具有迷向S-曲率的指数度量

On the Exponential Metrics with Isotropic S-curvature

研究了Finsler几何中一类特殊(α,β)-度量-指数度量F=αeks的S-曲率性质。笔者通过把指数度量的S-曲率与其特殊S-曲率的表达式进行比较,采用代数方程公式运算的方法,分析方程因式指数的变化,得到了指数度量具有迷向S-曲率的充要条件:指数度量具有迷向S-曲率当且仅当它具有迷向平均Berwald曲率。此时,该度量的S-曲率为零,且是弱Berwald度量。结论表明:对于这类特殊的(α,β)-度量来说,它的曲率性质较简单,即它有迷向S-曲率等价于它有迷向平均Berwald曲率,等价于它具有为零的S-曲率。

The writer studied the S-curvature of a special metric（α,β）-exponential metrics in the Finsler geometry with the form F =ae~ ,where s =β/α,α = √aij（x）y^i y^j is a Riemannian metric, β =bi（x）y^i is a non-zero 1-form , k is a constant that. The sufficiency and necessary conditions are given by comparing expressions of the S - curvature and of exponential metrics and it＇s special S-curvature, adopting the formula operations and analyzing the changes of the exponent of equation factors, i.e. the exponential metrics are of ispotropic S- curbature , if and only if they are of isoptropic mean Berwald metrics . In this case, it＇ s S-curvature vanishes, i.e. S ffi O, and it is of weakly-Berwald metric. The curvature characteristic of this class of （α,β）-metrics is not complex, i.e. they are of isotropic S -curvature which means they are of isotropic mean Berwald curvature or their S-curvatures are zero.