Noncommutative theories and general coordinate transformations

We study the class of noncommutative theories in $d$ dimensions whose spatial coordinates $(x_i{)}_{i=1}^d$ can be obtained by performing a smooth change of variables on $(y_i{)}_{i=1}^d$, the coordinates of a standard noncommutative theory, which satisfy the relation $[y_i,y_j]=i{\theta }_{ij}$, with a constant ${\theta }_{ij}$ tensor. The $x_i$ variables verify a commutation relation which is, in general, space-dependent. We study the main properties of this special kind of noncommutative theory and show explicitly that, in two dimensions, any theory with a space-dependent commutation relation can be mapped to another where that ${\theta }_{ij}$ is constant.