  There is a need in general relativity for a consistent and useful
mathematical theory defining the multiplication of tensor distributions in a
geometric (diffeomorphism invariant) way. Significant progress has been made
through the concept of Colombeau algebras, and the construction of full
Colombeau algebras on differential manifolds for arbitrary tensors. Despite the
fact that this goal was achieved, it does not incorporate clearly enough the
concept of covariant derivative and hence is of a limited use. We take a
different approach: we consider any type of preference for smooth distributions
(on a smooth manifold) as nonintuitive, which means all our approach must be
based fully on the Colombeau equivalence relation as the fundamental feature of
the theory. After taking this approach we very naturally obtain a canonical and
geometric theory defining tensorial operations with tensorial distributions,
including covariant derivative. This also happens because we no longer need any
explicit canonical geometric construction of Colombeau algebras. The big
advantage of our approach lies also in the fact that it brings a physical
insight into the mathematical concepts used and naturally leads to formulation
of physics on (what we call) piecewise smooth manifolds, rather than on smooth
manifold. This brings to the language of physics much higher symmetry (in the
same way as turning from Poincare invariance to diffeomorphism invariance), and
is compatible with our intuition, that "pointwise" properties in some
metaphorical sense "do not matter".
