Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis.
Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists in 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
The set of smooth vector fields is also called a local basis of . Note that the number may be different for different neighbourhoods. The notation is used to denote both the assignment and the subset .
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.
More compactly, a regular distribution is a vector subbundle 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).
Given a distribution , its sections consist of the vector fields which are tangent to , and they form a vector subspace of the space of all vector fields on . A distribution is called involutive if is also a Lie subalgebra: in other words, for any two vector fields , the Lie bracket belongs to .
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
An integrable 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 integrable manifold. This means that is the disjoint union of maximalconnected integrable manifolds, also called leaves; defines therefore a foliation.
Locally, integrability means that for every point there exists a local chart 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 , consider its associated Lie flag (note that some authors use a negative decreasing grading instead)
where , and . In other words, denotes the set of vector fields spanned by the -iterated Lie brackets of elements in .
Then is called weakly regular (or just regular by some authors) if there exists a sequence of nested vector subbundles such that (hence ). 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 is then called the grow vector of .
Any weakly regular distribution has an associated graded vector bundle
Moreover, the Lie bracket of vector fields descends, for any , to a -linear bundle morphism , 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 , 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.
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 of course the same.
Thanks to 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.
Examples of regular distributions
Any vector field on defines a rank 1 distribution, by setting , 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.
The Martinet distribution on is given by , for ; 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 , for ; 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 , for and ; 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.
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.
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, the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan and Sussmann. 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 singuar foliations according to which applications one has in mind, e.g. Poisson geometry or non-commutative geometry.
Given a Lie group action of a Lie group on a manifold , 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.
Given a Poisson manifold, the image of 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.
^Chow, Wei-Liang (1940-12-01). "Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung". Mathematische Annalen (in German). 117 (1): 98–105. doi:10.1007/BF01450011. ISSN 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. (in Russian). 2: 83–94.
^Engel, Friedrich (1889). "Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen". Leipz. Ber. (in German). 41: 157–176.
^Lavau, Sylvain (2018-12-01). "A short guide through integration theorems of generalized distributions". Differential Geometry and its Applications. 61: 42–58. doi:10.1016/j.difgeo.2018.07.005. ISSN 0926-2245.
^Stefan, P. (1974). "Accessibility and foliations with singularities". Bulletin of the American Mathematical Society. 80 (6): 1142–1145. doi:10.1090/S0002-9904-1974-13648-7. ISSN 0002-9904.
^Stefan, P. (1974). "Accessible Sets, Orbits, and Foliations with Singularities". Proceedings of the London Mathematical Society. s3-29 (4): 699–713. doi:10.1112/plms/s3-29.4.699. ISSN 1460-244X.
^Sussmann, Hector J. (1973). "Orbits of families of vector fields and integrability of systems with singularities". Bulletin of the American Mathematical Society. 79 (1): 197–199. doi:10.1090/S0002-9904-1973-13152-0. ISSN 0002-9904.
^Sussmann, Héctor J. (1973). "Orbits of families of vector fields and integrability of distributions". Transactions of the American Mathematical Society. 180: 171–188. doi:10.1090/S0002-9947-1973-0321133-2. ISSN 0002-9947.
^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. doi:10.1142/S0219887816410012. ISSN 0219-8878.
^Laurent-Gengoux, Camille; Lavau, Sylvain; Strobl, Thomas (2020). "The Universal Lie ∞-Algebroid of a Singular Foliation". eLibM – Doc. Math. 25: 1571–1652. doi:10.25537/dm.2020v25.1571-1652.
^Debord, Claire (2001-07-01). "Holonomy Groupoids of Singular Foliations". Journal of Differential Geometry. 58 (3). doi:10.4310/jdg/1090348356. ISSN 0022-040X.
^Androulidakis, Iakovos; Skandalis, Georges (2009-01-01). "The holonomy groupoid of a singular foliation". 2009 (626): 1–37. doi:10.1515/CRELLE.2009.001. ISSN 1435-5345. Cite journal requires |journal= (help)
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 Monographs91. 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.