某些芬斯拉空间的对称性质

<正> E.嘉当在他的黎曼几何教程中系统地讨论了对称的黎曼空间,并给出了充要条件的分析形式及一系列有趣的性质.本文在芬斯拉空间中引进了嘉当在黎曼几何中所定义的“对称”概念后(第一节),对这类芬斯拉空间的对称性质作了详尽的讨论.得到的结果如下:(一)在 F_n 的一区域Ω内,把任一向量关于0点(O∈Ω)作对称推移和沿经过0的极值曲线作平行推移(以后在不引起混淆的情祝下,简称为“向量经过平行推移及对称推移”),为使这时所得结果之差为三阶小量,充要条件是:挠率张量的共变导数在Ω中等于零.E.Cartan 对这种空间巳作了一些几何说明,而这里给了一个新的几何特征.我们称这样的芬斯拉空间为亚对称的,黎曼空间即口为其中最常见的一个.

ON THE SYMMETRIC PROPERTIES IN SOME FINSLER SPACES

Ia Riemannian space,E.Cartan had introduced the notions of symmetric transformation ofpoints and symmetric transpose of vectors reference to a fixed point O along the geodesics passingthrough O.Furthermore,he had calculated the difference between the parallel and symmetrictranspose of a vector,by means of which,a special class of Riemannian spaces so called sym-metric Riemannian space was defined in which the difference is an infinitesimal of order three,andmany geometric properties was inserted.In this note,we give a direct extension of these resultsin Finsler spaces,and establish the following theorem:THEOREM In order that the difference in question is an infinitesimal of order twoin a rigion D,it is necessary and sufficient that the covariant derivatives of torsion tensor vanishesin the rigion D.In this theorem, the family of geodesics in the definition of symmetric transpose cannot bereplaced by any other family of curves,though we can give a more general definition of symmetrictranspose of veclors independ on family of geodesics.We call the space in theorem the subsymmetric Finsler space.By a simple calculation,thegeometric definition of another special Finsler space——symmetric Finsler space with its tensorcharacter was given.