Adjacency preserving maps on symmetric tensors

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Let r and s be positive integers such that r⩾2. Let U and V be vector spaces over fields F and K, respectively, such that dim⁡U⩾3 and F has at least r+1 elements. In this paper, we characterize surjective maps ψ:SrU→SsV preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim⁡U⩾3 and the cardinality |F|⩾r+1 in our result.