In
mathematics, a 'foliation' is a geometric device used to study
manifolds. Informally speaking, a foliation is a kind of "clothing" worn on a manifold, cut from a striped fabric. On each sufficiently small piece of the manifold, these
stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches (i.e.,
well-defined globally): a stripe followed around long enough might return to a different, nearby
stripe.
Definition
More formally, a
dimension foliation
of an
-dimensional manifold
is a covering by
charts together with maps
:
such that on the overlaps
the
transition functions
defined by
:
take the form
:
where
denotes the first
co-ordinates, and
denotes the last ''p'' co-ordinates. That is,
:
and
:
.
In the chart
, the 'stripes'
constant match up with the stripes on other charts
. Technically, these stripes are called 'plaques' of the foliation. In each chart, the plaques are
dimensional
submanifolds. These submanifolds piece together from chart to chart to form maximal
connected injectively
immersed submanifolds called the
leaves of the foliation.
Examples
Flat space
Consider an
-dimensional space, foliated as a product by subspaces consisting of points whose first
co-ordinates are constant. This can be covered with a single chart. The statement is essentially that
:
with the leaves or plaques
being enumerated by
. The analogy is seen directly in three dimensions, by taking
and
: the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.
Covers
If
is a covering between manifolds, and
is a foliation on
, then it pulls back to a foliation on
. More generally, if the map is merely a branched covering, where the branch
locus is transverse to the foliation, then the foliation can be pulled back.
Lie groups
If
is a
Lie group, and
is a
subgroup obtained by exponentiating a closed
subalgebra of the
Lie algebra of
, then
is foliated by
cosets of
.
Foliations and integrability
There is a close relationship, assuming everything is
smooth, with
vector fields: given a vector field
on
that is never zero, its
integral curves will give a 1-dimensional foliation. (i.e. a codimension
foliation).
This observation generalises to a theorem of
Ferdinand Georg Frobenius (the
Frobenius theorem), saying that the
necessary and sufficient conditions for a distribution (i.e. an
dimensional
subbundle of the
tangent bundle of a manifold) to be tangent to the leaves of a foliation, are that the set of vector fields tangent to the distribution are closed under
Lie bracket. One can also phrase this differently, as a question of
reduction of the structure group of the
tangent bundle from
to a reducible subgroup.
The conditions in the Frobenius theorem appear as
integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required
block structure exist.
There is a global foliation theory, because topological constraints exist. For example in the
surface case, an everywhere non-zero vector field can exist on an
orientable compact surface only for the
torus. This is a consequence of the
Poincaré-Hopf index theorem, which shows the
Euler characteristic will have to be 0.
See also
★
G-structure
★
Classifying space for foliations
★
Reeb foliation
★
Taut foliation