Distribution (differential geometry) explained
is an assignment
x\mapsto\Deltax\subseteqTxM
of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the
tangent bundle
.
Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology.
Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis.
Definition
Let
be a smooth manifold; a
(smooth) distribution
assigns to any point
a vector subspace
in a smooth way. More precisely,
consists of a collection
of vector subspaces with the following property: Around any
there exist a
neighbourhood
and a collection of
vector fields
such that, for any point
,
span\{X1(y),\ldots,Xk(y)\}=\Deltay.
The set of smooth vector fields
is also called a
local basis of
. These need not be linearly independent at every point, and so aren't formally a vector space basis at every point; thus, the term
local generating set can be more appropriate. The notation
is used to denote both the assignment
and the subset
\Delta=\amalgx\Deltax\subseteqTM
.
Regular distributions
Given an integer
, a smooth distribution
on
is called
regular of rank
if all the subspaces
have the same dimension
. Locally, this amounts to ask that every local basis is given by
linearly independent vector fields.
of rank
(this is actually the most commonly used definition). A rank
distribution is sometimes called an
-plane distribution, and when
, one talks about
hyperplane distributions.
Special classes of distributions
Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above).
Involutive distributions
Given a distribution
, its sections consist of vector fields on
forming a vector subspace
\Gamma(\Delta)\subseteq\Gamma(TM)=ak{X}(M)
of the space of all vector fields on
. (Notation:
is the space of
sections of
) A distribution
is called
involutive if
\Gamma(\Delta)\subseteqak{X}(M)
is also a
Lie subalgebra: in other words, for any two vector fields
X,Y\in\Gamma(\Delta)\subseteqak{X}(M)
, the
Lie bracket
belongs to
\Gamma(\Delta)\subseteqak{X}(M)
.
Locally, this condition means that for every point
there exists a local basis
of the distribution in a neighbourhood of
such that, for all
, the Lie bracket
is in the span of
, i.e.
is a
linear combination of
Involutive distributions are a fundamental ingredient in the study of integrable systems. A related idea occurs in Hamiltonian mechanics: two functions
and
on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.Integrable distributions and foliations
An integral manifold for a rank
distribution
is a
submanifold
of dimension
such that
for every
. A distribution is called
integrable if through any point
there is an integral manifold. The base spaces of the bundle
are thus disjoint,
maximal,
connected integral manifolds, also called
leaves; that is,
defines an n-dimensional
foliation of
.
Locally, integrability means that for every point
there exists a local chart
(U,\{\chi1,\ldots,\chin\})
such that, for every
, the space
is spanned by the coordinate vectors
. In other words, every point admits a foliation chart, i.e. the distribution
is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a
-structures, when
is the group of real invertible upper-triangular
block matrices (with
and
-blocks).
It is easy to see that any integrable distribution is automatically involutive. The converse is less trivial but holds by Frobenius theorem.
Weakly regular distributions
Given any distribution
, the
associated Lie flag is a grading, defined as
\Delta(0)\subseteq\Delta(1)\subseteq\ldots\subseteq\Delta(i)\subseteq\Delta(i+1)\subseteq\ldots
where
\Delta(0):=\Gamma(\Delta)
,
\Delta(1):=\langle[\Delta(0),\Delta(0)]
and
\Delta(i+1):=\langle[\Delta(i),\Delta(0)]
. In other words,
\Delta(i)\subseteqak{X}(M)
denotes the set of vector fields spanned by the
-iterated Lie brackets of elements in
. Some authors use a negative decreasing grading for the definition.
Then
is called
weakly regular (or just regular by some authors) if there exists a sequence
of nested vector subbundles such that
(hence
).
[1] Note that, in such case, the associated Lie flag stabilises at a certain point
, since the ranks of
are bounded from above by
. The string of integers
(rank(\Delta(0)),rank(\Delta(1)),\ldots,rank(\Delta(m)))
is then called the
grow vector of
.
Any weakly regular distribution has an associated graded vector bundleMoreover, the Lie bracket of vector fields descends, for any
, to a
-linear bundle morphism
gri(TM) x grj(TM)\togri+j+1(TM)
, called the
-curvature. In particular, the
-curvature vanishes identically if and only if the distribution is involutive.
Patching together the curvatures, one obtains a morphism
l{L}:gr(TM) x gr(TM)\togr(TM)
, also called the
Levi bracket, which makes
into a bundle of nilpotent Lie algebras; for this reason,
is also called the
nilpotentisation of
.
The bundle
, however, is in general not locally trivial, since the Lie algebras
are not isomorphic when varying the point
. If this happens, the weakly regular distribution
is also called
regular (or strongly regular by some authors). Note that the names
(strongly, weakly) regular used here are completely unrelated with the notion of regularity discussed above (which is always assumed), i.e. the dimension of the spaces
being constant.
Bracket-generating distributions
A distribution
is called
bracket-generating (or
non-holonomic, or it is said to satisfy the
Hörmander condition) if taking a finite number of Lie brackets of elements in
is enough to generate the entire space of vector fields on
. With the notation introduced above, such condition can be written as
for certain
; then one says also that
is bracket-generating in
steps, or has
depth
.
Clearly, the associated Lie flag of a bracket-generating distribution stabilises at the point
. Even though being weakly regular and being bracket-generating are two independent properties (see the examples below), when a distribution satisfies both of them, the integer
from the two definitions is the same.
Thanks to the Chow-Rashevskii theorem, given a bracket-generating distribution
on a connected manifold, any two points in
can be joined by a path tangent to the distribution.
[2] [3] Examples of regular distributions
Integrable distributions
on
defines a rank 1 distribution, by setting
\Deltax:=\langleXx\rangle\subseteqTxM
, which is automatically integrable: the image of any
integral curve
is an integral manifold.
- The trivial distribution of rank
on
is generated by the first
coordinate vector fields
. It is automatically integrable, and the integral manifolds are defined by the equations
, for any constants
.
- In general, any involutive/integrable distribution is weakly regular (with
for every
), but it is never bracket-generating.
Non-integrable distributions
- The Martinet distribution on
is given by
\Delta=\ker(\omega)\subseteqTM
, for
\omega=dy-z2dx\in\Omega1(M)
; equivalently, it is generated by the vector fields
and
. It is bracket-generating since
, but it is not weakly regular:
has rank 3 everywhere except on the surface
.
- The contact distribution on
is given by
\Delta=\ker(\omega)\subseteqTM
, for
\omega=dz+
xidyi\in\Omega1(M)
; equivalently, it is generated by the vector fields
and
, for
. It is weakly regular, with grow vector
, and bracket-generating, with
. One can also define an abstract
contact structures on a manifold
as a hyperplane distribution which is maximally non-integrable, i.e. it is as far from being involutive as possible. An analogue of the
Darboux theorem shows that such structure has the unique local model described above.
- The Engel distribution on
is given by
\Delta=\ker(\omega1)\cap\ker(\omega2)\subseteqTM
, for
\omega1=dz-wdx\in\Omega1(M)
and
\omega2=dy-zdx\in\Omega1(M)
; equivalently, it is generated by the vector fields
and
. It is weakly regular, with grow vector
, and bracket-generating. One can also define an abstract
Engel structure on a manifold
as a weakly regular rank 2 distribution
such that
has rank 3 and
has rank 4; Engel proved that such structure has the unique local model described above.
[4] - In general, a Goursat structure on a manifold
is a rank 2 distribution which is weakly regular and bracket-generating, with grow vector
. For
and
one recovers, respectively, contact distributions on 3-dimensional manifolds and Engel distributions. Goursat structures are locally diffeomorphic to the Cartan distribution of the
jet bundles
.
Singular distributions
A singular distribution, generalised distribution, or Stefan-Sussmann distribution, is a smooth distribution which is not regular. This means that the subspaces
may have different dimensions, and therefore the subset
is no longer a smooth subbundle.
In particular, the number of elements in a local basis spanning
will change with
, and those vector fields will no longer be linearly independent everywhere. It is not hard to see that the dimension of
is lower semicontinuous, so that at special points the dimension is lower than at nearby points.
Integrability and singular foliations
The definitions of integral manifolds and of integrability given above applies also to the singular case (removing the requirement of the fixed dimension). However, Frobenius theorem does not hold in this context, and involutivity is in general not sufficient for integrability (counterexamples in low dimensions exist).
After several partial results,[5] the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan[6] [7] and Sussmann.[8] [9] It states that a singular distribution
is integrable if and only if the following two properties hold:
is generated by a family
of vector fields;
is invariant with respect to every
, i.e.
, where
is the
flow of
,
and
.
Similarly to the regular case, an integrable singular distribution defines a singular foliation, which intuitively consists in a partition of
into submanifolds (the maximal integral manifolds of
) of different dimensions.
The definition of singular foliation can be made precise in several equivalent ways. Actually, in the literature there is a plethora of variations, reformulations and generalisations of the Stefan-Sussman theorem, using different notion of singular foliations according to which applications one has in mind, e.g. Poisson geometry[10] [11] or non-commutative geometry.[12] [13]
Examples
, its infinitesimal generators span a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the orbits of the group action. The distribution/foliation is regular if and only if the action is free.
, the image of
\pi\sharp=\iota\pi:T*M\toTM
is a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the symplectic leaves of
. The distribution/foliation is regular If and only if the Poisson manifold is regular.
- More generally, the image of the anchor map
of any
Lie algebroid
defines a singular distribution which is automatically integrable, and the leaves of the associated singular foliation are precisely the leaves of the Lie algebroid. The distribution/foliation is regular if and only if
has constant rank, i.e. the Lie algebroid is regular. Considering, respectively, the action Lie algebroid
and the cotangent Lie algebroid
, one recovers the two examples above.
- In dynamical systems, a singular distribution arise from the set of vector fields that commute with a given one.
- There are also examples and applications in control theory, where the generalised distribution represents infinitesimal constraints of the system.
References
- Tanaka. Noboru. 1970-01-01. On differential systems, graded Lie algebras and pseudo-groups. Kyoto Journal of Mathematics. 10. 1. 10.1215/kjm/1250523814. 2156-2261. free.
- Chow. Wei-Liang. 1940-12-01. Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung. Mathematische Annalen. de. 117. 1. 98–105. 10.1007/BF01450011. 121523670 . 1432-1807.
- Rashevsky. P. K.. 1938. Any two points of a totally nonholonomic space may be connected by an admissible line.. Uch. Zap. Ped. Inst. Im. Liebknechta, Ser. Phys. Math.. Russian. 2. 83–94.
- Engel. Friedrich. 1889. Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Leipz. Ber.. German. 41. 157–176.
- Lavau. Sylvain. 2018-12-01. A short guide through integration theorems of generalized distributions. Differential Geometry and Its Applications. en. 61. 42–58. 10.1016/j.difgeo.2018.07.005. 1710.01627 . 119669163 . 0926-2245.
- Stefan. P.. 1974. Accessibility and foliations with singularities. Bulletin of the American Mathematical Society. en. 80. 6. 1142–1145. 10.1090/S0002-9904-1974-13648-7. 0002-9904. free.
- Stefan. P.. 1974. Accessible Sets, Orbits, and Foliations with Singularities. Proceedings of the London Mathematical Society. en. s3-29. 4. 699–713. 10.1112/plms/s3-29.4.699. 1460-244X.
- Sussmann. Hector J.. 1973. Orbits of families of vector fields and integrability of systems with singularities. Bulletin of the American Mathematical Society. en. 79. 1. 197–199. 10.1090/S0002-9904-1973-13152-0. 0002-9904. free.
- Sussmann. Héctor J.. 1973. Orbits of families of vector fields and integrability of distributions. Transactions of the American Mathematical Society. en. 180. 171–188. 10.1090/S0002-9947-1973-0321133-2. 0002-9947. free.
- Androulidakis. Iakovos. Zambon. Marco. 2016-04-28. Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves. International Journal of Geometric Methods in Modern Physics. 13. Supp. 1. 1641001–1641267. 10.1142/S0219887816410012. 2016IJGMM..1341001A . 0219-8878.
- Laurent-Gengoux. Camille. Lavau. Sylvain. Strobl. Thomas. 2020. The Universal Lie ∞-Algebroid of a Singular Foliation. ELibM – Doc. Math.. en. 25. 2020 . 1571–1652. 10.25537/dm.2020v25.1571-1652.
- Debord. Claire. 2001-07-01. Holonomy Groupoids of Singular Foliations. Journal of Differential Geometry. 58. 3. 10.4310/jdg/1090348356. 54714044 . 0022-040X. free.
- Androulidakis. Iakovos. Skandalis. Georges. 2009-01-01. The holonomy groupoid of a singular foliation. Journal für die reine und angewandte Mathematik (Crelle's Journal) . en. 2009. 626. 1–37. 10.1515/CRELLE.2009.001. 14450917 . 1435-5345. math/0612370.
Books, lecture notes and external links
- William M. Boothby. Section IV. 8 in An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
- John M. Lee, Chapter 19 in Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer-Verlag, 2003.
- Richard Montgomery, Chapters 2, 4 and 6 in A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. Amer. Math. Soc., Providence, RI, 2002.
- Álvaro del Pino, Topological aspects in the study of tangent distributions. Textos de Matemática. Série B, 48. Universidade de Coimbra, 2019.