The Study On Ricci Curvature Properties Of (α , β)-metrics

(α,β)-metrics form a very important class of Finsler metrics, whereαdenotes a Riemannian metric andβdenotes an 1-form on the manifold. In this paper, we mainly study the Ricci curvature of (α,β)-metrics. We first obtain the formulas of the Riemann curvature and Ricci curvature of (α,β)-metrics by a series of complex and meticulous computations. Based on these important formulas, in order to reveal the influence of Ricci curvature on the structure of (α,β)-spaces, we mainly study Einstein (α,β)-metrics and obtain the locally equivalent equations for an (α,β)-metric to be an Einstein metric. By use of these pivotal equations, we discuss the sufficient and necessary conditions for some important (α,β)-metrics F =αφ(β/α)to be Einstein metrics, whereφ=φ( s) is a C∞function. We emphatically study those (α,β)-metrics whoseφ( s) is either a polynomial in degree k ( k≥2) or an exponential function in the form ep ( s )( p ( s ) is a polynomial in degree k ( k≥1)). We obtain the sufficient and necessary conditions of such kinds of (α,β)-metrics to be Einstein metrics, and obtain the following important results: such two kinds of (α,β)-metrics are Einsteinian if and only if they are Ricci-flat. Finally, we locally characterize a class of Einstein (α,β)-metrics with 2φ( s ) = 1 +ε₁ s +ε₂s, whereε2≠0 andε₁2 ? 4ε2≠0. Our results indicate that this kind of Einstein (α,β)-metrics are not only Ricci-flat, but also whoseβis parallel with respect toα, i.e. they are Berwald metrics. Further, the Riemannian metricαis Ricci-flat too.