摘要
总可以在∧ｎＲ２ｎ上定义两个对称的双线性型（Ωαβ）和（Ｊαβ），它们分别由Ｒ２ｎ的体积元和Ｒ２ｎ上的辛形式确定．特别，当ｎ＝２时，视（Ωαβ）和（Ｊαβ）为Ｐ５中的两个配极，我们证明了：存在这两个配极的绝对形的交集和Ｌｉｅ’ｓ圆的集合之间的一一对应，并且，两个Ｌｉｅ′ｓ圆同向相切当且仅当它们在Ｐ５中的像点关于（Ｊαβ）彼此共轭，此外，Ｐ５中的射影变换Ｇ保持（Ωαβ）不变当且仅当Ｇ＝∧，∈ＰＧＬ（４，Ｒ），又如果Ｇ还保持（Ｊαβ）不变，则必∈ＰＧｓｐ（４）．于是，我们得到圆素几何的射影模式，这个几何空间的运动群是ＰＧｓｐ（４）．
Two bilinear symmetric forms (Ω αβ ) and (J αβ ) , are defined on ∧ n R 2n , they are determined by the element of volume in R 2n and the symplectic form on R 2n respectively. In particular, when n=2 , we consider (Ω αβ ) and (J αβ ) as two polars on P 5 . It is proved that: There exists one-to-one map between the cross set of absolute of these two polars and the set of Lie′s circle in the plane, and two Lie′s circles are tangential along the same direction if and only if their images in P 5 are conjugate each other with respect to the polar ( J αβ ) , and what is more, the projective transformation G in P 5 preserves the polar (Ω αβ ) invariant if and only if G= g∧ g, g∈ PGL (4,R), and if G preserves also the polar (J αβ ) , then g∈ PGsp(4). Thus, we obtain the projective model of the circle′s geometry, the motion group of this geometric space is PGsp(4) .
出处
《数学进展》
CSCD
北大核心
1997年第6期507-514,共8页
Advances in Mathematics（China）
