Let’s say we have a diffeomorphism 
from one 
-dimensional manifold to another. Since 
is both smooth and has a smooth inverse, we must find that the Jacobian is always invertible; the inverse of 
at 
is 
at 
. And so — assuming 
is connected — the sign of the determinant must be constant. That is, 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 
is either equivalent to or to 
. And I say that this equivalence comes out in integrals.
So further, let’s say we have a compactly-supported -form 
on . We can use to pull back from to . Then I say that
where the positive sign holds if is orientation-preserving and the negative if is orientation-reversing.
In fact, we just have to show the orientation-preserving side, since if is orientation-reversing from to then it’s orientation-preserving from to , and we already know that integrals over are the negatives of those over . Further, we can assume that the support of 
fits within some singular cube ![c:[0,1]^n\to M](http://s0.wp.com/latex.php?latex=c%3A%5B0%2C1%5D%5En%5Cto+M&bg=e6e6e6&fg=333333&s=0 "c:[0,1]^n\to M")
, for if it doesn’t we can chop it up into pieces that do fit into cubes 
, and similarly chop up into pieces that fit within corresponding singular cubes 
.
But now it’s easy! If is supported within the image of an orientation-preserving singular cube 
, then must be supported within 
, which is also orientation-preserving since both and 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}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cint%5Climits_N%5Comega%26%3D%5Cint%5Climits_%7Bf%5Ccirc+c%7D%5C%5C%26%3D%5Cint%5Climits_%7Bf%28c%28%5B0%2C1%5D%5En%29%29%7D%5Comega%5C%5C%26%3D%5Cint%5Climits_%7Bc%28%5B0%2C1%5D%5En%29%7Df%5E%2A%5Comega%5C%5C%26%3D%5Cint%5Climits_cf%5E%2A%5Comega%5C%5C%26%3D%5Cint%5Climits_Mf%5E%2A%5Comega%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\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 John Armstrong |
Differential Topology, Topology |
1 Comment
