  We construct an algebra of nonlinear generalized tensor fields on manifolds
in the sense of J.-F. Colombeau, i.e., containing distributional tensor fields
as a linear subspace and smooth tensor fields as a faithful subalgebra. The use
of a background connection on the manifold allows for a simplified construction
based on the existing scalar theory of full diffeomorphism invariant Colombeau
algebras on manifolds, still having a canonical embedding of tensor
distributions. In the particular case of the Levi-Civita connection on
Riemannian manifolds one obtains that this embedding commutes with pullback
along homotheties and Lie derivatives along Killing vector fields only.
