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{{Other uses|Distribution (disambiguation)#Mathematics}}

In [[differential geometry]], a discipline within [[mathematics]], a '''distribution''' on a [[differentiable manifold|manifold]] <math>M</math> is an assignment <math>x \mapsto \Delta_x \subseteq T_x M</math> of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the [[tangent bundle]] <math>TM</math>.

Distributions satisfying a further integrability condition give rise to [[foliation]]s, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including [[integrable system]]s, [[Poisson geometry]], [[Noncommutative 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 [[distribution (mathematics)|distribution]]s in the sense of analysis.

==Definition==
Let <math>M</math> be a smooth manifold; a '''(smooth) distribution''' <math>\Delta</math> assigns to any point <math>x \in M</math> a vector subspace <math>\Delta_x \subset T_xM</math> in a smooth way. More precisely, <math>\Delta</math> consists of a collection <math>\{ \Delta_x \subset T_xM \}_{x \in M}</math> of vector subspaces with the following property: Around any <math>x \in M</math> there exist a [[neighbourhood (mathematics)|neighbourhood]] <math>N_x \subset M</math> and a collection of [[vector field]]s <math>X_1,\ldots,X_k</math> such that, for any point <math>y \in N_x</math>, [[linear span|span]]<math>\{ X_1(y),\ldots,X_k(y) \} = \Delta_y.</math>

The set of smooth vector fields  <math>\{ X_1,\ldots,X_k \}</math> is also called a ''local basis'' of <math>\Delta</math>. 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 <math>\Delta</math> is used to denote both the assignment <math>x \mapsto \Delta_x</math> and the subset <math>\Delta = \amalg_{x \in M} \Delta_x \subseteq TM</math>.

=== Regular distributions ===
Given an integer <math>n \leq m = \mathrm{dim}(M)</math>, a smooth distribution <math>\Delta</math> on <math>M</math> is called '''regular''' of rank <math>n</math> if all the subspaces <math>\Delta_x \subset T_xM</math> have the same dimension <math>n</math>. Locally, this amounts to ask that every local basis is given by <math>n</math> [[linear independence|linearly independent]] vector fields.

More compactly, a regular distribution is a [[Subbundle|vector subbundle]] <math>\Delta \subset TM</math> of rank <math>n</math> (this is actually the most commonly used definition). A rank <math>n</math> distribution is sometimes called an <math>n</math>-plane distribution, and when <math>n = m-1</math>, 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 <math>\Delta</math>, its sections consist of vector fields on <math>M,</math> forming a [[vector subspace]] <math>\Gamma(\Delta) \subseteq \Gamma(TM) = \mathfrak{X}(M)</math> of the space of all vector fields on <math>M</math>. (Notation: <math>\Gamma(TM)</math> is the space of [[Section (fiber bundle)|sections]] of <math>TM.</math>) A distribution <math>\Delta</math> is called '''involutive''' if <math>\Gamma(\Delta) \subseteq \mathfrak{X}(M)</math> is also a [[Lie subalgebra]]: in other words, for any two vector fields <math>X, Y \in \Gamma(\Delta) \subseteq \mathfrak{X}(M)</math>, the [[Lie bracket of vector fields|Lie bracket]] <math>[X,Y]</math> belongs to <math>\Gamma(\Delta) \subseteq \mathfrak{X}(M)</math>.

Locally, this condition means that for every point <math>x \in M</math> there exists a local basis <math>\{ X_1,\ldots,X_n \}</math> of the distribution in a neighbourhood of <math>x</math> such that, for all <math>1 \leq i, j \leq n</math>, the Lie bracket <math>[X_i,X_j]</math> is in the span of <math>\{ X_1,\ldots,X_n \}</math>, i.e. <math>[X_i,X_j]</math> is a [[linear combination]] of <math>\{ X_1,\ldots,X_n \}.</math>

Involutive distributions are a fundamental ingredient in the study of [[integrable system]]s. A related idea occurs in [[Hamiltonian mechanics]]: two functions ''<math>f</math>'' and ''<math>g</math>'' 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 <math>n</math> distribution <math>\Delta</math> is a [[submanifold]] <math>N \subset M</math> of dimension <math>n</math> such that <math>T_xN = \Delta_x</math> for every <math>x \in N</math>. A distribution is called '''integrable''' if through any point <math>x \in M</math> there is an integral manifold. The base spaces of the bundle <math>\Delta \subset TM</math> are thus disjoint, ''maximal'', [[Connected space|connected]] integral manifolds, also called '''leaves'''; that is, <math>\Delta</math> defines an n-dimensional [[foliation]] of <math>M</math>.

Locally, integrability means that for every point <math>x \in M</math> there exists a local chart <math>(U, \{ \chi_1,\ldots,\chi_n \})</math> such that, for every <math>y \in U</math>, the space <math>\Delta_y</math> is spanned by the coordinate vectors <math>\frac{\partial}{\partial \chi_1}(y), \ldots, \frac{\partial}{\partial \chi_n}(y)</math>. In other words, every point admits a foliation chart, i.e. the distribution <math>\Delta</math> is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a [[G-structure on a manifold|<math>G</math>-structures]], when [[G-structure on a manifold|<math>G</math>]] is the group of real invertible upper-triangular [[block matrices]] (with <math>(n \times n)</math> and <math>(m-n,m-n)</math>-blocks).

It is easy to see that any integrable distribution is automatically involutive. The converse is less trivial but holds by [[Frobenius theorem (differential topology)|Frobenius theorem]].

=== Weakly regular distributions ===
Given any distribution <math>\Delta \subseteq TM</math>, the '''associated Lie flag''' is a grading, defined as

:<math>\Delta^{(0)} \subseteq \Delta^{(1)} \subseteq \ldots \subseteq \Delta^{(i)} \subseteq \Delta^{(i+1)} \subseteq \ldots</math>

where <math>\Delta^{(0)} := \Gamma(\Delta)</math>, <math>\Delta^{(1)} := \langle [ \Delta^{(0)}, \Delta^{(0)} ] \rangle_{\mathcal{C}^\infty(M)}</math> and <math>\Delta^{(i+1)} := \langle [ \Delta^{(i)}, \Delta^{(0)} ] \rangle_{\mathcal{C}^\infty(M)}</math>. In other words, <math>\Delta^{(i)} \subseteq \mathfrak{X}(M)</math> denotes the set of vector fields spanned by the <math>i</math>-iterated Lie brackets of elements in <math>\Gamma(\Delta)</math>. Some authors use a negative decreasing grading for the definition.

Then <math>\Delta</math> is called '''weakly regular''' (or just regular by some authors) if there exists a sequence [[G-structure on a manifold|<math>\{T^i M \subseteq TM \}_i</math>]] of nested vector subbundles such that [[G-structure on a manifold|<math>\Gamma(T^iM) = \Delta^{(i)}</math>]] (hence <math>T^0 M = \Delta</math>).<ref name=":0">{{Cite journal|last=Tanaka|first=Noboru|date=1970-01-01|title=On differential systems, graded Lie algebras and pseudo-groups|journal=Kyoto Journal of Mathematics|volume=10|issue=1|doi=10.1215/kjm/1250523814|issn=2156-2261|doi-access=free}}</ref> Note that, in such case, the associated Lie flag stabilises at a certain point <math>m \in \mathbb{N}</math>, since the ranks of [[G-structure on a manifold|<math>T^iM</math>]] are bounded from above by [[G-structure on a manifold|<math>\mathrm{rank}(TM) = \mathrm{dim}(M)</math>]]. The string of integers <math>(\mathrm{rank}(\Delta^{(0)}), \mathrm{rank}(\Delta^{(1)}), \ldots, \mathrm{rank}(\Delta^{(m)}))</math> is then called the '''grow vector''' of <math>\Delta</math>.

Any weakly regular distribution has an associated graded vector bundle<math display="block">\mathrm{gr}(TM):= T^0 M \oplus \Big( \bigoplus_{i=0}^{ m-1} T^{i+1}M/T^iM \Big) \oplus TM/T^m M.</math>Moreover, the Lie bracket of vector fields descends, for any <math>i,j = 0,\ldots,m</math>, to a <math>\mathcal{C}^{\infty}(M)</math>-linear bundle morphism <math>\mathrm{gr}_i(TM) \times \mathrm{gr}_j(TM) \to \mathrm{gr}_{i+j+1}(TM)</math>, called the '''<math>(i,j)</math>-curvature'''. In particular, the '''<math>(0,0)</math>'''-curvature vanishes identically if and only if the distribution is involutive.

Patching together the curvatures, one obtains a morphism <math>\mathcal{L}: \mathrm{gr}(TM) \times \mathrm{gr}(TM) \to \mathrm{gr}(TM)</math>, also called the '''Levi bracket''', which makes <math>\mathrm{gr}(TM)</math> into a bundle of nilpotent Lie algebras; for this reason, <math>(\mathrm{gr}(TM), \mathcal{L})</math> is also called the '''nilpotentisation''' of <math>\Delta</math>.<ref name=":0" />

The bundle <math>\mathrm{gr}(TM) \to M</math>, however, is in general not locally trivial, since the Lie algebras <math>\mathrm{gr}_i(T_xM) := T^i_x M/T^{i+1}_x M</math> are not isomorphic when varying the point <math>x \in M</math>. If this happens, the weakly regular distribution <math>\Delta</math> is also called '''regular''' (or strongly regular by some authors).{{clarify|date=May 2024|reason=When it says "if this happens", what is "this"? Is it trying to say that when the fibers are isomorphic, then the bundle is called strongly regular?}} 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 <math>\Delta_x</math> being constant.

=== Bracket-generating distributions ===
A distribution <math>\Delta \subseteq TM</math> 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 <math>\Gamma(\Delta)</math> is enough to generate the entire space of vector fields on <math>M</math>. With the notation introduced above, such condition can be written as <math>\Delta^{(m)} = \mathfrak{X}(M)</math> for certain <math>m \in \mathbb{N}</math>; then one says also that <math>\Delta</math> is bracket-generating in <math>m+1</math> ''steps'', or has ''depth'' <math>m+1</math>.

Clearly, the associated Lie flag of a bracket-generating distribution stabilises at the point <math>m</math>. 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 <math>m</math> from the two definitions is the same.

Thanks to the [[Chow–Rashevskii theorem|Chow-Rashevskii theorem]], given a bracket-generating distribution <math>\Delta \subseteq TM</math> on a connected manifold, any two points in <math>M</math> can be joined by a path tangent to the distribution.<ref>{{Cite journal|last=Chow|first=Wei-Liang|date=1940-12-01|title=Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung|url=https://doi.org/10.1007/BF01450011|journal=Mathematische Annalen|language=de|volume=117|issue=1|pages=98–105|doi=10.1007/BF01450011|s2cid=121523670 |issn=1432-1807|url-access=subscription}}</ref><ref>{{Cite journal|last=Rashevsky|first=P. K.|date=1938|title=Any two points of a totally nonholonomic space may be connected by an admissible line.|journal=Uch. Zap. Ped. Inst. Im. Liebknechta, Ser. Phys. Math.|language=Russian|volume=2|pages=83–94}}</ref>

== Examples of regular distributions ==

=== Integrable distributions ===

*Any vector field <math>X</math> on <math>M</math> defines a rank 1 distribution, by setting <math>\Delta_x := \langle X_x \rangle \subseteq T_x M</math>, which is automatically integrable: the image of any [[integral curve]] ''<math>\gamma: I \to M</math>'' is an integral manifold.
* The trivial distribution of rank <math>k</math> on <math>M = \mathbb{R}^n</math> is generated by the first <math>k</math> coordinate vector fields <math>\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_k}</math>. It is automatically integrable, and the integral manifolds are defined by the equations <math>\{ x_i = c_i \}_{i=k+1,\ldots,n}</math>, for any constants <math>c_i \in \mathbb{R}</math>.
* In general, any involutive/integrable distribution is weakly regular (with <math>\Delta^{(i)} = \Gamma(\Delta)</math> for every <math>i</math>), but it is never bracket-generating.

=== Non-integrable distributions ===
*The '''Martinet distribution''' on <math>M = \mathbb{R}^3</math> is given by <math>\Delta = \ker(\omega) \subseteq TM</math> , for <math>\omega = dy-z^2dx \in \Omega^1 (M)</math>; equivalently, it is generated by the vector fields <math>\frac{\partial}{\partial x} + z^2 \frac{\partial}{\partial y}</math> and <math>\frac{\partial}{\partial z}</math>. It is bracket-generating since <math>\Delta^{(2)} = \mathfrak{X}(M)</math>, but it is not weakly regular: <math>\Delta^{(1)}</math> has rank 3 everywhere except on the surface <math>z=0</math>.
*The '''contact distribution''' on <math>M = \mathbb{R}^{2n+1}</math> is given by <math>\Delta = \ker(\omega) \subseteq TM</math> , for <math>\omega = dz + \sum_{i=1}^n x_i dy_i \in \Omega^1 (M)</math>; equivalently, it is generated by the vector fields <math>\frac{\partial}{\partial y_i}</math> and <math>\frac{\partial}{\partial x_i} + y_i \frac{\partial}{\partial z}</math>, for <math>i=1,\ldots,n</math>. It is weakly regular, with grow vector <math>(2n, 2n+1)</math>, and bracket-generating, with <math>\Delta^{(1)} = \mathfrak{X}(M)</math>. One can also define an abstract [[contact structure]]s on a manifold <math>M^{2n+1}</math> 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's theorem|Darboux theorem]] shows that such structure has the unique local model described above.
*The '''Engel distribution''' on <math>M = \mathbb{R}^4</math> is given by <math>\Delta = \ker(\omega_1) \cap \ker(\omega_2) \subseteq TM</math>, for <math>\omega_1 = dz-wdx \in \Omega^1(M)</math> and <math>\omega_2 = dy - zdx \in \Omega^1 (M)</math>; equivalently, it is generated by the vector fields <math>\frac{\partial}{\partial x} + z \frac{\partial}{\partial y} + w \frac{\partial}{\partial z}</math> and <math>\frac{\partial}{\partial w}</math>. It is weakly regular, with grow vector <math>(2, 3, 4)</math>, and bracket-generating. One can also define an abstract '''Engel structure''' on a manifold <math>M^4</math> as a weakly regular rank 2 distribution <math>\Delta \subseteq TM</math> such that <math>\Delta^{(1)}</math> has rank 3 and <math>\Delta^{(2)}</math>has rank 4; Engel proved that such structure has the unique local model described above.<ref>{{Cite journal|last=Engel|first=Friedrich|date=1889|title=Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen|journal=Leipz. Ber.|language=German|volume=41|pages=157–176}}</ref> 
*In general, '''a Goursat structure''' on a manifold <math>M^{k+2}</math> is a rank 2 distribution which is weakly regular and bracket-generating, with grow vector <math>(2, 3, \ldots, k+1, k+2)</math>. For <math>k=1</math> and <math>k=2</math> one recovers, respectively, contact distributions on 3-dimensional manifolds and Engel distributions. Goursat structures are locally diffeomorphic to the [[Jet bundle#Contact structure|Cartan distribution]] of the [[jet bundle]]s <math>J^k (\mathbb{R}, \mathbb{R})</math>.

==Singular distributions==
{{Distinguish|Singular distribution}}
A '''singular distribution''', '''generalised distribution''', or '''Stefan-Sussmann distribution''', is a smooth distribution which is not regular. This means that the subspaces <math>\Delta_x \subset T_xM</math> may have different dimensions, and therefore the subset <math>\Delta \subset TM</math> is no longer a smooth subbundle.

In particular, the number of elements in a local basis spanning <math>\Delta_x</math> will change with <math>x</math>, and those vector fields will no longer be linearly independent everywhere. It is not hard to see that the dimension of <math>\Delta_x</math> 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,<ref>{{Cite journal|last=Lavau|first=Sylvain|date=2018-12-01|title=A short guide through integration theorems of generalized distributions|url=https://www.sciencedirect.com/science/article/abs/pii/S0926224518302730|journal=Differential Geometry and Its Applications|language=en|volume=61|pages=42–58|doi=10.1016/j.difgeo.2018.07.005|arxiv=1710.01627 |s2cid=119669163 |issn=0926-2245}}</ref> the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan<ref>{{Cite journal|last=Stefan|first=P.|date=1974|title=Accessibility and foliations with singularities|url=https://www.ams.org/bull/1974-80-06/S0002-9904-1974-13648-7/|journal=Bulletin of the American Mathematical Society|language=en|volume=80|issue=6|pages=1142–1145|doi=10.1090/S0002-9904-1974-13648-7|issn=0002-9904|doi-access=free}}</ref><ref>{{Cite journal|last=Stefan|first=P.|date=1974|title=Accessible Sets, Orbits, and Foliations with Singularities|url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-29.4.699|journal=Proceedings of the London Mathematical Society|language=en|volume=s3-29|issue=4|pages=699–713|doi=10.1112/plms/s3-29.4.699|issn=1460-244X|url-access=subscription}}</ref> and Sussmann.<ref>{{Cite journal|last=Sussmann|first=Hector J.|date=1973|title=Orbits of families of vector fields and integrability of systems with singularities|url=https://www.ams.org/bull/1973-79-01/S0002-9904-1973-13152-0/|journal=Bulletin of the American Mathematical Society|language=en|volume=79|issue=1|pages=197–199|doi=10.1090/S0002-9904-1973-13152-0|issn=0002-9904|doi-access=free}}</ref><ref>{{Cite journal|last=Sussmann|first=Héctor J.|date=1973|title=Orbits of families of vector fields and integrability of distributions|url=https://www.ams.org/tran/1973-180-00/S0002-9947-1973-0321133-2/|journal=Transactions of the American Mathematical Society|language=en|volume=180|pages=171–188|doi=10.1090/S0002-9947-1973-0321133-2|issn=0002-9947|doi-access=free}}</ref> It states that a singular distribution <math>\Delta</math> is integrable if and only if the following two properties hold:

* <math>\Delta</math> is generated by a family <math>F \subseteq \mathfrak{X}(M)</math> of vector fields;
* <math>\Delta</math> is invariant with respect to every <math>X \in F</math>, i.e. <math>(\phi^t_X)_* (\Delta_y) \subseteq \Delta_{\phi^t_X(y)}</math>, where <math>\phi^t_X</math> is the [[Vector flow|flow]] of <math>X</math>, <math>t \in \mathbb{R}</math> and <math>y \in \mathrm{dom}(X)</math>.

Similarly to the regular case, an integrable singular distribution defines a '''singular foliation''', which intuitively consists in a partition of <math>M</math> into submanifolds (the maximal integral manifolds of <math>\Delta</math>) 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]]<ref>{{Cite journal|last1=Androulidakis|first1=Iakovos|last2=Zambon|first2=Marco|date=2016-04-28|title=Stefan–Sussmann singular foliations, singular subalgebroids and their associated sheaves|url=https://www.worldscientific.com/doi/abs/10.1142/S0219887816410012|journal=International Journal of Geometric Methods in Modern Physics|volume=13|issue=Supp. 1|pages=1641001–1641267|doi=10.1142/S0219887816410012|bibcode=2016IJGMM..1341001A |issn=0219-8878|url-access=subscription}}</ref><ref>{{Cite journal|last1=Laurent-Gengoux|first1=Camille|last2=Lavau|first2=Sylvain|last3=Strobl|first3=Thomas|date=2020|title=The Universal Lie ∞-Algebroid of a Singular Foliation|url=https://www.elibm.org/article/10012064|journal=ELibM – Doc. Math.|language=en|volume=25|issue=2020 |pages=1571–1652|doi=10.25537/dm.2020v25.1571-1652}}</ref> or [[Noncommutative geometry|non-commutative geometry]].<ref>{{Cite journal|last=Debord|first=Claire|date=2001-07-01|title=Holonomy Groupoids of Singular Foliations|journal=Journal of Differential Geometry|volume=58|issue=3|doi=10.4310/jdg/1090348356|s2cid=54714044 |issn=0022-040X|doi-access=free}}</ref><ref>{{Cite journal|last1=Androulidakis|first1=Iakovos|last2=Skandalis|first2=Georges|date=2009-01-01|title=The holonomy groupoid of a singular foliation|journal=Journal für die reine und angewandte Mathematik (Crelle's Journal) |url=https://www.degruyter.com/document/doi/10.1515/CRELLE.2009.001/html|language=en|volume=2009|issue=626|pages=1–37|doi=10.1515/CRELLE.2009.001|s2cid=14450917 |issn=1435-5345|arxiv=math/0612370}}</ref>

=== Examples ===

* Given a [[Lie group action]] of a [[Lie group]] on a manifold <math>M</math>, its infinitesimal generators span a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the [[Orbit (group theory)|orbits]] of the group action. The distribution/foliation is regular if and only if the action is free.
* Given a [[Poisson manifold]] <math>(M,\pi)</math>, the image of <math>\pi^\sharp = \iota_\pi:  T^*M \to TM</math> is a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the symplectic leaves of <math>(M,\pi)</math>. The distribution/foliation is regular If and only if the Poisson manifold is regular.
* More generally, the image of the anchor map <math>\rho: A \to TM</math> of any [[Lie algebroid]] <math>A \to M</math> 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 <math>\rho</math> has constant rank, i.e. the Lie algebroid is regular. Considering, respectively, the action Lie algebroid <math>M \times \mathfrak{g}</math> and the cotangent Lie algebroid <math>T^* M</math>, 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==
<references />

== Books, lecture notes and external links ==
* William M. Boothby. Section IV. 8 in [https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/63/suppl/C An Introduction to Differentiable Manifolds and Riemannian Geometry], Academic Press, San Diego, California, 2003.
*John M. Lee, Chapter 19 in [https://link.springer.com/book/10.1007/978-1-4419-9982-5 Introduction to Smooth Manifolds], Graduate Texts in Mathematics, Springer-Verlag, 2003.
*Richard Montgomery, Chapters 2, 4 and 6 in [https://www.ams.org/books/surv/091/ 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. ''[https://www.uc.pt/fctuc/dmat/seccoes/publicacoes/textosDeMatematica Textos de Matemática. Série B],'' '''48'''. Universidade de Coimbra, 2019.
* {{springer|title=Involutive distribution|id=p/i052550}}

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[[Category:Differential geometry]]
[[Category:Foliations]]