Geodesic
Transformations of n-Space which Preserve a Fixed Square-Distance
Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 392-395

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1. Introduction. Our interest here lies in the following theorem:THEOREM 1. Assume there is defined on R n (n ≧ 3) a “square-distance” of the form where (g ij) is a given symmetric non-singular matrix over the reals and x = (x 1, ..., x n ), y = (y 1, ..., y n ) ∈ R n . Assume further that f is a bijection ofR n which preserves a given fixed square-distance ρ, i.e. d(x, y) = ρ if and only if d(ƒ(x),ƒ(y)) = ρ. Then (unless ρ = 0 and (gij) is positive or negative definite) ƒ(x) = Lx + ƒ(0), where L is a linear bijection ofR nsatisfying d(Lx, Ly) = ±d(x, y) for all x, y ∈ R n (the – sign is possible if and only if ρ = 0 and (g ij ) has signature 0). 
Lester, J. A. Transformations of n-Space which Preserve a Fixed Square-Distance. Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 392-395. doi: 10.4153/CJM-1979-043-6
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