-curvature. Keywords S-curvature · Kropina metric · Projectively flat · Ricci curvature · Riemann curvature. Mathematics Subject Classification (2010) 53B40 · 53C60. 1 Introduction A Finsler metric L (x , y ) on an n-dimensional manifold M n is called an (α, β)-metric [3] L (α, β) if L is positively homogeneous function of α and β of degree one, where α 2 = a ij (x ) y i y j is a Riemannian metric and β = b i (x ) y i is a 1-form on M n . Kropina metrics are special (α, β) metrics defined in the form F = α 2 β . They form an important class of Finsler metrics investigated by Kropina [2] in 1961. Since then many authors have investigated the geometric properties of Kropina metrics. There are several interesting curvatures in Finsler geometry, among them two important and interesting curvatures are Riemann curvature and the Ricci curvature. The Ricci curvature plays an important role in the geometry of Finsler manifolds and is defined as the trace of the Riemannian curvature on each tangent space. Shen [4] introduced the notion of S- curvature, a non-Riemannian quantity which measures the rate of change of the volume form of a Finsler space along the geodesics. The S-curvature This work is supported by CSIR, Govt. of India under the grant 25(0179)/10/EMR-II. H. G. Nagaraja (B ) Department of Mathematics, Bangalore University, Central College Campus, Bangalore 577451, Karnataka, India e-mail: hgnraj@yahoo.com 123