Foliations
A 
-dimensional “foliation” 
of an 
-dimensional manifold 
is a partition of into -dimensional connected immersed submanifolds, which are called the “leaves” of the foliation. We also ask that the tangent spaces to the leaves define a -dimensional distribution 
on , which we say is “induced” by , and that any connected integral submanifold of should be contained in a leaf of . It makes sense, then, that we should call a leaf of a “maximal integral submanifold” of .
One obvious family of foliations arises as follows: let 
, and pick some -dimensional vector subspace 
. The quotient space 
consists of all the -dimensional affine spaces “parallel” to 
— if we pick a representative point 
then 
is one of the cosets in . The decomposition of into the collection of is a foliation. At any point the induced distribution is the subspace 
, which is the image of under the standard identification of with 
.
Now we have another theorem of Frobenius (prolific guy, wasn’t he?) about foliations: every integrable distribution of on comes from a foliation of .
Around any point we know we can find some chart 
so that the slices 
are all integrable submanifolds of . By the assumption that is second-countable we can find a countable cover of consisting of these patches.
We let 
be the collection of all the slices from all the patches in this cover, and define an equivalence relation 
on . We say that 
if there is a finite sequence 
of slices so that 
. Since each 
is a manifold, it can only intersect another chart 
in countably many slices, and from here it’s straightforward to show that each equivalence class of 
can only contain countably many slices. Taking the (countable) union of each equivalence class gives us a bunch of immersed connected integral manifolds of , and any two of these are either equal or disjoint, thus giving us a partition. And since any connected integral manifold of must align with one of the slices in any of our coordinate patches it meets, it must be contained in one of these leaves. Thus we have a foliation, which induces .
Like this:
Like Loading...
Related
July 1, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
[…] a nontrivial example of a foliation, I present the “Hopf fibration”. The name I won’t really explain quite yet, but […]
Pingback by The Hopf Fibration « The Unapologetic Mathematician | July 4, 2011 |