不可度量化的射影平坦Spray的构造

Construction of non-metrizable projectively flat Spray

刻画射影平坦Finsler度量是著名的Hilbert第四问题正则性情形,且任意一个Finsler度量可以通过它的测地线方程诱导一个Spray,因此研究射影平坦Spray的可度量化问题令人关注.本文研究一类射影平坦Spray的可度量化问题,通过欧氏度量|y|和内积(27)x, y (29)的线性组合,构造两类射影平坦Spray;其次利用反证法和具有迷向曲率Spray的定义,证明以上两类Spray均不由任意Finsler度量诱导,且不具有迷向曲率.

The regularity of Hilbert’s fourth problem concerns characterization of the projectively flat Finsler metric,and every Finsler metric induces a spray by its geodesic formula.This arouses attention to study the measurable problems of projectively flat spray.In this paper,we study the measurable problems of a class of projectively flat sprays,and construct two groups of projectively flat sprays by the linear combination of Euclidean metric and inner product.Next,using the proofs by contradiction and the definition of isotropic curvature for Spray,it is proved that the above Sprays are not induced by any Finsler metric and they are not of isotropic curvature.