Diffeomorphism Invariance

Diffeomorphism invariance refers to the form invariance of tensor(-equations)s under diffeomorphisms ([5], see also ► covariance).

A diffeomorphism Φ is a one-to-one mapping of a differentiable manifold M (or an open subset) onto another differentiable manifold N (or an open subset). Moreover, Φ (and its inverse Φ⁻¹) is differentiable. The concept of a diffeomorphism is intrinsically tied to the concept of a differentiable manifold. Here, we are mainly concerned with the four-dimensional spacetime manifold. The curves in Fig. 1 correspond to coordinate lines. There are two interpretations of the action of a diffeomorphism. A passive diffeomorphism changes one coordinate system to another one, like a cartesian to a polar coordinate system. Thus, one just changes the description of one and the same manifold (M = N). An active diffeomorphism corresponds to a transformation of the manifold which may be visualized as a smooth deformation of a continuous medium.