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.