论射影平坦空间的一个特征

<正> §1.如所周知,黎曼空间中关于平面公理的嘉当(E.Cartan)定理可以拓广到更一般的空间中去,满足平面公理的 m 维黎曼空间在它的每点容有∞~(m-1)张全测地超曲面.柏尔特拉米(Beltrami)给出常曲率空间的另一特征,只有常曲率空间才能与欧氏空

A CHARACTERIZATION OF A PROJECTIVE FLAT SPACE

The aim of the present paper is to give a characterization of a projectiveflat space.It has an intimate connection with Cartan's theorem on the axiomof plane and Beltrami's on the geodesic correspondence of a Riemannian spaceand a fiat space.The main result is as follows:Theorem:If an m-dimensional affine connected space without torsioncontains m+2 families of totally geodesic hypersurfaces,such that there existsa mapping of V_m into affine space A_m which brings the m+2 families ofhypersurfaces onto m+2 families of planes in general position,then the spacemust be projective flat and conversely.Especially,in the case of a Riemannian space,we obtain a characterizationof a space of constant curvature.It is also shown that the number m+2 cannot be replaced by a smaller one.Even when the images of m+1 families among m+2 families are hyperplanesand the remaining one is not,the space is not always projectively flat.