Diameter preserving bijections between Grassmann spaces over Bezout domains

作者：Huang; Li Ping

来源：Geometriae Dedicata, 2009, 138(1): 1-12.

DOI：10.1007/s10711-008-9295-4

• Grassmann space
• Bezout domain
• Adjacency
• Diameter
• Preserving
• Projective space
• Collineation

Let R, S be Bezout domains. Assume that n is an integer >= 3, 1 <= k <= n - 2. Denoted by G(k) (R-R(n)) the k-dimensional Grassmann space on R-R(n). Let phi : G(k) (R-R(n)) -> Gk (S-S(n)) be a map. This paper proves the following are equivalent: ( i) phi is an adjacency preserving bijection in both directions. (ii) phi is a diameter preserving bijection in both directions. Moreover, Chow's theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If phi : G(k) (R-R(n)) -> G(k) (S-S(n)) is an adjacency preserving bijection in both directions, then phi is induced by either a collineation or the duality of a collineation.

• 出版日期2009-2
• 单位长沙学院