The problem is connected with the existence of more than one differential structure for the Exotic R4.

The unit tangent bundle carries a variety of differential geometric structures.

As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold.

For large Betti numbers in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure.

With the help of this procedure one can produce countably infinite many differential structures.

But even for simple spaces like one doesn't know the construction of other differential structures.

For non-compact 4-manifolds there are many examples like having uncountably many differential structures.

Among other things, this work generalized the notion of differential structure to a noncommutative setting.

Anisotropic diffusion includes a variable conductance term that, in turn, depends on the differential structure of the image.

These criteria of differentiability can be applied to the transition functions of a differential structure.