Ah, I see. Makes sense now. Thank you! One more question. The way you play with the coordinates is independent from a point to another? I.e. can you at a point keep the coordinates still while at a nearby point on the manifold you rotate the coordinate system by 90 degrees? Or the transformation should be smooth along the manifold? So if the basis vectors at a point transform using a matrix M, is this matrix a smooth function M(p), or it can take any value at any point on the manifold?