On the Class of(q,α,β)-Metrics

Ahmad Alimohammadi

doi:10.1155/2014/596090

Abstract

We study a class of Finsler metrics in the formF=α+βq/αq-1, whereαis a Riemannian metric,βis a1-form, and1<q<2. Fis called(q,α,β)-metrics. We find the necessary and sufficient conditions under which the class of(q,α,β)-metrics is locally projectively flat and Douglas metrics, respectively.

Highlights

It is Hilbert’s Fourth Problem in the regular case to study and characterize Finsler metrics on an open domain U ⊂ Rn whose geodesics are straight lines
It is easy to see that a Finsler metric F = F(x, y) on an open subset U ⊂ Rn is projectively flat if and only if the spray coefficients are in the form Gi = Pyi, where P is a positively homogeneous function of degree one in y
In [5], it is proved that a Randers metric F = α+β is a Douglas metric if and only if β is closed. For another example a Matsumoto metric F = α2/(α − β) and the exponential metric F = α exp(β/α) + εβ are Douglas metrics if and only if β is parallel with respect to α

Summary

Introduction

It is Hilbert’s Fourth Problem in the regular case to study and characterize Finsler metrics on an open domain U ⊂ Rn whose geodesics are straight lines. The Beltrami Theorem tells us that a Riemannian metric α = √aij(x)yiyj is locally projectively flat if and only if it is of constant sectional curvature. This problem has been solved in Reimannian Geometry. F is locally projectively flat if and only if the following conditions hold: ISRN Geometry (1) β is parallel with respect to α;. It is known that a locally projectively flat Finsler metric is a Douglas metric, but the converse is not necessarily true. By Theorems 1 and 2, if F = (α + β)q/αq−1 (1 < q < 2) is a Douglas metric and α is a locally projectively flat Riemannian metric, F is a projectively flat Finsler metric.

Preliminaries

Douglas Metric