Abstract
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.
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Project 10671026 supported by National Natural Science Foundation of China.
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Huang, LP. Diameter preserving bijections between Grassmann spaces over Bezout domains. Geom Dedicata 138, 1–12 (2009). https://doi.org/10.1007/s10711-008-9295-4
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DOI: https://doi.org/10.1007/s10711-008-9295-4