# The Unapologetic Mathematician

## 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

$\displaystyle\int\limits_Mf^*\omega=\pm\int\limits_N\omega$

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