The Unapologetic Mathematician

Mathematics for the interested outsider

Integrals and Diffeomorphisms

Let’s say we have a diffeomorphism f:M^n\to N^n from one n-dimensional manifold to another. Since f is both smooth and has a smooth inverse, we must find that the Jacobian is always invertible; the inverse of J_f at p\in M is J_{f^{-1}} at f(p)\in N. And so — assuming M is connected — the sign of the determinant must be constant. That is, f is either orientation preserving or orientation-reversing.

Remembering that diffeomorphism is meant to be our idea of what it means for two smooth manifolds to be “equivalent”, this means that N is either equivalent to M or to -M. And I say that this equivalence comes out in integrals.

So further, let’s say we have a compactly-supported n-form \omega on N. We can use f to pull back \omega from N to M. Then I say that


where the positive sign holds if f is orientation-preserving and the negative if f is orientation-reversing.

In fact, we just have to show the orientation-preserving side, since if f is orientation-reversing from M to N then it’s orientation-preserving from -M to N, and we already know that integrals over -M are the negatives of those over M. Further, we can assume that the support of f^*\omega fits within some singular cube c:[0,1]^n\to M, for if it doesn’t we can chop it up into pieces that do fit into cubes c_i, and similarly chop up N into pieces that fit within corresponding singular cubes f\circ c_i.

But now it’s easy! If f^*\omega is supported within the image of an orientation-preserving singular cube c, then \omega must be supported within f\circ c, which is also orientation-preserving since both f and c are, by assumption. Then we find

\displaystyle\begin{aligned}\int\limits_N\omega&=\int\limits_{f\circ c}\\&=\int\limits_{f(c([0,1]^n))}\omega\\&=\int\limits_{c([0,1]^n)}f^*\omega\\&=\int\limits_cf^*\omega\\&=\int\limits_Mf^*\omega\end{aligned}

In this sense we say that integrals are preserved by (orientation-preserving) diffeomorphisms.

September 12, 2011 Posted by | Differential Topology, Topology | 1 Comment