Noncommutative Euclidean spaces

We give a definition of noncommutative finite-dimensional Euclidean spaces. We then remind our definition of noncommutative products of Euclidean spaces. We solve completely the conditions defining the noncommutative products of the Euclidean spaces R^N1 and R^N2 and prove that the corresponding noncommutative unit spheres S^(N1+N2−1) are noncommutative spherical manifolds. We then apply these concepts to define ‘‘noncommutative’’ quaternionic planes and noncommutative quaternionic tori on which acts the classical quaternionic torus T^2 = U1(H) × U1(H).