A foliation of a manifold is a decomposition into submanifolds. These submanifolds are called the leaves of the foliation and one says that is foliated by the leaves. In order to have a useful notion, leaves are required to behave sufficiently well locally. In particular if all leaves have the same dimension then one speaks of a regular foliation, which is the case discussed here. If the dimension of leaves is allowed to vary one speaks instead a singular foliation, see there for more details.
For smooth manifolds , foliations arise (and this was the historical motivation for introducing them in (Ehresmann), (Reeb)) from subbundles of the tangent bundle which are integrable distributions (in that the Lie bracket of vector fields that are sections of is again a section of ): the leaves are the submanifolds whose tangent vectors are sections of . If one thinks of as encoding a differential equation then the leaves are the solution spaces to this equation.
Expressed in terms of higher Lie theory such an integrable distribution is a sub-Lie algebroid of the tangent Lie algebroid of . Accordingly, under Lie integration of this structure foliations of are also equivalently encodes as Lie groupoids whose space of objects is and whose orbits are the leaves of the foliation.
Moreover, foliations are classified by Cech cohomology cocycles with coefficients in a topological groupoid/Lie groupoid called the Haefliger groupoid. These relations make foliation theory of sub-topic of Lie groupoid-theory. See also at motivation for higher differential geometry.
The Haefliger groupoids in fact classifies structures slightly more general than foliations: Haefliger structures.
There are several equivalent definitions of foliations.
is called a foliation if there is a cover of by a collection of “special” charts of the form , such that for each “special” chart and each there is a number , called the dimension of the foliation, such that the intersection of any given leaf with is one of the level sets, i.e. the solution of the system for all .
If the manifold is a smooth manifold, the charts may be required to be smooth too, to obtain the notion of a smooth foliation or folitation in differential geometry. In this case, the -dimensional foliations with underlying manifold are in 1-1 correspondence with integrable distributions of hyperplanes of dimension in the tangent bundle of .
The following equivalent definitions and their relation are discussed for instance in (IfLg, 1.2).
with respect to the canonical decomposition .
satisfying on each the condition
This is called the Haefliger cocycle of the foliation atlas.
The Lie groupoids which under Lie differentiation give rise to Lie algebroids with injective anchors as in def. 4 are precisely those which are Morita-equivalent to étale groupoids (hence are the foliation groupoids, see there for more details) (Crainic-Moerdijk 00, theorem 1).
Among all Lie groupoids that integrate a given foliation of a manifold , the two special extreme
Then there are maps
which are surjective local diffeomorphisms and such that the composite is the holonomy morphism (…).
This is (Crainic-Moerdijk 00, prop. 1).
The following is a suggestion for an axiomatization of foliations in higher differential geometry in the formalization of differential cohesion, followed by some considerations showing how these axioms reproduce traditional theory.
As usual, we write for the adjoint triple of modalities that defines the cohesion (shape modality flat modality sharp modality) and we write for the adjoint triple of modalities that defines the differential cohesion (reduction modality infinitesimal shape modality infinitesimal flat modality).
Below we are going to axiomatize aspects of the traditional description of foliations by Lie groupoids/foliation groupoids as discussed above, so we start by briefly setting up some terminology on groupoid objects in differential cohesion.
Hence we use notation where omitting the subscript decorationon a groupoid object refers to its realization
We have the following “geometricity” constraints on groupoid objects.
For any morphism in , write
For , a foliation of is a morphism in which is
Given a foliation on we say that the leaf decomposition of induced by the foliation is the (∞,1)-pullback
Now let a braided ∞-group. Write
Given a closed 2-form
hence such that the top composite morphism in the diagram
factors through the point.
In the following we regard smooth manifolds canonically under the embedding
This is discussed at SynthDiff∞Grpd. The idea of the proof is to use the ∞-cohesive site of definition CartSp and evaluate the homotopy pullback in def. 8 first on all representables of the form where ranges over Cartesian spaces and where is the first order ininfitesimal neighbourhood of the origin on (whose smooth algebra of fucntions is the ring of dual numbers). Then the homotopy pullback is represented as an ordinary pullback of sheaves over Cartesian spaces and the naturality diagram in question is the diagram of tangent bundles
Let be a smooth groupoid which has a presentation by a simplicial presheaf with values in 2-coskeletak Kan complexes where objects and morphisms are represented by a smooth manifold each, and consider it equipped with the induced atlas . Then
in the sense of def. 8.
This follows by the corresponding discussion at SynthDiff∞Grpd. The idea of the proof is that one presents the atlas in the projective model structure on simplicial presheaves by the décalage fibration resolution, schematically
Then the homotopy pullback is presented by an ordinary pullback and so example 3 applies degreewise. In degree 0 the above resolution is the target map in the groupoid and so by example 3 this is a submersion or local diffeomorphism, respectively, as claimed.
Then by the discussion at synthetic differential ∞-groupoid, this projection map is also a formally smooth morphism in according to def. 8. Moreover, being a quotient projection it is a 1-epimorphism and hence exhibits the corresponding foliation groupoid
for every leaf labeled by the point in leaf space, which exhibits the leaf decomposition of under according to def. 9 as the disjoint union of the leaves of in the traditional sense, injected into in the canonical way.
Consider now any Lie groupoid, hence in particular a smooth groupoid equipped with an atlas , which hence by example 4 exhibits a geometric ∞-groupoid in the sense of def. 8, hence a foliation in the sense of def. 9.
Computation of the homotopy pullback
smoothyl -parameterized collection of objects are smoothly -parameterized collections of morphisms in with held fixed;
morphisms are given by precomposing these collections with a fixed (not varying with ) morphism in .
This means that if is an étale groupoid to start with, then is the disjoint union of all its orbit leaves (as smooth manifolds), hence that the abstractly defined reproduces the decomposition of by the foliation encoded by the foliation groupoid as in traditional theory.
We may suggestively summarize example 6 in words as:
“In cohesive higher geometry, every foliation is a simple foliation.”
We might decide to rule out this possibility by adding to the axioms in def. 9 the clause that (here ) be étale.
However, we might also keep that case and regard it as the first instance of what is certainly a natural phenomenon as we pass to higher geometry, namely that leaves of a foliation no longer need to manifolds but will be (higher) groupoids themselves.
Finally, given the above it is clear how isotropic appear in the cohesive axiomatics.
Then for a traditional foliation of regarded as a foliation in by example 6, it follows with the discussion there that is precisely the collection of restriction of to each of the leaves of the foliation. Therefore this is a foliation by isotropic submanifolds in the traditional sense precisely if it is an -isotropc foliation in the sense of def. 10.
Every Lie groupoid gives a folitation on its space of objects: the leaves are the orbits. Conversely, every regular foliation gives rise to its holonomy groupoid. This is a (not necessarily Hausdorff) Lie groupoid whose orbits are the leaves of the original foliation, and which in some sense is minimal with this condition.
There is a theory of characteristic classes for foliations. A most well known example is the Godbillon-Vey characteristic class.
The notion of foliated manifolds was introduced in the 1950s, motivated from partial differential equation theory, in
Eli Cartan, Sur l’intégration des équations différentiels completement intégrable, Oeuvres Complètes, Pt. II, Vol. I, 555-561.
A discussion in differential geometry is in
A textbook account with a view to the modern formulation in Lie groupoid theory is
Foliations in Lie groupoid theory are discussed in more detail in
The corresponding groupoid algebras are discussed in chapter 2, section 8 of
A survey by Fuks in Russian Itogi:
Cohomology of formal vector fields and characteristic classes of foliations were originally studied in the papers
D. B. Fuks, Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations (book, Rus. and Eng. versions)
I. M. Gelʹfand, B. L. Feĭgin, D. B. Fuks, Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations, Funkcional. Anal. i Priložen. 8 (1974), no. 2, 13–29 (Russian original mathnet.ru, pdf)
Claude Godbillon, Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 421, pp. 69–87. Lecture Notes in Math. 383, Springer 1974.
И. М. Гельфанд, Д. Б. Фукс, Когомологии алгебры Ли формальных векторных полей, Изв. АН СССР. Сер. матем., 1970, 34, в. 2, стр. 322–-337, pdf
In a series of works of Connes and Moscovici, the local index formulas in the context of transverse geometry of foliations has been studied in connection to a new cyclic homology of a Hopf algebra arising in this context:
More general issues of index theory in noncommutative geometry applied to foliations is in
Yu. A. Kordyukov, Noncommutative geometry of foliations, J. K-Theory, 2:2, Special issue in memory of Y. P. Solovyev, Part 1 (2008), 219–327 MR2009m:58018; Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 64:2 (2009), 273–391 (original: Ю. А. Кордюков, УМН, 64:2(386) (2009), 73–202); Формула следов для трансверсально-эллиптических операторов на римановых слоениях, Алгебра и анализ, 12:3 (2000), 81–105 pdf
W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268 (doi); Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307 (pdf); The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974), 214–231 (link)