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. Later on, the concept of a Covariant derivative will be
explored in detail. The primitive idea is that a Covariant derivative is a
constrained or restricted process which when acting on a tensor produces another
tensor.
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 interior product operation may
look like the scalar product of ordinary vector analysis, but the idea is a bit
more complex. It is best to think of the interior product as an operation that
linearly takes a p element into a p1 element of the exterior algebra. Given a
p form on the final state, and operation that takes a p form to a p1 form on
the final state, it must follow that the pullback of the p1 form is well
defined. However, a contravariant vector field on the final state is not well
defined on the initial state if the inverse Jacobian matrix does not exist.
However a contravaraint vector density on the final state can be pulled back to
the initial state by means of the adjoint of the Jacobian matrix, even though
the inverse is not defined. Hence, the inner product of a pform and a
contravariant vector density pulled back is well defined in terms of the
interior product of the pullback of the pform and the pull back of the
contravariant vector density.
The interior product of a contravariant vector density or current with a 1form
creates a 0form, or function. The interior product is often utilized in terms
of a contravariant vectors, not contravariant vector densities, but then the
result is only well defined with respect to diffeomorphisms, for which the
inverse Jacobian is defined.
It is best to think of the interior product as being with respect to "currents",
and not with respect to vectors. Currents are N1 form structures on the N
dimensional exterior algebra dual to the 1form. 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 pforms.
