The theory of metrical distortions and turbulences [1-3] is improved by some further argumentation details concerning volume and length preservation. Especially, an example in the situation of radially symmetric manifolds shows that even in simple situations, classical differentiability is not covering the problem. Moreover, as an immediate consequence, it is shown that the Lie algebra of a Lie group is a synthetical coordinate system, what is an essential theorem for a complete substantial survey within the differential topology of Lie groups.

general transformation groups, metrical distortions, radial symmetric manifolds, Lie groups.