On a Class of Projectively Flat Metrics with Constant Flag Curvature

Abstract

In this paper, we find equations that characterize locally projectively flat Finsler metrics in the form F = (α + β) 2 /α, where α = q a i j y i y j is a Riemannian metric and β = b i y i is a 1-form. Then we completely determine the local structure of those with constant flag curvature.

Projectively flat general (α,β)-metrics with constant flag curvature

“…(1 − |x| 2 ) 2 (1 − |x| 2 )|y| 2 + x, y 2 (1.2) are also locally projectively flat with constant flag curvature K = 0 [10]. (1.2) belong to the so-called square metrics given in the form F = (α+β) 2 α .…”

On projectively flat Finsler spaces

“…In the last time especially the projectively flatness of Randers and Einstein spaces came into the lime light. See T. Q. Binh and X. Cheng [5], Z. Shen, and C. Yildrim [23], B. Li and Z. Shen [14], Z. Shen [20,21,22], X. Cheng and M. Li [9].…”

The flag curvature of invariant (α, β)-metrics of type \frac{(\alpha+\beta)^2}{\alpha}

“…(α, β)-metrics are interesting Finsler metrics which have been studied by many Finsler geometers. F = (α+β) 2 α is a special (α, β)-metric which has been studied by Z. Shen and G. C. Yildirim (see [15].). In this paper we study flag curvature of these metrics on homogeneous spaces G/H which are invariant under the action G. We suppose that α is induced by an invariant Riemannian metric g on the homogeneous space and the Chern connection of F coincides to the Levi-Civita connection of g. Also we study the special cases when (G/H, g) is naturally reductive or when H is trivial (H = {e}).…”