The Unapologetic Mathematician

Mathematics for the interested outsider

Orientation-Preserving Mappings

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 f:M\to N be an embedding of manifolds where each of M and N has dimension n. Since their dimensions are the same, the codimension of this embedding — the difference between the dimension of N and that of M — is 0. If M and N are both oriented, then we say that f preserves the orientation if the pullback of any n-form on N which gives the chosen orientation gives us an n-form on M which gives its chosen orientation. We easily see that this concept wouldn’t even make sense if M and N didn’t have the same dimension.

More specifically, let M and N be oriented by n-forms \omega_M and \omega_N, respectively. If f^*\omega_N=\lambda\omega_M for some smooth, everywhere-positive \lambda\in\mathcal{O}(M), we say that f is orientation-preserving. The specific choices of \omega_M and \omega_N don’t matter; if \omega_M' gives the same orientation on M then we must have \omega_M=\phi\omega_M' for some smooth, everywhere-positive \phi, and f^*\omega_N=\lambda\phi\omega_M'; if \omega_N' gives the same orientation on N then we must have \omega_N'=\phi\omega_N for some smooth, everywhere-positive \phi, and f^*\omega_N'=\phi f^*\omega_N=\phi\lambda\omega_M.

In fact, we have a convenient way of coming up with test forms. Let (U,x) be a coordinate patch on M around p whose native orientation agrees with that of M, and let (V,y) be a similar coordinate patch on N around f(p). Now we have neighborhoods of p and f(p) between which f is a diffeomorphism, and we have top forms dx^1\wedge\dots\wedge dx^n and dy^1\wedge\dots\wedge dy^n in U and V, respectively. Pulling back the latter form we find

\displaystyle\begin{aligned}f^*(dy^1\wedge\dots\wedge dy^n)&=d(y^1\circ f)\wedge\dots\wedge d(y^n\circ f)\\&=\left(\sum\limits_{i_1=1}^n\frac{\partial(y^1\circ f)}{\partial x^{i_1}}dx^{i_1}\right)\wedge\dots\wedge\left(\sum\limits_{i_n=1}^n\frac{\partial(y^n\circ f)}{\partial x^{i_n}}dx^{i_n}\right)\\&=\det\left(\frac{\partial(y^j\circ f)}{\partial x^{i}}\right)dx^1\wedge\dots\wedge dx^n\end{aligned}

That is, the pullback of the (local) orientation form on N differs from the (local) orientation form on M by a factor of the Jacobian determinant of the function f 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 M and N we find an everywhere-positive Jacobian determinant of f, then f preserves orientation.

September 1, 2011 Posted by | Differential Topology, Topology | 7 Comments

   

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