| In this paper, we study a new class of general (α, β)-metrics F defined by aRiemannian metric α, a1-form β andC∞function φ(b2, s).In Finsler Geometry, general (α, β)-metrics are an important and good theoreticalvalue class of Finsler metrics. In his doctoral dissertation, Yu Changtao made clearthe geomtric meaning of (α, β)-norms by considering the symmetry of the indica-trixes about Minkowski norms. These metrics not only generalize (α, β)-metricsnaturally, but also include some metrics structured by R. Bryant which are of con-stant positive flag curvature and have all the great cycles as their geodesics. Onthis basis, we discuss locally projectively flat generalized (α, β)-metrics and locallydually flat generalized (α, β)-metrics.This paper are divided into three parts: In the first part, we review the devel-opment history and research background of Finsler geometry, and make a statementabout the research results of domestic and international Finsler geometry in recentyears; In the second part, we study general (α, β)-metrics in the formwhere α is a Riemannian metric, β is a non-zero1-form, b=βαis a norms withrespect to α, ε is a non-zero constant. We verify that its is locally projectively flat,and apply these formulae to obtaining its flag curvature expression. In the third part, we characterization locally dually flat general (α, β)-metrics, and obtain itis necessary and sufcient conditions of locally dually flat, which generalizes someresults in Cheng et al.[1] and Xia [2][3]. |
