Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution

Let D be any division ring with an involution, . n (D) be the space of all n . n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank(A . B) = 1. It is proved that if . is a bijective map from . n (D)(n . 2) to itself such that . preserves the adjacency, then . .1 also preserves the adjacency. Moreover, if . n (D . S 3(F 2), then . preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe.Xian is answered for geometry of symmetric and hermitian matrices.