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A foliation of a manifold $X$ is a decomposition into submanifolds. These submanifolds are called the leaves of the foliation and one says that $X$ 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 smooth foliations are decomposition into immersed submanifolds such that locally this is by the fibers of a submersion (the projection to the space of leaves).
For smooth manifolds $X$, foliations arise (and this was the historical motivation for introducing them in (Ehresmann), (Reeb)) from subbundles of the tangent bundle $E \hookrightarrow T X$ which are integrable distributions (in that the Lie bracket of vector fields that are sections of $E$ is again a section of $E$): the leaves are the submanifolds whose tangent vectors are sections of $E$. If one thinks of $E$ 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 $X$. Accordingly, under Lie integration of this structure foliations of $X$ are also equivalently encodes as Lie groupoids whose space of objects is $X$ 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.
Let $M$ be an $n$-dimensional topological manifold. A decomposition of $M$ as a disjoint union of connected subsets $V_\alpha$, called leaves,
is called a foliation if there is a cover of $M$ by a collection of “special” charts of the form $(U, \phi)$, $\phi = (\phi_1,\ldots,\phi_n) : U \to \mathbb{R}^n$ such that for each “special” chart and each $\alpha$ there is a number $p\leq n$, called the dimension of the foliation, such that the intersection of any given leaf $V_\alpha$ with $U$ is one of the level sets, i.e. the solution of the system $\phi_r(x) = const = const(r,U,\alpha)$ for all $r\gt p$.
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 $p$-dimensional foliations with underlying manifold $X$ are in 1-1 correspondence with integrable distributions of hyperplanes of dimension $p$ in the tangent bundle of $X$.
The following equivalent definitions and their relation are discussed for instance in (IfLg, 1.2).
A foliation atlas of a manifold $X$ of dimension $n$ and leaf-codimension $q$ is an atlas $\{\phi_i^{-1}: R^n \to X\}_i$ such that the transition functions are globally of the form
with respect to the canonical decomposition $\mathbb{R}^n = \mathbb{R}^{n-q} \times \mathbb{R}^q$.
A foliation atlas of a manifold $X$ of dimension $n$ and leaf-codimension $q$ is an open cover $\{U_i \to X\}_i$ of $X$ equipped with submersions $\{ s_i \colon U_i \to \mathbb{R}^q \}$ such that there exists diffeomorphisms
satisfying on each $U_i \cap U_j$ the condition
Given a foliation atlas as in def. 2, the diffeomorphisms $\{\gamma_{i j}\}_{i,j}$ satisfy the Cech cocycle condition
This is called the Haefliger cocycle of the foliation atlas.
A smooth foliation of a smooth manifold $X$ is equivalently an integrable distribution (or an integrable subbundle) $E \hookrightarrow T X$.
Definition 3 above is immediately reformulated equivalently as the following statement in higher Lie theory.
For $X$ a smooth manifold, a foliation of $X$ is equivalently a Lie algebroid over $X$ such that the anchor map is an injection.
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).
One says:
A Lie groupoid integrates a given foliation, if it Lie integrates the coresponding Lie algebroid, according to def. 4.
For a simple foliation $\mathcal{D}$ of a manifold $X$, example 8, hence one where there is a submersion
to the leaf space, that map itself is the atlas of a Lie groupoid $\mathcal{G}$ which integrates the foliation, which is the Cech nerve
Among all Lie groupoids that integrate a given foliation $\mathcal{F}$ of a manifold $X$, the two special extreme
holonomy groupoid $Hol(X,\mathcal{F})_\bullet$
monodromy groupoid $Monod(X,\mathcal{F})_\bullet$
Let $\mathcal{G}_\bullet$ be a Lie groupoid with (for simplicity) connected source-fibers.
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).
One can consider the generalization of the notion of foliation of manifolds to foliations of structures in higher differential geometry such as Lie groupoids and Lie algebroids. See at
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.
Under construction.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos equipped with differential cohesion.
As usual, we write $(\int \dashv \flat \dashv \sharp)$ for the adjoint triple of modalities that defines the cohesion (shape modality $\dashv$ flat modality $\dashv$ sharp modality) and we write $(Red \dashv \int_{inf} \dashv \flat_{inf})$ for the adjoint triple of modalities that defines the differential cohesion (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ 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.
By the Giraud-Rezk-Lurie axioms we may think of a 1-epimorphism $\mathcal{G}_0 \to \mathcal{G}$ in $\mathbf{H}$ as an atlas of the cohesive $\infty$-groupoid $\mathcal{G} \in \mathbf{H}$, exhibiting equivalently the corresponding groupoid object which we write
Hence we use notation where omitting the subscript decorationon a groupoid object $\mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}$ refers to its realization
We have the following “geometricity” constraints on groupoid objects.
For $f \colon X \to Y$ any morphism in $\mathbf{H}$, write
for the canonical morphism induced by the naturality of the $\int_{inf}$-unit. We say that
$f$ is a formally smooth morphism (or submersion) if $L(f)$ is a 1-epimorphism;
$f$ is a formally étale morphism (or local diffeomorphism) if $L(f)$ is an equivalence.
Now if $\pi \colon \mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism, hence an atlas for the cohesive $\infty$-groupoid $\mathcal{G}$, then we say about the corresponding groupoid object as in def. 7, that
$\mathcal{G}_\bullet$ is an geometric ∞-groupoid if its atlas $\pi$ is a formally smooth morphism/submersion.
$\mathcal{G}_\bullet$ is an étale ∞-groupoid if its atlas $\pi$ is a formally étale morphism/local diffeomorphism.
For $X \in \mathbf{H}$, a foliation of $X$ is a morphism $\mathcal{D} \colon X \to X//\mathcal{D}$ in $\mathbf{H}$ which is
Equivalently a foliation of $X$ is a map that exhibits $X$ as an atlas for a geometric ∞-groupoid, def. 8.
Given a foliation $\mathcal{D}$ on $X$ we say that the leaf decomposition of $X$ induced by the foliation is the (∞,1)-pullback
in
where the bottom map is the counit of the flat modality.
Now let $\mathbb{G} \in Grp_2(\mathbf{H})$ a braided ∞-group. Write
for the corresponding coefficient object for curvature forms of $\mathbb{G}$-principal ∞-connections (as discussed there).
Given a closed 2-form
a foliation of $X$ by $\omega$-isotropic subspaces is a foliation $\mathcal{D} \colon X \to X//\mathcal{D}$ as in def. 9 such that the restriction of $\omega$ to the leaf decomposition is equivalent to the 0-form
hence such that the top composite morphism in the diagram
factors through the point.
We now discuss how low-degree examples of this axiomatics interpreted in $\mathbf{H} \coloneqq$ SynthDiff∞Grpd reproduces the traditional notions of foliations and isotropic submanifolds of pre-symplectic manifolds.
In the following we regard smooth manifolds canonically under the embedding
SmoothMfd $\hookrightarrow$ Smooth∞Grpd $\stackrel{i_!}{\hookrightarrow}$ SynthDiff∞Grpd $= \mathbf{H}$
as reduced synthetic differential ∞-groupoids.
A smooth function $f \colon X \to Y$ between smooth manifolds is
a local diffeomorphism in the traditional sense precisely if it is a formally étale morphism in the sense of def. 8;
a submersion in the traditional sense precisely if it is a formally smooth morphism in the sense of def. 8.
This is discussed at SynthDiff∞Grpd. The idea of the proof is to use the ∞-cohesive site of definition CartSp${}_{synthdiff}$ and evaluate the homotopy pullback in def. 8 first on all representables of the form $U \times D_1$ where $U$ ranges over Cartesian spaces and where $D_1$ is the first order ininfitesimal neighbourhood of the origin on $\mathbb{R}^1$ (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
With this now the claim is reduced to the traditional characterization of submersions and local diffeomorphisms.
Let $\mathcal{G}$ be a smooth groupoid which has a presentation by a simplicial presheaf $\mathcal{G}_\bullet$ 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 $\mathcal{G}_0 \to \mathcal{G}$. Then
if the presentation $\mathcal{G}_\bullet$ is a Lie groupoid then $\mathcal{G}_0 \to \mathcal{G}$ is a geometric ∞-groupoid
if the presentation $\mathcal{G}_\bullet$ is an étale groupoid then $\mathcal{G}_0 \to \mathcal{G}$ is an étale ∞-groupoid
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 $\mathcal{G} \underset{\int_{inf}\mathcal{G}} {\times}\int_{inf} X$ 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 $\mathcal{G}$ and so by example 3 this is a submersion or local diffeomorphism, respectively, as claimed.
Let $X$ be a smooth manifold and let $\mathcal{D}$ be a traditional foliation on $X$ which is a simple foliation, example 8, in that the leaf space $X/\mathcal{D}$ exists as a smooth manifold and the projection map $X \to X/\mathcal{D}$ is a submersion.
Then by the discussion at synthetic differential ∞-groupoid, this projection map is also a formally smooth morphism in $\mathbf{H}$ according to def. 8. Moreover, being a quotient projection it is a 1-epimorphism and hence exhibits the corresponding foliation groupoid
as a geometric ∞-groupoid in the sense of def. 8.
Now $\flat ( X// \mathcal{D})$ is the underlying set of points of the leaf space, regarded as a discrete ∞-groupoid. So we have the pasting diagram of pullbacks
for every leaf $L_l$ labeled by the point $l \in X/\mathcal{D}$ in leaf space, which exhibits the leaf decomposition of $X$ under $\mathcal{D}$ according to def. 9 as the disjoint union of the leaves of $(X,\mathcal{D})$ in the traditional sense, injected into $X$ in the canonical way.
Consider now $\mathcal{G}_\bullet$ any Lie groupoid, hence in particular a smooth groupoid $\mathcal{G} \in \mathbf{H}$ equipped with an atlas $\mathcal{G}_0 \to \mathcal{G}$, which hence by example 4 exhibits a geometric ∞-groupoid in the sense of def. 8, hence a foliation $\mathcal{D} \;\colon\; \mathcal{G}_0 \to \mathcal{G}$ in the sense of def. 9.
Computation of the homotopy pullback
by the method as in example 4 shows that $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the smooth groupoid presented by the presheaf of groupoids whose
smoothyl $U$-parameterized collection of objects are smoothly $U$-parameterized collections of morphisms $\{g_0 \to g(u)\}_{u \in U}$ in $\mathcal{G}_\bullet$ with $g_0$ held fixed;
morphisms are given by precomposing these collections with a fixed (not varying with $U$) morphism in $\mathcal{G}_\bullet$.
This means that if $\mathcal{G}_\bullet$ is an étale groupoid to start with, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the disjoint union of all its orbit leaves (as smooth manifolds), hence that the abstractly defined $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ reproduces the decomposition of $\mathcal{G}_0$ by the foliation encoded by the foliation groupoid $\mathcal{G}_\bullet$ as in traditional theory.
We may suggestively summarize example 6 in words as:
“In cohesive higher geometry, every foliation is a simple foliation.”
Because the quotient map to the leaf space of a general foliation is always a submersion/formally smooth morphism, just not always onto a manifold, but onto a higher space.
If the $\mathcal{G}_\bullet$ in example 6 is not an étale groupoid to start with but a more general Lie groupoid, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ in general retains information of non-discrete isotropy groups of $\mathcal{G}_\bullet$.
We might decide to rule out this possibility by adding to the axioms in def. 9 the clause that $X//\mathcal{G}$ (here $\mathcal{G}_0//\mathcal{D}$) 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.
For $\mathbb{G} = U(1)$ the smooth circle group, $\Omega^2_{cl}$ is the ordinary sheaf of closed differential 2-forms under the canonical embedding
Then for $X$ a smooth manifold a morphism $\omega \;\colon\; X \to \Omega^2_{cl}$ is equivalently a differential 2-form.
Then for $\mathcal{D} \;\colon\; X \to X//\mathcal{D}$ a traditional foliation of $X$ regarded as a foliation in $SynthDiff\infty Grpd$ by example 6, it follows with the discussion there that $\iota_{\mathcal{D}}^* \omega$ is precisely the collection of restriction of $\omega$ 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 $\omega$-isotropc foliation in the sense of def. 10.
For $X \to Y$ a submersion of smooth manifolds, the connected fibers of the submersion constitute a foliation of $X$ whose codimension is the dimension of $Y$. Foliations of this form are called simple foliations.
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.
Every Poisson manifold has a canonical structure of a foliation whose leaves are its maximal symplectic submanifolds, called symplectic leaves.
The set of components of a foliation is typically non-Hausdorff, which is one of the motivations of the Connes-style noncommutative geometry.
Folitation are classified by the Haefliger groupoid. See at Haefliger theorem.
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
Reeb, …
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
See also
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)