Unit-circle-preserving mappings

If f is an isometry, then every distance r > 0 is conserved by f , and vice versa. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → X preserving a distance r > 0 is an isometry, which is now known to us as the Aleksandrov problem. Without loss of generality, we may assume r = 1 when X is a normed space (see [16]). Beckman and Quarles [2] solved the Aleksandrov problem for finite-dimensional real Euclidean spaces X =Rn (see also [3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 17, 18, 19, 20]). Theorem 1.1 (Beckman andQuarles). If amapping f :Rn→Rn (2≤n<∞) preserves a distance r > 0, then f is a linear isometry up to translation.