In
mathematics (
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mul ...
), a foliation is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on an
''n''-manifold, the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es being connected,
injectively immersed submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
s, all of the same dimension ''p'', modeled on the
decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ...
of the
real coordinate space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
R
''n'' into the
cosets ''x'' + R
''p'' of the standardly
embedded subspace R
''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a
piecewise-linear,
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in i ...
(of class ''C
r''), or
analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''C
r'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''
0 is a topological foliation). The number ''p'' (the dimension of the leaves) is called the dimension of the foliation and is called its
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals t ...
.
In some papers on
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant
Lorentz manifold (a (''p''+1)-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
) has been decomposed into
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
s of dimension ''p'', specified as the level sets of a real-valued
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
(
scalar field) whose
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere
time-like
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally,
as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.
Foliated charts and atlases
In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.
A ''rectangular
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
'' in R
''n'' is an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (Y ...
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the form ''B'' = ''J''
1 × ⋅⋅⋅ × ''J''
n, where ''J
i'' is a (possibly unbounded) relatively open interval in the ''i''th coordinate axis. If ''J''
1 is of the form (''a'',0], it is said that ''B'' has
boundary (topology), boundary
:
In the following definition, coordinate charts are considered that have values in R
''p'' × R
''q'', allowing the possibility of manifolds with boundary and (
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polyto ...
) corners.
A ''foliated chart'' on the ''n''-manifold ''M'' of codimension ''q'' is a pair (''U'',''φ''), where ''U'' ⊆ ''M'' is open and
is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
,
being a rectangular neighborhood in R
''q'' and
a rectangular neighborhood in R
''p''. The set ''P
y'' = ''φ''
−1(''B
τ'' × ), where
, is called a ''plaque'' of this foliated chart. For each x ∈ ''B
τ'', the set ''S
x'' = ''φ''
−1( ×
) is called a ''
transversal'' of the foliated chart. The set ''∂
τU'' = ''φ''
−1(''B
τ'' × (''∂''
)) is called the ''tangential boundary'' of ''U'' and
= ''φ''
−1((''∂B
τ'') ×
) is called the ''transverse boundary'' of ''U''.
The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation ''B
τ'' is read as "''B''-tangential" and
as "''B''-transverse". There are also various possibilities. If both
and ''B
τ'' have empty boundary, the foliated chart models codimension-''q'' foliations of ''n''-manifolds without boundary. If one, but not both of these rectangular neighborhoods has boundary, the foliated chart models the various possibilities for foliations of ''n''-manifolds with boundary and without corners. Specifically, if ''∂''
≠ ∅ = ''∂B
τ'', then ''∂U'' = ''∂
τU'' is a union of plaques and the foliation by plaques is tangent to the boundary. If ''∂B
τ'' ≠ ∅ = ''∂''
, then ''∂U'' =
is a union of transversals and the foliation is transverse to the boundary. Finally, if ''∂''
≠ ∅ ≠ ''∂B
τ'', this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.
A ''foliated
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographi ...
'' of codimension ''q'' and class ''C
r'' (0 ≤ ''r'' ≤ ∞) on the ''n''-manifold ''M'' is a ''C
r''-atlas
of foliated charts of codimension ''q'' which are ''coherently foliated'' in the sense that, whenever ''P'' and ''Q'' are plaques in distinct charts of
, then ''P'' ∩ ''Q'' is open both in ''P'' and ''Q''.
A useful way to reformulate the notion of coherently foliated charts is to write for ''w'' ∈ ''U''
α ∩ ''U''
β
:
:
The notation (''U''
α,''φ''
α) is often written (''U''
α,''x''
α,''y''
α), with
:
:
On ''φ''
β(''U''
α ∩ ''U''
β) the coordinates formula can be changed as
:
The condition that (''U''
''α'',''x''
''α'',''y''
''α'') and (''U''
''β'',''x''
''β'',''y''
''β'') be coherently foliated means that, if ''P'' ⊂ ''U''
''α'' is a plaque, the connected components of ''P'' ∩ ''U''
''β'' lie in (possibly distinct) plaques of ''U''
''β''. Equivalently, since the plaques of ''U''
''α'' and ''U''
''β'' are level sets of the transverse coordinates ''y''
''α'' and ''y''
''β'', respectively, each point ''z'' ∈ ''U''
''α'' ∩ ''U''
''β'' has a neighborhood in which the formula
:
is independent of ''x''
''β''.
The main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation. For this and other purposes, the general definition of foliated atlas above is a bit clumsy. One problem is that a plaque of (''U''
α,''φ''
α) can meet multiple plaques of (''U''
β,''φ''
β). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality is lost in assuming the situation to be much more regular as shown below.
Two foliated atlases
and
on ''M'' of the same codimension and
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
class ''C
r'' are ''coherent''
if
is a foliated ''C
r''-atlas. Coherence of foliated atlases is an equivalence relation.
:
Plaques and transversals defined above on open sets are also open. But one can speak also of closed plaques and transversals. Namely, if (''U'',''φ'') and (''W'',''ψ'') are foliated charts such that
(the
closure of ''U'') is a subset of ''W'' and ''φ'' = ''ψ'', ''U'' then, if
it can be seen that
, written
, carries
diffeomorphically onto
A foliated atlas is said to be ''regular'' if
# for each α ∈ ''A'',
is a
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
of a foliated chart (''W''
α,''ψ''
α) and ''φ''
α = ''ψ''
α, ''U''
α;
# the
cover is
locally finite;
# if (''U''
α,''φ''
α) and (''U''
β,''φ''
β) are elements of the foliated atlas, then the interior of each closed plaque ''P'' ⊂
meets at most one plaque in
By property (1), the coordinates ''x''
α and ''y''
α extend to coordinates
and
on
and one writes
Property (3) is equivalent to requiring that, if ''U''
α ∩ ''U''
β ≠ ∅, the transverse coordinate changes
be independent of
That is
:
has the formula
:
Similar assertions hold also for open charts (without the overlines). The transverse coordinate map ''y''
α can be viewed as a
submersion
:
and the formulas ''y''
α = ''y''
α(''y''
β) can be viewed as
diffeomorphisms
:
These satisfy the
cocycle conditions. That is, on ''y''
δ(''U''
α ∩ ''U''
β ∩ ''U''
δ),
:
and, in particular,
:
:
Using the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent
refinement that is regular.
:
Foliation definitions
Several alternative definitions of foliation exist depending on the way through which the foliation is achieved. The most common way to achieve a foliation is through
decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ...
reaching to the following
Definition. A ''p''-dimensional, class ''C
r'' foliation of an ''n''-dimensional manifold ''M'' is a decomposition of ''M'' into a union of
disjoint connected submanifolds
α∈''A'', called the ''leaves'' of the foliation, with the following property: Every point in ''M'' has a neighborhood ''U'' and a system of local, class ''C
r'' coordinates ''x''=(''x''
1, ⋅⋅⋅, ''x
n'') : ''U''→R
''n'' such that for each leaf ''L''
α, the components of ''U'' ∩ ''L''
α are described by the equations ''x''
''p''+1=constant, ⋅⋅⋅, ''x
n''=constant. A foliation is denoted by
=
α∈''A''.
The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, -dimensional foliation
of an -manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed -dimensional submanifolds (the leaves of the foliation) of , such that for every point in , there is a chart
with homeomorphic to containing such that every leaf, , meets in either the empty set or a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
collection of subspaces whose images under
in
are -dimensional
affine subspaces whose first coordinates are constant.
Locally, every foliation is a
submersion allowing the following
Definition. Let ''M'' and ''Q'' be manifolds of dimension ''n'' and ''q''≤''n'' respectively, and let ''f'' : ''M''→''Q'' be a submersion, that is, suppose that the rank of the function differential (the
Jacobian) is ''q''. It follows from the
Implicit Function Theorem that ''ƒ'' induces a codimension-''q'' foliation on ''M'' where the leaves are defined to be the components of ''f''
−1(''x'') for ''x'' ∈ ''Q''.
This definition describes a
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
- foliation
of an -dimensional manifold that is a covered by
charts
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent t ...
together with maps
:
such that for overlapping pairs the
transition functions In mathematics, a transition function may refer to:
* a transition map between two charts of an atlas of a manifold or other topological space
* the function that defines the transitions of a state transition system in computing, which may refer mo ...
defined by
:
take the form
:
where denotes the first = coordinates, and denotes the last co-ordinates. That is,
:
The splitting of the transition functions ''φ
ij'' into
and
as a part of the submersion is completely analogous to the splitting of
into
and
as a part of the definition of a regular foliated atlas. This makes possible another definition of foliations in terms of regular foliated atlases. To this end, one has to prove first that every regular foliated atlas of codimension ''q'' is associated
to a unique foliation
of codimension ''q''.
:
As shown in the proof, the leaves of the foliation are equivalence classes of plaque chains of length ≤ ''p'' which are also topologically immersed Hausdorff -dimensional
submanifolds. Next, it is shown that the equivalence relation of plaques on a leaf is expressed in equivalence of coherent foliated atlases in respect to their association with a foliation. More specifically, if
and
are foliated atlases on ''M'' and if
is associated to a foliation
then
and
are coherent if and only if
is also associated to
.
:
It is now obvious that the correspondence between foliations on ''M'' and their associated foliated atlases induces a one-to-one correspondence between the set of foliations on ''M'' and the set of coherence classes of foliated atlases or, in other words, a foliation
of codimension ''q'' and class ''C
r'' on ''M'' is a coherence class of foliated atlases of codimension ''q'' and class ''C
r'' on ''M''.
By
Zorn's lemma, it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas. Thus,
Definition. A foliation of codimension ''q'' and class ''C
r'' on ''M'' is a maximal foliated ''C
r''-atlas of codimension ''q'' on ''M''.
In practice, a relatively small foliated atlas is generally used to represent a foliation. Usually, it is also required this atlas to be regular.
In the chart , the stripes match up with the stripes on other charts . These submanifolds piece together from chart to chart to form maximal
connected injectively
immersed submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
s called the leaves of the foliation.
If one shrinks the chart it can be written as , where ,
is homeomorphic to the plaques, and the points of parametrize the plaques in . If one picks in , then is a submanifold of that intersects every plaque exactly once. This is called a local
''transversal'' ''section'' of the foliation. Note that due to
monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
global transversal sections of the foliation might not exist.
The case ''r'' = 0 is rather special. Those ''C''
0 foliations that arise in practice are usually "smooth-leaved". More precisely, they are of class ''C''
''r'',0, in the following sense.
Definition. A foliation
is of class ''C
r,k'', ''r'' > ''k'' ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas
α∈''A'' such that the change of coordinate formula
:
is of class ''C
k'', but ''x''
α is of class ''C
r'' in the coordinates ''x''
β and its mixed ''x''
β partials of orders ≤ ''r'' are ''C
k'' in the coordinates (''x''
β,''y''
β).
The above definition suggests the more general concept of a ''foliated space'' or ''abstract lamination''. One relaxes the condition that the transversals be open, relatively compact subsets of R
''q'', allowing the transverse coordinates ''y''
α to take their values in some more general topological space ''Z''. The plaques are still open, relatively compact subsets of R
''p'', the change of transverse coordinate formula ''y''
α(''y''
β) is continuous and ''x''
''α''(''x''
''β'',''y''
''β'') is of class ''C
r'' in the coordinates ''x''
''β'' and its mixed ''x''
''β'' partials of orders ≤ ''r'' are continuous in the coordinates (''x''
''β'',''y''
''β''). One usually requires ''M'' and ''Z'' to be locally compact,
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
and metrizable. This may seem like a rather wild generalization, but there are contexts in which it is useful.
Holonomy
Let (''M'',
) be a foliated manifold. If ''L'' is a leaf of
and ''s'' is a path in ''L'', one is interested in the behavior of the foliation in a neighborhood of ''s'' in ''M''. Intuitively, an inhabitant of the leaf walks along the path ''s'', keeping an eye on all of the nearby leaves. As
they
In Modern English, ''they'' is a third-person pronoun relating to a grammatical subject.
Morphology
In Standard Modern English, ''they'' has five distinct word forms:
* ''they'': the nominative (subjective) form
* ''them'': the accusat ...
(hereafter denoted by ''s''(''t'')) proceed, some of these leaves may "peel away", getting out of visual range, others may suddenly come into range and approach ''L'' asymptotically, others may follow along in a more or less parallel fashion or wind around ''L'' laterally, ''etc''. If ''s'' is a loop, then ''s''(''t'') repeatedly returns to the same point ''s''(''t''
0) as ''t'' goes to infinity and each time more and more leaves may have spiraled into view or out of view, ''etc''. This behavior, when appropriately formalized, is called the ''holonomy'' of the foliation.
Holonomy is implemented on foliated manifolds in various specific ways: the total holonomy group of foliated bundles, the holonomy pseudogroup of general foliated manifolds, the germinal holonomy groupoid of general foliated manifolds, the germinal holonomy group of a leaf, and the infinitesimal holonomy group of a leaf.
Foliated bundles
The easiest case of holonomy to understand is the ''total holonomy'' of a foliated bundle. This is a generalization of the notion of a ''
Poincaré map
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional ...
''.
The term
"first return (recurrence) map" comes from the theory of dynamical systems. Let Φ
''t'' be a nonsingular ''C
r'' flow (''r'' ≥ 1) on the compact ''n''-manifold ''M''. In applications, one can imagine that ''M'' is a
cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Jan ...
or some closed loop with fluid flow. If ''M'' has a boundary, the flow is assumed to be tangent to the boundary. The flow generates a 1-dimensional foliation
. If one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, ''etc''.), the underlying foliation
is said to be oriented. Suppose that the flow admits a global cross section ''N''. That is, ''N'' is a compact, properly embedded, ''C
r'' submanifold of ''M'' of dimension ''n'' – 1, the foliation
is transverse to ''N'', and every flow line meets ''N''. Because the dimensions of ''N'' and of the leaves are complementary, the transversality condition is that
:
Let ''y'' ∈ ''N'' and consider the ''ω''-
limit set
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...
ω(''y'') of all accumulation points in ''M'' of all sequences
, where ''t
k'' goes to infinity. It can be shown that ω(y) is compact, nonempty, and a union of flow lines. If
there is a value ''t''* ∈ R such that Φ
''t''*(''z'') ∈ ''N'' and it follows that
:
Since ''N'' is compact and
is transverse to ''N'', it follows that the set is a monotonically increasing sequence
that diverges to infinity.
As ''y'' ∈ ''N'' varies, let ''τ''(''y'') = ''τ''
1(''y''), defining in this way a positive function ''τ'' ∈ ''C
r''(''N'') (the first return time) such that, for arbitrary ''y'' ∈ ''N'', Φ
''t''(''y'') ∉ ''N'', 0 < ''t'' < ''τ''(''y''), and Φ
''τ''(''y'')(''y'') ∈ ''N''.
Define ''f'' : ''N'' → ''N'' by the formula ''f''(''y'') = Φ
''τ''(''y'')(''y''). This is a ''C''
''r'' map. If the flow is reversed, exactly the same construction provides the inverse ''f''
−1; so ''f'' ∈ Diff
''r''(''N''). This diffeomorphism is the first return map and τ is called the ''
first return time''. While the first return time depends on the parametrization of the flow, it should be evident that ''f'' depends only on the oriented foliation
. It is possible to reparametrize the flow Φ
''t'', keeping it nonsingular, of class ''C
r'', and not reversing its direction, so that ''τ'' ≡ 1.
The assumption that there is a cross section N to the flow is very restrictive, implying that ''M'' is the total space of a fiber bundle over ''S''
1. Indeed, on R × ''N'', define ~
''f'' to be the equivalence relation generated by
:
Equivalently, this is the orbit equivalence for the action of the additive group Z on R × ''N'' defined by
:
for each ''k'' ∈ Z and for each (''t'',''y'') ∈ R × ''N''. The mapping cylinder of ''f'' is defined to be the ''C''
''r'' manifold
:
By the definition of the first return map ''f'' and the assumption that the first return time is τ ≡ 1, it is immediate that the map
:
defined by the flow, induces a canonical ''C''
''r'' diffeomorphism
:
If we make the identification ''M''
''f'' = ''M'', then the projection of R × ''N'' onto R induces a ''C''
''r'' map
:
that makes ''M'' into the total space of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...
over the circle. This is just the projection of ''S''
1 × ''D''
2 onto ''S''
1. The foliation
is transverse to the fibers of this bundle and the bundle projection , restricted to each leaf ''L'', is a covering map : ''L'' → ''S''
1. This is called a ''foliated bundle''.
Take as basepoint ''x''
0 ∈ ''S''
1 the equivalence class 0 + Z; so π
−1(''x''
0) is the original cross section ''N''. For each loop ''s'' on ''S''
1, based at ''x''
0, the homotopy class
's''∈ π
1(''S''
1,''x''
0) is uniquely characterized by deg ''s'' ∈ Z. The loop ''s'' lifts to a path in each flow line and it should be clear that the lift ''s
y'' that starts at ''y'' ∈ ''N'' ends at ''f
k''(''y'') ∈ ''N'', where ''k'' = deg ''s''. The diffeomorphism ''f
k'' ∈ Diff
''r''(''N'') is also denoted by ''h
s'' and is called the ''total holonomy'' of the loop ''s''. Since this depends only on
's'' this is a definition of a homomorphism
:
called the ''total holonomy homomorphism'' for the foliated bundle.
Using fiber bundles in a more direct manner, let (''M'',
) be a foliated ''n''-manifold of codimension ''q''. Let : ''M'' → ''B'' be a fiber bundle with ''q''-dimensional fiber ''F'' and connected base space ''B''. Assume that all of these structures are of class ''C
r'', 0 ≤ ''r'' ≤ ∞, with the condition that, if ''r'' = 0, ''B'' supports a ''C''
1 structure. Since every maximal ''C''
1 atlas on ''B'' contains a ''C''
∞ subatlas, no generality is lost in assuming that ''B'' is as smooth as desired. Finally, for each ''x'' ∈ ''B'', assume that there is a connected, open neighborhood ''U'' ⊆ ''B'' of ''x'' and a local trivialization
:
where ''φ'' is a ''C
r'' diffeomorphism (a homeomorphism, if ''r'' = 0) that carries
to the product foliation
''y'' ∈ ''F''. Here,
is the foliation with leaves the connected components of ''L'' ∩ π
−1(''U''), where ''L'' ranges over the leaves of
. This is the general definition of the term "foliated bundle" (''M'',
,π) of class ''C
r''.
is transverse to the fibers of π (it is said that
is transverse to the fibration) and that the restriction of π to each leaf ''L'' of
is a covering map π : ''L'' → ''B''. In particular, each fiber ''F
x'' =
−1(''x'') meets every leaf of
. The fiber is a cross section of
in complete analogy with the notion of a cross section of a flow.
The foliation
being transverse to the fibers does not, of itself, guarantee that the leaves are covering spaces of ''B''. A simple version of the problem is a foliation of R
2, transverse to the fibration
:
:
but with infinitely many leaves missing the ''y''-axis. In the respective figure, it is intended that the "arrowed" leaves, and all above them, are asymptotic to the axis ''x'' = 0. One calls such a foliation incomplete relative to the fibration, meaning that some of the leaves "run off to infinity" as the parameter ''x'' ∈ ''B'' approaches some ''x''
0 ∈ ''B''. More precisely, there may be a leaf ''L'' and a continuous path ''s'' : [0,''a'') → ''L'' such that lim
''t''→''a''−π(''s''(''t'')) = ''x''
0 ∈ ''B'', but lim
''t''→''a''−''s''(''t'') does not exist in the manifold topology of ''L''. This is analogous to the case of incomplete flows, where some flow lines "go to infinity" in finite time. Although such a leaf ''L'' may elsewhere meet π
−1(''x''
0), it cannot evenly cover a neighborhood of ''x''
0, hence cannot be a covering space of ''B'' under . When ''F'' is compact, however, it is true that transversality of
to the fibration does guarantee completeness, hence that
is a foliated bundle.
There is an atlas
=
α∈A on ''B'', consisting of open, connected coordinate charts, together with trivializations ''φ''
''α'' : ''π''
−1(''U''
''α'') → ''U''
''α'' × ''F'' that carry
, π
−1(''U''
''α'') to the product foliation. Set ''W''
''α'' = ''π''
−1(''U''
''α'') and write ''φ''
''α'' = (''x''
''α'',''y''
''α'') where (by abuse of notation) ''x''
α represents ''x''
''α'' ∘ ''π'' and ''y''
''α'' : ''π''
−1(''U''
α) → ''F'' is the submersion obtained by composing ''φ''
α with the canonical projection ''U''
α × ''F'' → ''F''.
The atlas
=
''α''∈''A'' plays a role analogous to that of a foliated atlas. The plaques of ''W''
''α'' are the level sets of ''y''
''α'' and this family of plaques is identical to ''F'' via ''y''
''α''. Since ''B'' is assumed to support a ''C''
∞ structure, according to the Whitehead theorem one can fix a Riemannian metric on ''B'' and choose the atlas
to be geodesically convex. Thus, ''U''
α ∩ ''U''
''β'' is always connected. If this intersection is nonempty, each plaque of ''W''
''α'' meets exactly one plaque of ''W''
''β''. Then define a ''holonomy cocycle''
by setting
:
Examples
Flat space
Consider an -dimensional space, foliated as a product by subspaces consisting of points whose first coordinates 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 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.
Bundles
A rather trivial example of foliations are products , foliated by the leaves . (Another foliation of is given by .)
A more general class are flat -bundles with for a manifold . Given a
representation
Representation may refer to:
Law and politics
* Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
, the flat -bundle with monodromy is given by
, where acts on the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
by
deck transformations and on by means of the representation .
Flat bundles fit into the framework of
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...
s. A map between manifolds is a fiber bundle if there is a manifold F such that each has an open neighborhood such that there is a homeomorphism
with
, with projection to the first factor. The fiber bundle yields a foliation by fibers
. Its space of leaves L is homeomorphic to , in particular L is a Hausdorff manifold.
Coverings
If is a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
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 In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology
In topology, a map is a ''branched covering'' if it is a covering map everywhere except for a nowhere dense set known as the branch set. ...
, where the branch
locus is transverse to the foliation, then the foliation can be pulled back.
Submersions
If is a
submersion of manifolds, it follows from the
inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at t ...
that the connected components of the fibers of the submersion define a codimension foliation of .
Fiber bundles
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...
are an example of this type.
An example of a submersion, which is not a fiber bundle, is given by
:
This submersion yields a foliation of which is invariant under the -actions given by
:
for and . The induced foliations of are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliation (of the Möbius band). Their leaf spaces are not Hausdorff.
Reeb foliations
Define a submersion
:
where are cylindrical coordinates on the -dimensional disk . This submersion yields a foliation of which is invariant under the -actions given by
:
for . The induced foliation of is called the -dimensional
Reeb foliation. Its leaf space is not Hausdorff.
For , this gives a foliation of the solid torus which can be used to define the
Reeb foliation of the 3-sphere by gluing two solid tori along their boundary. Foliations of odd-dimensional spheres are also explicitly known.
Lie groups
If is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
, and is a
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
, then is foliated by
cosets of . When is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
in , the
quotient space / is a smooth (
Hausdorff) manifold turning into a fiber bundle with fiber and base /. This fiber bundle is actually
principal, with structure group .
Lie group actions
Let be a Lie group acting smoothly on a manifold . If the action is a
locally free action or
free action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
, then the orbits of define a foliation of .
Linear and Kronecker foliations
If
is a nonsingular (''i.e.'', nowhere zero) vector field, then the local flow defined by
patches together to define a foliation of dimension 1. Indeed, given an arbitrary point ''x'' ∈ ''M'', the fact that
is nonsingular allows one to find a coordinate neighborhood (''U'',''x''
1,...,''x
n'') about ''x'' such that
:
and
:
Geometrically, the flow lines of
are just the level sets
:
where all
Since by convention manifolds are second countable, leaf anomalies like the "long line" are precluded by the second countability of ''M'' itself. The difficulty can be sidestepped by requiring that
be a complete field (''e.g.'', that ''M'' be compact), hence that each leaf be a flow line.
An important class of 1-dimensional foliations on the torus ''T''
2 are derived from projecting constant vector fields on ''T''
2. A constant vector field
:
on R
2 is invariant by all translations in R
2, hence passes to a well-defined vector field ''X'' when projected on the torus . It is assumed that ''a'' ≠ 0. The foliation
on R
2 produced by
has as leaves the parallel straight lines of slope θ = ''b''/''a''. This foliation is also invariant under translations and passes to the foliation
on ''T''
2 produced by ''X''.
Each leaf of
is of the form
:
If the slope is
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
then all leaves are closed curves
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
. In this case, one can take ''a'',''b'' ∈ Z. For fixed ''t'' ∈ R, the points of
corresponding to values of ''t'' ∈ ''t''
0 + Z all project to the same point of ''T''
2; so the corresponding leaf ''L'' of
is an embedded circle in ''T''
2. Since ''L'' is arbitrary,
is a foliation of ''T''
2 by circles. It follows rather easily that this foliation is actually a fiber bundle π : ''T''
2 → ''S''
1. This is known as a ''linear foliation''.
When the slope θ = ''b''/''a'' is
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, the leaves are noncompact, homeomorphic to the non-compactified
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, and
dense in the torus (cf
Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map
: T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1
where ''θ'' is an irrational number. Under the identification of a circle with ...
). The trajectory of each point (''x''
0,''y''
0) never returns to the same point, but generates an "everywhere dense" winding about the torus, i.e. approaches arbitrarily close to any given point. Thus the closure to the trajectory is the entire two-dimensional torus. This case is named ''Kronecker foliation'', after
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
and his
Kronecker's Density Theorem. If the real number θ is distinct from each rational multiple of π, then the set is dense in the unit circle.
:
A similar construction using a foliation of by parallel lines yields a 1-dimensional foliation of the -torus associated with the
linear flow on the torus.
Suspension foliations
A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are
:
where
is the canonical projection. This foliation is called the suspension of the representation .
In particular, if and
is a homeomorphism of , then the suspension foliation of
is defined to be the suspension foliation of the representation given by . Its space of leaves is , where whenever for some .
The simplest example of foliation by suspension is a manifold ''X'' of dimension ''q''. Let ''f'' : ''X'' → ''X'' be a bijection. One defines the suspension ''M'' = ''S''
1 ×
''f'' ''X'' as the quotient of
,1× ''X'' by the equivalence relation (1,''x'') ~ (0,''f''(''x'')).
:''M'' = ''S''
1 ×
''f'' ''X'' =
,1× ''X''
Then automatically ''M'' carries two foliations:
2 consisting of sets of the form ''F''
2,''t'' = and
1 consisting of sets of the form ''F''
2,''x''0 = , where the orbit O
x0 is defined as
:O
''x''0 = ,
where the exponent refers to the number of times the function ''f'' is composed with itself. Note that O
''x''0 = O
''f''(''x''0) = O
''f''−2(''x''0), etc., so the same is true for ''F''
1,''x''0. Understanding the foliation
1 is equivalent to understanding the dynamics of the map ''f''. If the manifold ''X'' is already foliated, one can use the construction to increase the codimension of the foliation, as long as ''f'' maps leaves to leaves.
The Kronecker foliations of the 2-torus are the suspension foliations of the rotations by angle
More specifically, if Σ = Σ
2 is the two-holed torus with C
1,C
2 ∈ Σ the two embedded circles let
be the product foliation of the 3-manifold ''M'' = Σ × ''S''
1 with leaves Σ × , ''y'' ∈ ''S''
1. Note that ''N
i'' = ''C
i'' × ''S''
1 is an embedded torus and that
is transverse to ''N
i'', ''i'' = 1,2. Let Diff
+(''S''
1) denote the group of orientation-preserving diffeomorphisms of ''S''
1 and choose ''f''
1,''f''
2 ∈ Diff
+(''S''
1). Cut ''M'' apart along ''N''
1 and ''N''
2, letting
and
denote the resulting copies of ''N
i'', ''i'' = 1,2. At this point one has a manifold ''M = Σ' × ''S''
1 with four boundary components
The foliation
has passed to a foliation
transverse to the boundary ∂''M' '', each leaf of which is of the form Σ' × , ''y'' ∈ ''S''
1.
This leaf meets ∂''M' '' in four circles
If ''z'' ∈ ''C
i'', the corresponding points in
are denoted by ''z''
± and
is "reglued" to
by the identification
:
Since ''f''
1 and ''f''
2 are orientation-preserving diffeomorphisms of ''S''
1, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic to ''M''. The leaves of
, however, reassemble to produce a new foliation
(''f''
1,''f''
2) of ''M''. If a leaf ''L'' of
(''f''
1,''f''
2) contains a piece Σ' × , then
:
where ''G'' ⊂ Diff
+(''S''
1) is the subgroup generated by . These copies of Σ' are attached to one another by identifications
:(''z''
−,''g''(''y''
0)) ≡ (''z''
+,''f''
1(''g''(''y''
0))) for each ''z'' ∈ ''C''
1,
:(''z''
−,''g''(''y''
0)) ≡ (''z''
+,''f''
2(''g''(''y''
0))) for each ''z'' ∈ ''C''
2,
where ''g'' ranges over ''G''. The leaf is completely determined by the ''G''-orbit of ''y''
0 ∈ ''S''
1 and can he simple or immensely complicated. For instance, a leaf will be compact precisely if the corresponding ''G''-orbit is finite. As an extreme example, if ''G'' is trivial (''f''
1 = ''f''
2 = id
S1), then
(''f''
1,''f''
2) =
. If an orbit is dense in ''S''
1, the corresponding leaf is dense in ''M''. As an example, if ''f''
1 and ''f''
2 are rotations through rationally independent multiples of 2π, every leaf will be dense. In other examples, some leaf ''L'' has closure
that meets each factor × ''S''
1 in a
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Th ...
. Similar constructions can be made on Σ × ''I'', where ''I'' is a compact, nondegenerate interval. Here one takes ''f''
1,''f''
2 ∈ Diff
+(''I'') and, since ∂''I'' is fixed pointwise by all orientation-preserving diffeomorphisms, one gets a foliation having the two components of ∂''M'' as leaves. When one forms ''M' '' in this case, one gets a foliated manifold with corners. In either case, this construction is called the ''suspension'' of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.
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 the
Frobenius theorem, saying that the
necessary and sufficient conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for a distribution (i.e. an dimensional
subbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces U_xof the fibers V_x of V at x in X, that make up a vector bundle in their own right.
In connection with foliation theory, a subbundl ...
of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
. One can also phrase this differently, as a question of
reduction of the structure group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes vari ...
of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as , for some (non-canonical) (i.e. a non-zero co-vector field). A given is integrable iff everywhere.
There is a global foliation theory, because topological constraints exist. For example, in the
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
case, an everywhere non-zero vector field can exist on an
orientable compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Briti ...
surface only for the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not t ...
. This is a consequence of the
Poincaré–Hopf index theorem, which shows the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
will have to be 0. There are many deep connections with
contact topology, which is the "opposite" concept, requiring that the integrability condition is ''never'' satisfied.
Existence of foliations
gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. showed that any compact manifold with a distribution has a foliation of the same dimension.
See also
*
* closed under taking pullbacks.
*
* of the 3-sphere.
*
Notes
References
*
*
*
*
*
*
*
*
*
*
External links
Foliationsat the Manifold Atlas
{{Manifolds
Structures on manifolds