We’ve defined how to integrate forms over chains made up of singular cubes, but we still haven’t really defined integration on manifolds. We’ve sort of waved our hands at the idea that integrating over a cube is the same as integrating over its image, but this needs firming up. In particular, we will restrict to oriented manifolds.
To this end, we start by supposing that an -form is supported in the image of an orientation-preserving singular -cube . Then we will define
Indeed, here the image of is some embedded submanifold of that even agrees with its orientation. And since is zero outside of this submanifold it makes sense to say that the integral over the submanifold — over the singular cube — is the same as the integral over the whole manifold.
What if we have two orientation-preserving singular cubes and that both contain the support of ? It only makes sense that they should give the same integral. And, indeed, we find that
where we use to reparameterize our integral. Of course, this function may not be defined on all of , but it’s defined on , where is supported, and that’s enough.
Of course now that we have more structure, we have more structured maps. But this time it’s not going to be quite so general; we will only extend our notion of an embedding, and particularly of an embedding in codimension zero.
That is, let be an embedding of manifolds where each of and has dimension . Since their dimensions are the same, the codimension of this embedding — the difference between the dimension of and that of — is . If and are both oriented, then we say that preserves the orientation if the pullback of any -form on which gives the chosen orientation gives us an -form on which gives its chosen orientation. We easily see that this concept wouldn’t even make sense if and didn’t have the same dimension.
More specifically, let and be oriented by -forms and , respectively. If for some smooth, everywhere-positive , we say that is orientation-preserving. The specific choices of and don’t matter; if gives the same orientation on then we must have for some smooth, everywhere-positive , and ; if gives the same orientation on then we must have for some smooth, everywhere-positive , and .
In fact, we have a convenient way of coming up with test forms. Let be a coordinate patch on around whose native orientation agrees with that of , and let be a similar coordinate patch on around . Now we have neighborhoods of and between which is a diffeomorphism, and we have top forms and in and , respectively. Pulling back the latter form we find
That is, the pullback of the (local) orientation form on differs from the (local) orientation form on by a factor of the Jacobian determinant of the function with respect to these coordinate maps. This repeats what we saw in the case of transition functions between coordinates. And so if whenever we pick local coordinates on and we find an everywhere-positive Jacobian determinant of , then preserves orientation.