Noncommutative Euclidean spaces

We give a definition of noncommutative finite-dimensional Euclidean spaces Rn. We then remind our definition of noncommutative products of Euclidean spaces RN1 and RN2 which produces noncommutative Euclidean spaces RN1+N2. We solve completely the conditions defining the noncommutative products of the Euclidean spaces RN1 and RN2 and prove that the corresponding noncommutative unit spheres SN1+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 TH2=U1(H)×U1(H).