Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment 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

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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 

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  .

Regular distributions

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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

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Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above).

Involutive distributions

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Given a distribution  , its sections consist of vector fields on   forming a vector subspace   of the space of all vector fields on  . (Notation:   is the space of sections of  ) 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  

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

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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   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

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Given any distribution  , the associated Lie flag is a grading, defined as

 

where  ,   and  . In other words,   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   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  .[1]

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).[clarification needed] 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

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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

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Integrable distributions

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  • 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.

Non-integrable distributions

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  • 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.[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

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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

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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

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  • 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.

References

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  1. ^ a b Tanaka, Noboru (1970-01-01). "On differential systems, graded Lie algebras and pseudo-groups". Kyoto Journal of Mathematics. 10 (1). doi:10.1215/kjm/1250523814. ISSN 2156-2261.
  2. ^ 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. S2CID 121523670.
  3. ^ 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.
  4. ^ Engel, Friedrich (1889). "Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen". Leipz. Ber. (in German). 41: 157–176.
  5. ^ Lavau, Sylvain (2018-12-01). "A short guide through integration theorems of generalized distributions". Differential Geometry and Its Applications. 61: 42–58. arXiv:1710.01627. doi:10.1016/j.difgeo.2018.07.005. ISSN 0926-2245. S2CID 119669163.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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.
  10. ^ 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. Bibcode:2016IJGMM..1341001A. doi:10.1142/S0219887816410012. ISSN 0219-8878.
  11. ^ Laurent-Gengoux, Camille; Lavau, Sylvain; Strobl, Thomas (2020). "The Universal Lie ∞-Algebroid of a Singular Foliation". ELibM – Doc. Math. 25 (2020): 1571–1652. doi:10.25537/dm.2020v25.1571-1652.
  12. ^ Debord, Claire (2001-07-01). "Holonomy Groupoids of Singular Foliations". Journal of Differential Geometry. 58 (3). doi:10.4310/jdg/1090348356. ISSN 0022-040X. S2CID 54714044.
  13. ^ Androulidakis, Iakovos; Skandalis, Georges (2009-01-01). "The holonomy groupoid of a singular foliation". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2009 (626): 1–37. arXiv:math/0612370. doi:10.1515/CRELLE.2009.001. ISSN 1435-5345. S2CID 14450917.
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