None of the last two concepts, the concept of wedge exterior product, and the concept of the exterior derivative depend upon the notion of a metric space. The exterior operations are defined on a variety for which the constraints of metric have not been defined. The exterior derivative is also independent of a connection. It is not equivalent to the tensor covariant derivative except under special circumstances.

Another operation in the Cartan exterior algebra is useful. It is the concept of the interior product. The interior product is essentially the concept of projective transversality or collinearity. The operations may look like the scalar product of ordinary vector analysis. However it is best to think of the interior product as an operation that linearly takes a p element into a p-1 element of the exterior algebra. The interior product of a (contravariant density) vector field with a 1-form creates a 0-form, or function. The interior product is often utilized in terms of a contravariant vector, but then the result is only well defined with respect to diffeomorphisms.

It is best to think of the interior product as being with respect to "currents", and not with respect to vectors. Currents are N-1 form structures on the N dimensional exterior algebra dual to the 1-form. Currents on the final state are well defined on the initial state with respect to the processes of adjoint pullback and functional substitution. The process is dual to the transpose pullback and functional substitution of p-forms.

As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.