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
.
[…] a nontrivial example of a foliation, I present the “Hopf fibration”. The name I won’t really explain quite yet, but […]
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