I'm fairly new to differential geometry and have been reading up on integration on manifolds. All the texts/lecture notes that I've read so far always consider integrating an $n$-form over an $n$-dimensional manifold, what confuses me about this is why one can't integrate a $k$-form (with $k<n$) over an $n$-dimensional manifold? Is it only possible to do so by introducing mappings from the $n$-dimensional manifold to $k$-dimensional manifolds (integration on a chain)?
Edit: I've also read that in order for integration to be well-defined on a manifold, the integrand must be a top-form (is this to do with ensuring coordinate independence of integration on a manifold?). Is this the case, and if so, why?
Apologies if this is a really rudimentary question, but it's proving a real snagging point in my understanding.
Best Answer
It's a matter of principle.[*] A $k$-form eats $k$ vectors and spits out a number. If you want to integrate a $k$-form over a $m$-dimensional submanifold, how do you choose which $k$ of the tangent vectors you're going to plug in to the form on each tangent space? If $k<m$, you'll have leftover vectors and if $k>m$ you won't have enough vectors. Either way, you have to make some kind of arbitrary choice and there's no natural way of doing so, so to begin with we limit integration of to same-dimensional submanifolds.
This is actually important in differential topology because de Rham's theorem identifies the $k^{th}$ de Rham cohomology ring with the $k^{th}$ singular cohomology ring via integration over $k$-dimensional submanifolds. If you think of a submanifold as representing a homology class, then because you integrate forms over submanifolds instead of the whole space, you can think of a form as representing a cohomology class. There's quite a bit more detail work, of course, but that's the basic idea.
If you specify a way of making the choice if tangent vectors, on the other hand, for instance by talking about $k$-dimensional distributions, then you've started in on the theory of currents.
[*] The principle is that you don't want to make any choices without a good reason. If you do make a choice, you want to be aware that you're making a choice and think about all the different ways you can make that choice.