Geometriae Dedicata

, Volume 138, Issue 1, pp 1–12 | Cite as

Diameter preserving bijections between Grassmann spaces over Bezout domains

Original Paper


Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by \({\mathbb{G}_k(_RR^n)}\) the k-dimensional Grassmann space on \({_RR^n}\). Let \({\varphi: \mathbb{G}_k(_RR^n)\rightarrow \mathbb{G}_k(_SS^n)}\) be a map. This paper proves the following are equivalent: (i) \({\varphi}\) is an adjacency preserving bijection in both directions. (ii) \({\varphi}\) 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 \({\varphi: \mathbb{G}_k(_RR^n)\rightarrow \mathbb{G}_k(_SS^n)}\) is an adjacency preserving bijection in both directions, then \({\varphi}\) is induced by either a collineation or the duality of a collineation.


Grassmann space Bezout domain Adjacency Diameter Preserving Projective space Collineation 

Mathematics Subject Classification (2000)

51A10 51A05 51M35 16D40 06C05 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.College of Mathematics and Computing ScienceChangsha University of Science and TechnologyChangshaChina

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