  We extend the construction of [19] by introducing spaces of generalized
tensor fields on smooth manifolds that possess optimal embedding and
consistency properties with spaces of tensor distributions in the sense of L.
Schwartz. We thereby obtain a universal algebra of generalized tensor fields
canonically containing the space of distributional tensor fields. The canonical
embedding of distributional tensor fields also commutes with the Lie
derivative. This construction provides the basis for applications of algebras
of generalized functions in nonlinear distributional geometry and, in
particular, to the study of spacetimes of low differentiability in general
relativity.
