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BOYER: “SASBOOK” — 2007/9/26 — 15:55 — PAGE 51 — #63

CHAPTER 2

Foliations

Foliations were defined in Example 1.4.13. It is the purpose of this chapter togather together those well-known facts concerning foliations that we shall need inthe sequel. The theory of foliations is very well developed, there being many bookson the subject [CC00, CC03, Rei83, CL85, HH86, Mol88, God91, Tam92,Ton97]. The modern theory began with the work of Ehresmann and Reeb in the1940s (cf. [ER44]), but as with much in mathematics its roots go back to thenineteenth century. The type of foliations that we are mainly concerned with inthis book are the so-called Riemannian foliations. They have been developed in thebooks of Reinhart [Rei83], Molino [Mol88], and Tondeur [Ton97].

2.1. Examples of Foliations

We give here several examples of foliations which should provide insight later on.Only those examples which illustrate concepts pertinent to our development aregiven.Example 2.1.1: Submersions. As mentioned previously a submersion f :M−−→N,where M and N are smooth manifolds, is a very special case of a foliation, called asimple foliation. A fiber bundle is a special case of a submersion; however, not everysubmersion satisfies the local triviality condition. If one removes a point from a fiberof a fiber bundle, one still has a submersion, but local triviality fails. For a moreinteresting examples of submersions that are not locally trivial see [CC00] pp. 6–7.The Ehresmann Fibration Theorem states that if f is proper then a submersion isa fiber bundle.

The next example illustrates phenomena that will re-occur in our work.Example 2.1.2: Linear flows on tori. For simplicity we consider the 2-dimensionaltorus T 2 defined as the quotient space by the integer lattice, viz. R2/(Z×Z).A linear flow is the flow generated by the vector fields

X = a1∂

∂x1+ a2

∂

∂x2

where ai ∈ R for i = 1, 2 are not both zero. For each such pair (a1, a2) ∈ R2−(0, 0)the vector fieldX generates a subbundle E of TT 2, and thus defines a 1-dimensionalfoliation F on T 2. (Frobenius integrability is automatic for 1-dimensional distri-butions.) The leaf passing through the point (x1(0), x2(0)) ∈ T 2 is given by theimage of the flow φ : R−−→R2 defined by

φ(t) = (x1(0) + a1t, x2(0) + a2t).

51


52 2. FOLIATIONS

We assume that a1 = 0. There are two cases to consider:(i) The ratio a2

a1is rational. In this case the vector field X is periodic and

the leaves of the foliation are circles. The foliation F is simple, and infact describes T 2 as an S1 bundle over S1.

(ii) The ratio a2a1is irrational. In this case the leaves are diffeomorphic to R,

and each leaf is dense in T 2. Hence, the topology of the quotient spaceT 2/F is not even a T1 space (referring to the separation axioms of point-set topology).

Exercise 2.1: Prove the assertions (i) and (ii) of Example 2.1.2.Our next example generalizes a well-known result in Lie theory.

Example 2.1.3: A locally free action of a Lie group. Let G be a Lie groupwhich acts locally freely on a smooth manifold M, i.e., there is a smooth mapG ×M−−→M such that the isotropy subgroup at any point is a discrete subgroupof G. Let g denote the Lie algebra of G. The action induces a monomorphismψ : g−−→dim(g) of Lie algebras, and since this action is locally free the image ψ(g)generates a subbundle E of TM of dimension X (M). Furthermore, since ψ(g) issubalgebra of X (M), the subbundle E is integrable and defines a foliation F on M.As mentioned in the paragraph following Corollary 1.6.29, if G is compact and actslocally freely on M, then the quotient space M/G has the structure of an orbifold.This can be shown by constructing a Riemannian foliation by averaging over G andusing Molino’s Theorem 2.5.11 given below. This generalizes to the case of locallyfree action the well-known theorem of Lie theory that says that if G acts properlyand freely on M, then the space of leaves of F , i.e., the orbit space M/G of theaction is a smooth manifold such that M is the total space of a principal G-bundleover M/G.

2.2. Haefliger Structures

Let (Uα;φα) be a foliated atlas with coordinates φα = (x(α)1 , . . . , x

(α)p ;

y(α)1 , . . . , y

(α)q ). Then there are local submersions fα : Uα−−→Rq defined to be the

composition of φα : Uα−−→Rp+q = Rp × Rq with projection onto the second factor.The inverse image f−1(y) of a point y ∈ Rq defines a plaque of F , that is a com-ponent of Uα ∩ Lα. In the overlap of foliated charts Uα ∩ Uβ the local submersionsfα, fβ are related by

(2.2.1) fβ = τβαfα,

where the transition functions τβα : fα(Uα∩Uβ)−−→fβ(Uα∩Uβ) are diffeomorphismsthat satisfy Haefliger’s cocycle conditions

(2.2.2) ταβ = τ−1βα on Uα ∩ Uβ , τγα = τγβτβα on Uα ∩ Uβ ∩ Uγ .

This point of view has particular interest in regard to transverse G-structures whichare introduced below in Definition 2.5.2.

Haefliger [Hae62] has generalized this notion in order to construct a correcthomotopy theory for foliations. It has also been generalized by using other pseudo-groups, cf. [Hae71].


2.2. HAEFLIGER STRUCTURES 53

Definition 2.2.1: Let Γ be a subpseudogroup of the pseudogroup ΓGL(q,R) of localdiffeomorphisms of Rq, and let X be a topological space. A Haefliger cocycle onX is given by the following data:

(i) an open cover U = Uαα∈I of X,(ii) continuous maps fα : Uα−−→Rq, called local projections,(iii) for each α, β ∈ I and x ∈ Uα ∩ Uβ , a diffeomorphism τγβ ∈ Γq from

a neighborhood of fα(x) onto a neighborhood of fβ(x), where the germof the map fα at the point fα(x) varies continuously with x, such thatfβ = τβαfα, and for all x ∈ Uα ∩ Uβ ∩ Uγ one has τγα = τγβτβα.

We remark that if the local projections fα are open maps, then the secondcondition in (iii) follows from the first. It is also clear that if X is a smooth manifoldand the local projections fα are local submersions we recover Haefliger’s descriptionof a foliation given above. In this case we say that the Haefliger cocycle is a Haefligercocycle of a foliation, and in the case of a proper subpseudogroup Γq ⊂ ΓGL(q,R),it is a Haefliger cocycle of a foliation with an additional transverse structure. Weshall consider this case in more detail in Section 2.5. Definition 2.2.1 also makesperfect sense if we replace Rq by an appropriate q-dimensional “model space” N,and Γq by any pseudogroup of transformations on N.

Next we give an alternative definition of a Haefliger cocycle in terms of groupoids.We refer to Appendix A for a brief discussion of groupoids. Actually this seconddefinition is a bit more general since it holds for any topological groupoid G. WhenG is a groupoid of germs this definition is completely equivalent to Definition 2.2.1by Proposition 1.5.4.

Definition 2.2.2: Let G be a topological groupoid with X as its space of units. AHaefliger cocycle on X consists of an open cover Uα together with a family ofcontinuous maps gαβ : Uα ∩ Uβ−−→G such that for every x ∈ Uα ∩ Uβ ∩ Uγ thecondition gαγ(x) = gαβ(x)gβγ(x) holds. Two such cocycles are equivalent if theycan be extended to a cocycle on the union of the covers.

Notice that local projection maps are implicit in this definition. They areobtained by the composition fα = s gαα : Uα−−→X. We now take the groupoid Gto be a groupoid of germs G(Γq) of a pseudogroup Γq. Haefliger’s cocycle condition,of Definition 2.2.2, is a cocycle for the cohomology set H1(U,G(Γq)). In this settwo cocycles gβα and g′βα are equivalent if there exist germs of diffeomorphismsζα, ζβ ∈ Γq such that on fα(Uα ∩ Uβ) the relation g′βα = ζβ gβα ζ−1

α holds. Theequivalence stated in Definition 2.2.2 is the following: we say that the Haefligercocycle gαβ corresponding to the cover U is equivalent to the Haefliger cocycle g′α′β′

corresponding to the cover U′ if there is a Haefliger cocycle g′′α′′β′′ on the disjointunion U U′ that restricts to gαβ on U and to g′α′β′ on U′. As usual by takingthe inductive limit over finer covers we obtain the cohomology set H1(X,G(Γq))whose elements we refer to as Γq-structures or Haefliger structures. Unfortunately,the classes in H1(X,G(Γq)) are not invariant under homotopy, so we need todefine the set Γq(X) of homotopy classes of elements in H1(X,G(Γq)). We say thetwo cocycles g1, g2 ∈ H1(X,G(Γq)) are homotopic if there exists an element Υ ∈H1(X × I,G(Γq)) such that ι∗jΥ = gj , where I = [0, 1], and ιj : X−−→X × I are thenatural face maps.

Exercise 2.2: Show that the property of being homotopic defines an equivalencerelation on H1(X,G(Γq)).


54 2. FOLIATIONS

Now we are interested in the case when the Haefliger structure comes from afoliation F on a smooth manifold M. In this case we haveProposition 2.2.3: Let M be a smooth manifold with a foliation F of codimensionq.ThenF determines a unique Haefliger structure (g)∈H1(M,G(Γq)). Furthermore,any two foliations that determine the same Haefliger structure are isomorphic.

Proof. Given a foliated atlas U = (Uα;φα) we obtain local submersionsfα : Uα−−→Rq as above. This uniquely determines a cocycle gβα by Equation (2.2.1)which determines a unique germ gxβα, and this germ determines a unique Haefligerstructure (g) ∈ H1(U,G(Γq)). Suppose that V = (Vα′ ;ψα′) is another foliatedatlas for F , and gα′ : Vα′−−→Rq are the corresponding local submersions. This deter-mines a unique element (υ) ∈ H1(V,G(Γq)). Let W = (Wα;φα) be a commonrefinement of U and V. By abuse of notation we let fα, gα denote their correspond-ing restrictions toWα. Then for each α there is a diffeomorphism ζα ∈ Γq such thatgα = ζα fα. So by the uniqueness of the diffeomorphism in Definition 2.2.1 andwe have υαβ = ζα gαβ ζ−1

β . Then by passing to germs this gives a unique classin H1(W,G(Γq)), and hence, in H1(M,G(Γq)).

Suppose two foliations F and F ′ determine the same Haefliger structure, thenthey determine the same element (g) ∈ H1(U,G(Γq)) for some common foliatedatlas U.

2.3. Leaf Holonomy and the Holonomy Groupoid

In this section we describe an important invariant of foliations, the holonomygroupoid. We begin by describing the leaf holonomy of a given leaf, a conceptwhich generalizes the classical “first return map” of Poincare from dynamical sys-tems theory. Let (M,F) be a foliated manifold with dimension of M equal to n,

and the dimension of F equal to p. A submanifold Nι→M is called a transversal or

transverse submanifold if at each point x ∈ N we have

Tι(x)M =TLx + TxN.

In order to construct leaf holonomy, we now describe, following [Mol88], a pro-cedure known as “sliding along the leaves.” Consider two points x0 and x′0 lyingon the same leaf L0. Choose transversals N and N ′ through x0 and x′0, respectively,and a piecewise smooth path γ:[0, 1]−−→L0 such that γ(0)=x0 and γ(1)=x′0. Subdi-vide the interval [0, 1] into subintervals [tα−1, tα] with 0= t0 <t1 < · · ·<tk =1 suchthat each γ([tα−1, tα]) lies in a simple open set Uα of a foliated atlas. Moreover, sinceγ([tα−1, tα]) is connected it lies on a unique plaque of L0 ∩Uα. Let Nα be transver-sals at γ(tα). Then since γ(tα−1) lies on a single plaque in Uα−1 ∩ Uα, for eachα=1, . . . , k there is a diffeomorphism φα from an open neighborhood Vi−1 ⊂Nα−1about γ(tα−1) onto an open neighborhood Vα⊂Nα about γ(tα) which is constructedfrom the local submersions fα : Uα−−→Rq by the commutative diagram

(2.3.1)Vα−1

φα−−−−→ Vα,

Rq

where the diagonal arrows are fα−1|Vα−1 and fα|Vα , respectively. By looking at thelocal foliated coordinate charts (Uα−1;x

(α−1)1 , . . . , x

(α−1)p , y

(α−1)1 , . . . , y

(α−1)q ) and


2.3. LEAF HOLONOMY AND THE HOLONOMY GROUPOID 55

(Uα;x(α)1 , . . . , x

(α)p , y

(α)1 , . . . , y

(α)q ) one can easily identify the coordinate represen-

tative of φα with the Haefliger cocycle g(α−1)α. It follows that the compositionφk1 = φk · · · φ1 : V0−−→Vk is a diffeomorphism that depends on the subdivisionand the intermediate transversals only through their domains. It does, however,depend on the choice of foliated atlas, and the transversals at x0 and x′0. The set ofall local diffeomorphisms obtained in this way form a pseudogroup ΓU,N,N ′ , calledthe holonomy pseudogroup of F . This is a bit of a misnomer; however, a differentchoice of foliated atlas, and different choices of transversals give pseudogroups thatare locally conjugate, so it makes sense to speak of “the” holonomy pseudogroup.

Since the diffeomorphism φk1 described above depends on the path, the subdivision, and intermediate transversals through their domains, it makes sense topass to germs. We denote the germ of the diffeomorphism φk1 at x0 by hγ . Thisis now independent of the subdivision and transversals. Notice also that if γ′ isanother path with the same endpoints and Γ : [0, 1]× [0, 1]−−→∪αUα is a homotopybetween γ and γ′, then hγ only depends on the homotopy class [γ] of γ (cf. [CC00]).Now suppose that x′0 = x0 in which case we can take N ′ = N. Then hγ defines thegerm of a local diffeomorphism of N which leaves x0 fixed. We let Gx0

x0denote the

set of all such germs obtained by sliding along leaves. (The notation will be clearshortly). This set forms a group under composition. Notice that if γ and γ′ are twoloops at x0, then hγ′γ = hγ′ hγ , so one gets a group epimorphism

(2.3.2) hx0 : π1(L0, x0)−−−→Gx0x0

→ Diffx0 ,

where Diffx0 denotes the group of germs of diffeomorphisms of N that fix x0. Itcan be identified with the group of germs of local diffeomorphisms of Rq which fixthe origin.Definition 2.3.1: The group Gx0

x0is called the holonomy group of the leaf L0

at x0.

If a different transversal N ′ is chosen at x0, then one can identify the germ ofthe transversal N ′ with the germ of the transversal N. So the leaf holonomy groupis well-defined up to this identification. Furthermore, if x1 ∈ L0 is another point ofthe same leaf, any path γ : [0, 1]−−→L0 satisfying γ(0) = x0 and γ(1) = x1 inducesan isomorphism

γ∗ : Gx0x0−−−→Gx1

x1,

where γ∗(hx0) = hγ hx0 hγ−1 . Thus, it makes sense to talk about the holonomygroup Hol(L) of a leaf L.

Proposition 2.3.2: Let F be a simple foliation. Then Hol(L) is trivial for allleaves L of F .

Proof. Since the foliation is simple the local submersions in diagram (2.3.1)are the restrictions of a global submersion f . Thus, after identifying Vα−1 with Vαby translation along the leaves, the map φα in (2.3.1) is the identity, so slidingalong the leaves produces the germ of the identity diffeomorphism.

The following proposition says that for any foliation the “generic” leaf hastrivial holonomy.Proposition 2.3.3: For any foliated manifold the leaves having a trivial holonomygroup form a residual (i.e., a countable intersection of dense open sets) set.

Proof. Cf. [CC00].


56 2. FOLIATIONS

The kernel of the homomorphism hx of Equation (2.3.2) is a normal subgroupof the fundamental group of the leaf called the holonomy kernel, and denote byHolker(L). A space (leaf) whose fundamental group is Holker(L) is called the holo-nomy covering. We consider several examples regarding holonomy. The first is acontinuation of Example 2.1.2.Example 2.3.4: Linear flows on tori. Again for simplicity we consider only T 2.In both the irrational and rational cases the holonomy groups are all trivial, butfor different reasons. In the rational case the foliation is simple, so the holonomy istrivial. In the irrational case the leaves are simply connected, so the holonomy isagain trivial. However, in this latter case the holonomy pseudogroup is non-trivial.

The next example is more interesting and plays an important role in the sequel.Example 2.3.5: Linear flows on odd-dimensional spheres. Consider the unitsphere in Cn+1 defined by

S2n+1 = z = (z0, . . . , zn) ∈ Cn+1 | |z0|2 + · · ·+ |zn|2 = 1.By linear flow we mean linear in the complex Cartesian coordinates z = (z0, . . . , zn).Again for simplicity we consider the case n = 1. Let w0 and w1 be two non-zeroreal numbers, and consider the real vector field on C2

ξ = i

(w0z0

∂

∂z0+ w1z1

∂

∂z1− c.c

).

Restricted to S3 this vector field is everywhere tangent to S3, and defines a nowherevanishing vector field on S3. Thus, ξ generates a 1-dimensional foliation on S3 whoseflow is given by

(z0, z1) → (e2πiw0tz0, e2πiw1tz1).

Again there are two cases to consider depending on whether the ratio w1w0is rational

or irrational. If the ratio is rational by a reparameterization and considering thecomplex conjugate coordinates if necessary, we can take the pair (w0, w1) to becoprime positive integers. In this case all the leaves are circles. The leaf throughx0 = (1, 0) is a circle which returns to x0 at t = 1

w0, whereas the nearby leaves with

z1 = 0 take w0 times as long to return. So the holonomy group Hol(Lx0) ≈ Zw0 .Similarly we see for x1 = (0, 1) the holonomy through the leaf Lx1 is Hol(Lx1) ≈Zw1 . We also see that Holker(Li) ≈ wiZ. All other leaves have trivial holonomy.

Now consider the case when the ratio w1w0is irrational. Consider the leaf through

a point with z0z1 = 0, say the point ( 1√2, 1√

2). This leaf is non-compact; in fact,

it is the irrational flow on the torus T 2 in S3 defined by |z0|2 = |z1|2 = 12 . So this

leaf is not closed in S3. Its closure is just the 2-torus T 2. The same phenomenonoccurs for any point (z0, z1) ∈ S3 with z0z1 = 0. However, the two leaves throughx0 = (1, 0) and x1 = (0, 1) are circles for they return after a time t = 1

w0and t = 1

w1,

respectively. Nevertheless, all leaves have trivial holonomy which can be seen from(v) of Theorem 2.5.10 below, but as in Example 2.3.4 the holonomy pseudogroupis non-trivial.

The holonomy groups attached to a foliation can be studied together by form-ing a more global object, the holonomy groupoid. Although the ideas go back toEhresmann, Reeb, and Thom, it appears that its first precise formulation was givenby Winkelnkemper [Win83]. See also [Rei83].


2.3. LEAF HOLONOMY AND THE HOLONOMY GROUPOID 57

Proposition 2.3.6: Let (M,F) be a foliated manifold. The collection of all triples(x, y, [γ]), where x, y ∈ M lie on the same leaf L, γ is a piecewise smooth pathfrom x to y, and [γ] denotes the holonomy equivalence class of γ, i.e., γ ∼ γ′

if γ′γ−1 = e ∈ Gyy , is a Lie groupoid of dimension n + p called the holonomygroupoid1 of (M,F) and denoted by G = G(M,F). Furthermore, if every holonomygerm is conjugate to an analytic germ, G(M,F) is Hausdorff.

Proof. In Appendix A.1 we give a brief review of groupoids. Clearly, the setof objects G0 is M and this is a smooth Hausdorff manifold, while G1 is the setof triples (x, y, [γ]). We show that G1 is a smooth manifold by following Connes[Rei83] and exhibiting a system of coordinates on it. We then check that thestructure maps are smooth. Write the coordinates as (v, w) and (v′, w) having thesame transverse coordinate w at the beginning and end of a path, respectively. Herev and v′ are coordinates along the leaf. Then (v′, v, w) is a coordinate system onG1, and a change of coordinates is of the form

F (v′, v, w) = (f ′(v′, w), f(v, w), g(w)),

where (v, w) → (f(v, w), g(w)) is a change of coordinates on M. This gives G1 alocally Euclidean topology of dimension n+p that is not necessarily Hausdorff. Thistopology is also second countable (see [Rei83], p. 137). Moreover, G1 is Hausdorffif and only if each holonomy germ at x ∈ M which is the identity on some open setwhose closure contains x is the identity germ at x [Rei83, Win83]. In particular,if every holonomy germ is conjugate to an analytic germ, the groupoid G(M,F)will be Hausdorff.

The canonical source and target maps s : G−−→M and t : G−−→M, aredefined by

(2.3.3) s(x, y, [γ]) = x, t(x, y, [γ]) = y,

while the unit is u(x) = (x, x, 0), where 0 denotes the constant loop. Furthermore,the associative multiplication on G is defined when the range of one element coin-cides with the source of the other, specifically in our case we have

(2.3.4) (x, y, [γ]) · (x′, y′, [γ′]) = (x, y′, [γ γ′]) if y = x′

The inverse map is defined by ι(x, y, [γ]) = (y, x, [γ−1]). To see that the structuremaps are smooth, we write in coordinates:

s(v′, v, w) = (v, w), t(v′, v, w) = (v′, w), u(v, w) = (v, v, w),

ι(v′, v, w) = (v, v′, w), (v′′, v′, w) · (v′, v, w) = (v′′, v, w).So the structure maps are smooth. Furthermore, both s and t are submersions. Thiscompletes the proof.

The fact that the holonomy groupoid G(M,F) of a foliation is, in general, notHausdorff is of very little consequence for us, since it known to be Hausdorff whenthe foliation F is Riemannian (see Definition 2.5.4 and Theorem 2.5.10 below)which is the case of interest to us. Notice that F induces a foliation2 of dimension2p on G(M,F) by pulling back the foliation F to G(M,F) by the source map s. Wedenote this foliation on G(M,F) by FG . If E denotes the integrable subbundle ofTM corresponding to the foliation F , then the integrable subbundle of TG(M,F)

1This is called the graph of the foliation in [Win83].2A different 2p-dimensional foliation having trivial holonomy groups is described in [Win83].


58 2. FOLIATIONS

corresponding to FG is s∗E ⊕ V, where V is the vertical bundle consisting of thetangent vectors to the fibers of the submersion s. So the leaf L(x,y,[α]) of FG through(x, y, [α]) ∈ G(M,F) projects under s and t to the leaf Lx of F . Furthermore,these projections induce an isomorphism of the corresponding holonomy groups(see [Rei83], Proposition IV.2.44). Notice that the both subbundles s∗E and V ofTG(M,F) are separately integrable. Summarizing we haveProposition 2.3.7: Let F be a p-dimensional foliation on a manifold M. ThenF pulls back under either s or t to a 2p-dimensional foliation FG on the holonomygroupoid G(M,F) that is invariant under the involution ι. If L(x,y,[α]) denotes theleaf of FG through (x, y, [α]) ∈ G(M,F) and Lx denotes the leaf of F through x ∈ M,then there is an isomorphism Hol(L(x,y,[α])) ≈ Hol(Lx).

More information about the holonomy groupoid G(M,F) can be ascertainedwhen F is a Riemannian foliation. This will be described later in this section. Nowwe consider another important groupoid introduced by Haefliger [Hae84] which hecalled the transverse holonomy groupoid. It is closely related to, but different fromG(M,F). Here is the definition.Definition 2.3.8: Let N be a complete transversal. The transverse holonomygroupoid associated to N is the full subgroupoid GNN of G(M,F).

GNN is defined in the Appendix by Equations (A.1.2). Here complete means thatN intersects each leaf at least once. Complete transversals always exist since wedo not require them to be connected. Of course, GNN depends on N, but Haefliger[Hae84] shows that GNN is independent of N up to Morita equivalence. In factTheorem 2.3.9: Let F be a p-dimensional foliation on a manifold of dimensionp+q, and let N be any complete transversal. Then the transverse holonomy groupoidGNN is an etale Lie groupoid of dimension q that is Morita equivalent to the holonomygroupoid G(M,F).

Proof. GN = t−1(N) is a submanifold of G1 such that

codim(GN → G1) = codim(N → G0) = p.

Since G1 has dimension 2p + q we see that dim (GN ) = p + q = dim (G0). Nowconsider the source map sN = s|GN restricted to GN . If we show that this mapsN : GN−−→G0 is an immersion, it will be a local diffeomorphism. Then GNN =s−1N (N) will be a submanifold of GN of dimension q. So the restriction of sN to GNNwill be a local diffeomorphism. Thus, GNN will be etale. To see that the differentialdsn is injective at a morphism (g : x−−→y) ∈ GN ⊂ G1, we consider a tangentvector v ∈ TgG1. It will be tangent to GN if (dt)g(v) ∈ TyN. Since s : G1−−→G0

is a submersion, it defines a p-dimensional foliation F on G1 whose leaves are theconnected components of the fibers of the submersion. Thus, if dsN (v) = 0 with vtangent to GN , then v is both tangent to the fibers of the source submersion anddtg(v) is tangent to N. But (dt)g maps the tangent space to the leaf Lg of F at g tothe tangent space to the leaf Ly of F at y. Thus, v = 0 which proves the injectivityof dsN , or equivalently, that sN is a local diffeomorphism.

To prove the last statement we consider the smooth functor F : GNN−−→G(M,F)defined by F (x, y, [g]) = (ιN (x), ιN (y), [g]).We see from the fact that the transversalN is complete that sN is surjective. This shows that the map sπ2 : N×G0G1−−→G0on pairs (x, g) satisfying ιN (x) = t(g) is surjective. Moreover, one easily checks


2.4. BASIC COHOMOLOGY 59

that corresponding square is a pullback. Thus, F : GNN−−→G(M,F) is an essentialequivalence, and hence GNN and G(M,F) are Morita equivalent.

In particular, Theorem 2.3.9 says that the etale groupoid GNN is independent ofN up to Morita equivalence. We shall see later that Morita equivalent groupoidshave homotopy equivalent classifying spaces. We will consider GNN further in Chap-ter 4.

We are most interested in the case where the foliation arises from a locally freeaction of a Lie group (Example 2.1.3) with finite isotropy groups. So we recall theaction groupoid from the Appendix (Definition A.1.5). For G a Lie group and M asmooth manifold, we consider the action groupoid M G. According to Reinhart([Rei83], p. 138) the space of morphisms of the holonomy groupoid of a foliationcoming from a locally free action of a Lie group G is just M × G. So in this casethese two groupoids are isomorphic.

2.4. Basic Cohomology

Basic cohomology was introduced by Reinhart in [Rei59b]. We begin with a briefreview following [Ton97]. Let F be a foliation of a smooth manifold M. A differ-ential r-form ω on M is said to be basic if for all vector fields V on M that aretangent to the leaves of F the following conditions hold:

(2.4.1) V ω = 0, £V ω = 0.

In a local foliated coordinate chart (U ;φ) with φ = (x1, . . . , xp; y1, . . . , yq) a basicr-form ω takes the form

(2.4.2) ω =∑

ωi1...ir (y1, . . . , yq)dyi1 ∧ · · · ∧ dyir ,

where the sum is taken over all repeated indices.Let Ωr

B(F) denote sheaf of germs of basic r-forms on M, and ΩrB(F) the set ofits global sections. The direct sum ΩB(F) = ⊕rΩrB(F) is closed under addition andexterior multiplication, and so is a subalgebra of the algebra of exterior differentialforms on M. Furthermore, exterior derivation takes basic forms to basic forms, i.e.,

(2.4.3) £V dω = d£V ω = 0 and V dω = £V ω − d(V ω) = 0.

Thus, the subalgebra ΩB(F) forms a subcomplex of the de Rham complex, andits cohomology ring H∗

B(F) = H∗B(F , dB) is called the basic cohomology ring of

F . Here dB denotes the restriction of the exterior derivative d to the basic formsΩB(F).

The groups HrB(F) are defined for all 0 ≤ r ≤ q, where q is the codimension

of the foliation, but generally they may be infinite dimensional for r ≥ 2. However,they are finite dimensional for Riemannian foliations on compact manifolds whichis mainly the case of interest to us. Let us look at the groups Hr

B(F) for r = 0, 1.The set Ω0

B(F) is just the set of functions that are constant along the leaves of F ,so the cocycles with respect to dB are just the constant functions. Thus, if M isconnected H0

B(F) ≈ R. Let us now consider H1B(F). We claim that H1

B(F) injectsinto H1(M,R) induced by the natural inclusion ΩB(F)−−→Ω(M). Let ω be a 1-cocycle, i.e., dBω = 0, where ω is a smooth section of Ω1

B(F). So if ω = df for somesmooth function on M, then we have V f = V df = V dω = 0 for any vector field


60 2. FOLIATIONS

V tangent to the leaves of F . But this says that f ∈ Ω1B(F), so ω is a coboundary

with respect to dB . This proves the claim. We have arrived atProposition 2.4.1: For any foliation F on a connected manifold M,

(i) H0B(F) ≈ H0(M,R) ≈ R.

(ii) H1B(F) is a subgroup of H1(M,R).

The appropriate notion of basic for vector fields isDefinition 2.4.2: A vector field X on a foliated manifold (M,F) is said to befoliate with respect to F if for every vector field V tangent to the leaves of F , theLie bracket [X,V ] is tangent to the leaves of F .

Let us set some notation. We denote by XF (M) the Lie subalgebra of X (M)consisting of those vector fields that are tangent to the leaves of the foliation F .These are just the smooth sections of the integrable vector bundle E defining F .The set of foliate vector fields also forms a Lie subalgebra of X (M). It is denotedby fol(M,F). From Definition 2.4.2 one sees that it is precisely the normalizer ofXF (M) in X (M). Equivalently, its local 1-parameter group preserves the foliation.We denote the subgroup of the diffeomorphism group Diff(M) that preserve thefoliation F by Fol(M,F), i.e.,(2.4.4) Fol(M,F) = φ ∈ Diff(M) | φ∗E ⊂ E.Exercise 2.3: Show that in local foliated coordinates (x1, . . . , xp; y1, . . . , yq) a foli-ate vector field X takes the form

(2.4.5) X =q∑

a=1

Aa(y1, . . . , yq)∂

∂ya+

p∑i=1

Bi(x1, . . . , xp; y1, . . . , yq)∂

∂xi,

where Aa and Bi are smooth functions of the indicated variables.

2.5. Transverse Geometry

We are interested in foliations with a “transverse structure.” These were introducedby Conlon [Con74] and further developed by Molino [Mol75, Mol88] in a slightlydifferent guise. Here we follow [Mol88], but our notation is somewhat different.Let F be a p-dimensional foliation on the manifold M with integrable subbundleE. There is an exact sequence of vector bundles, viz.

(2.5.1) 0−−−→E−−−→TMπQ−−−→Q−−−→0.

The quotient bundle Q is called the normal bundle of the foliation, and it is oftendenoted by ν(F) to emphasize the foliation. Notice that any foliate vector field Xprojects to a section X of ν(F) that is independent of the coordinates along theleaves. This follows easily from (2.4.5) together with a change in foliated coordinatecharts. The set of all such sections forms a Lie algebra, denoted by trans(M,F),and we have an exact sequence of Lie algebras

(2.5.2) 0−−−→XF (M)−−−→fol(M,F)−−−→trans(M,F)−−−→0.Here the Lie bracket on trans(M,F) is defined by [X, Y ] = [X,Y ].We call elementsof trans(M,F) transverse vector fields. Note also that the Lie derivative of basictensor fields with respect to transverse vector fields is well-defined [Mol88].


2.5. TRANSVERSE GEOMETRY 61

A transverse frame at the point x ∈ M is a basis (Y1, . . . , Yq) for the fiber Qx ofQ at x. (Yi is not a tangent vector at x, but an equivalence class of tangent vectorsmodulo vectors in Ex.) The set LT (M,F) of all transverse frames at all points ofM forms a principal GL(q,R) bundle on M, called the transverse frame bundle on(M,F). Let G ⊂ GL(q,R) be a Lie subgroup and let πT : PT (M,G,F)−−→M denotethe corresponding principal G subbundle of LT (M,F). As with the ordinary framebundle a point z ∈ LT (M,F) can be viewed as the linear map z : Rq−−→QπT (z)that assigns to the standard basis of Rq the frame z = (Y1, . . . , Yq). Also as in theusual case LT (M,F) and its subbundles PT (M,G,F) have a canonical Rq-valued1-form θT defined by

(2.5.3) 〈θT , X〉 = z−1πQπT∗X.

The foliation F on M lifts to a foliation FT on LT (M,F) as follows: we define afoliation on LT (M,F) by the p-dimensional subbundle ET ⊂ TLT (M,F) generatedby the vectors in TzLT (M,F) that satisfy(2.5.4) ETz = Xz ∈ TzLT (M,F) | Xz θT = Xz dθT = 0 for all z ∈ LT (M,F).We now haveProposition 2.5.1: Let (M,F) be a foliated manifold and let LT (M,F) be itstransverse frame bundle. The subbundle ET defined by Equation (2.5.4) is inte-grable, and thus defines a foliation FT on LT (M,F). The leaves of this foliationproject under πT to the leaves of the foliation F .

Proof. Let X,Y be vector fields on LT (M,F) that are sections of the sub-bundle ET . Then

0 = dθT (X,Y ) = XθT (Y )− Y θT (X)− θT ([X,Y ]) = −θT ([X,Y ]),

so [X,Y ] θT = 0. But also we have for any section X of ET that

£XθT = d(X θT ) +X dθT = 0.

Thus,

0 = [£X ,£Y ]θT = £[X,Y ]θT = d([X,Y ] θT ) + [X,Y ] dθT = [X,Y ] dθT

which implies that ET is integrable by Frobenius’ Theorem.Now suppose that LT is a leaf of FT . Let U be a simple open set of M such

that the quotient map of the foliation restricted to U is a submersion fU : U−−→Rq.Then on ET |U = π−1

T (U), the canonical 1-form θT is just the pullback f∗Uθ, whereθ is the canonical 1-form on the frame bundle L(Rq) on Rq. So in local coordinates(x1, . . . , xp; y1, . . . , yq;A1

1, . . . , Aq1, . . . , A

qq) on ET , Equation (2.5.4) implies that the

tangent vectors to ET that are tangent to the leaves of the foliation FT are spannedby the ∂

∂xjand ∂

∂Aij. Thus, these vectors project under πT to vectors that are tangent

to the leaves of the foliation F on M. But since πT is surjective this implies thatthe leaf LT of FT projects to a leaf of F .

The foliation FT is called the lifted foliation. We are now ready forDefinition 2.5.2: Let G be a Lie subgroup of GL(q,R), and let PT (M,G,F) bea principal G subbundle of LT (M,F). We say that PT (M,G,F) is a transverseG-structure if at each point z ∈ PT (M,G,F) the tangent space TzPT (M,G,F)contains the subspace ETz tangent to the leaves of the lifted foliation FT .


62 2. FOLIATIONS

Important examples for us will be when G = O(q,R), G = GL(m,C), andG = U(m), where q = 2m. These correspond to transverse Riemannian structures,transverse almost complex structures, and transverse almost Hermitian structures,respectively. The following result [Mol88] gives a characterization of transverseG-structures in terms of the local submersions.Proposition 2.5.3: Let PT (M,G,F) be a principal G subbundle of LT (M,F).Then PT (M,G,F) is a transverse G-structure if and only if for every simple openset U, the bundle PT (M,G,F)|U is the pullback by πT : LT (U,F|U )−−→L(Rq) of aG-structure P (Rq, G) on Rq.

Proof. The argument is purely local, and the details are left to the reader.

2.5.1. Transverse Riemannian Geometry. The foliations that interest usmost are those with a transverse Riemannian structure and actually more, a trans-verse Kahler structure. It turns out that allowing for a transverse Riemannianstructure adds quite a bit of tractability to the subject which was begun by Rein-hart [Rei59a], and subsequently developed by Molino and others. See [Mol88] andreferences therein.Definition 2.5.4: A Riemannian foliation is a foliation with a transverseRiemannian structure, i.e., a transverse G-structure with G = O(q,R).

A Riemannian foliation (M,F) defines a transverse Riemannian metric gT onM by

(2.5.5) gT (X,Y ) = 〈z−1πQX, z−1πQY 〉,whereX,Y ∈ TMπT (z) and πQ is the projection in the exact sequence (2.5.1). Noticethat gT defines a non-negative symmetric bilinear form on TM, and viewed as asection of Sym2TM has kernel E⊗TM +TM ⊗E, and thus defines a Riemannianmetric on the quotient bundle Q.

Proposition 2.5.5: Let F be a codimension q foliation on M. There is a one-to-one correspondence between transverse O(q,R)-structures on M and transverseRiemannian metrics on M.

The second definition is that of a bundle-like metric which is due to Reinhart[Rei59a].Definition 2.5.6: A Riemannian metric g is said to be bundle-like with respectto a foliation F if for any foliate horizontal vector fields X,Y , the function g(X,Y )is basic, i.e., for any vector field V along the leaves of F we have V g(X,Y ) = 0.

The following is well-known [Mol88]:Proposition 2.5.7: If g is a bundle-like metric with respect to a foliation F , thenF is a Riemannian foliation. Conversely, if F is a Riemannian foliation with trans-verse metric gT , then there exist bundle-like metrics g whose associated transversemetric is gT .

Proof. Let g be a bundle-like metric on M. This metric splits the exactsequence (2.5.1) as

(2.5.6) TM = E ⊕ E⊥,



where E⊥ denotes the subbundle of TM consisting of all vectors perpendicular tothe vectors of E with respect to the Riemannian metric g. Thus, any vector fieldX on M can be decomposed as

X = X +X⊥,

where X is a section of E and X⊥ is a section of E⊥. Define a transversemetric on (M,F) by gT (X,Y ) = g(X⊥, Y ⊥). Since g is bundle-like, this makesF a Riemannian foliation.

Conversely, given a Riemannian foliation, we have a transverse metric gT definedby Equation (2.5.5), and we can choose a Riemannian metric in the vector bundleE to obtain a bundle-like metric on M.

In terms of Haefliger structures of Section 2.2, the transverse metric gT can beobtained by pulling back Riemannian metrics on Rq by the local submersions fα. Inthis case the transition functions ταβ are local isometries between local Riemannianmetrics. In fact one can give an equivalent definition of a Riemannian foliation interms Haefliger cocycles:Proposition 2.5.8: Let (M,F) be a foliated manifold, and let τ be the associatedHaefliger cocyle. Then (M,F) is a Riemannian foliation if and only if there areRiemannian metrics gα on each fα(Uα) such that gα = τ∗βαgβ .

Proof. Given a foliation F whose associated Haefliger structure satisfies gα =τ∗βαgβ with Riemannian metrics gα on each open set fα(Uα) ⊂ Rq, we obtain a trans-verse Riemannian metric gT,α on each open set Uα and the cocycle conditions implythat these metrics agree on each overlap Uα∩Uβ giving a global transverse Riemann-ian metric gT . Hence, the foliation is Riemannian. Conversely, if F is Riemannianwe have a transverse Riemannian metric gT which extends to a bundle-like metric gon M. Since for any foliate vector fields X,Y the function g(X,Y ) is basic, we candefine Riemannian metrics gα on each fα(Uα) by gα(fα∗X, fα∗Y ) = g|Uα(X,Y ). Itis then easy to check that gα = τ∗βαgβ .

For a Riemannian foliation on a compact manifold we have [Rei83, Mol88]Proposition 2.5.9: Let (M,F) be a Riemannian foliation on a compact connectedmanifold M. Then the leaves of F all have the same universal cover.

In fact a common covering of the leaves appears naturally in the holonomygroupoid G(M,F) of a Riemannian foliation. Generally, for a Riemannian foliationG(M,F) has some nice special properties. We say that a Riemannian submersionis horizontally complete if any horizontal geodesic can be extended for all values ofits affine parameter.Theorem 2.5.10: Let F be a p-dimensional Riemannian foliation on the manifoldM. Then

(i) The holonomy groupoid G(M,F) of F is a Hausdorff Lie groupoid.(ii) There is a unique Riemannian metric gG on G(M,F) such that both s

and t are Riemannian submersions.(iii) The inverse map ι is an isometry with respect to gG .(iv) The 2p dimensional foliation FG on G(M,F) is a Riemannian foliation

such that each leaf is locally the Riemannian product of leaves of thep-dimensional foliations.


64 2. FOLIATIONS

(v) If the source submersion s : G(M,F)−−→M is horizontally complete, thenit is a locally trivial fiber bundle whose fibers are precisely the holonomycovers of the leaves of F .

(vi) If M is compact and all leaves are compact, then G(M,F) is compact.Proof. (i): The holonomy groupoid is a Lie groupoid by Proposition 2.3.6.

Since the transverse geometry of F is orthogonal, so are the leaf holonomy groups.Thus, the holonomy is linear, and hence, analytic. So the result follows by Pro-position 2.3.6. For the remainder see Proposition IV.4.23 of [Rei83].

Next we give a theorem of Molino [Mol88] that is of great importance to us inthe sequel. It says that a Riemannian foliation with compact leaves has a tractablespace of leaves, namely an orbifold which we discuss in detail in Chapter 4.Theorem 2.5.11: Let F be a p-dimensional Riemannian foliation with compactleaves on the manifold M. Then the space of leaves M/F admits the structure ofa p-dimensional orbifold such that the canonical projection π : M−−→M/F is anorbifold Riemannian submersion.

Proof. The proof can be found in [Mol88], Proposition 3.7.

Now we discuss the relation between the curvature of (M, g) and the curvatureof the transverse metric gT , where g is a bundle-like metric whose transverse com-ponent is gT . Accordingly we split TM as in Equation (2.5.6). This relationship isessentially that of O’Neill for Riemannian submersions [O’N66, Rei83, Ton97].We first define the induced connection ∇T in the bundle E⊥.

(2.5.7) ∇TXY =

πE⊥(∇XY ) if X is a smooth section of E⊥,πE⊥ [V, Y ] if X = V is a smooth section of E,

where Y is a smooth section of E. It is left as an exercise to mimic the standardproof of the first fundamental theorem of Riemannian geometry that ∇T is theunique torsion-free connection such that the transverse metric gT is parallel, i.e.,∇TXgT = 0.

Exercise 2.4: Show that the connection ∇T is the unique torsion-free metric con-nection with respect to the transverse metric gT .

We let RT denote the Riemannian curvature tensor field with respect to gT .Then if Z,W are any vector fields on M, and X is a vector field in E⊥, we have

(2.5.8) RT (Z,W )X = ∇TZ∇T

WX −∇TW∇T

ZX −∇T[Z,W ].

Proposition 2.5.12: For any smooth section V of E the following hold:(i) £V∇T = 0.(ii) V RT = 0.(iii) £V RT = 0.

Proof. (i) follows as usual from the fact that ∇T is determined uniquely interm the transverse metric gT and £V gT = 0. (ii): If W is a vector field on M andX is a smooth section of E⊥ we have

RT (V,W )X = ∇TV∇T

WX −∇TW∇T

V X −∇T[V,W ]X

= £V∇TWX −∇T

W£V X −∇T£VWX = (£V∇T )W (X) = 0.

We leave the proof of (iii) as an exercise.



Exercise 2.5: Prove (iii) of Proposition 2.5.12.Let us fix some common notation [Bes87, Ton97]: we let Ei denote any vector

fields onM, while we let U, V,W denote vector fields in the integrable subbundle E,and X,Y, Z be vector fields in E⊥. We also let π : TM−−→E, and π⊥ : TM−−→E⊥

denote the corresponding natural projections. Following O’Neill [O’N83] (see alsoGray [Gra67]) we introduce the type (2, 1) tensor field

(2.5.9) TE1E2 = π(∇π(E1)π⊥(E2)) + π⊥(∇π(E1)π(E2)).

Exercise 2.6: Show that T defined by Equation (2.5.9) is a tensor field of type(2, 1).

Clearly, T only depends of projection of E1 along the leaves, so

(2.5.10) TXU = TXY = 0.

We also have

TUV = π⊥(∇UV ), TUX = π(∇UX),(2.5.11)

TUV = TV U,(2.5.12)

T is anti-symmetric, i.e., g(TUV,X) = −g(V, TUX).(2.5.13)

The symmetry property of Equation (2.5.12) follows by taking U and V ascoordinate vector fields along the leaves of the foliation F (valid since T is atensor field), and using the fact that ∇ is torsion-free, while the anti-symmetryproperty of Equation (2.5.13) follows since ∇ is a metric connection. Notice that byEquation (2.5.13) T vanishes if and only if TUV = 0 for all vector fields tangent tothe leaves of F . Since any leaf L is an immersed submanifold of M, we see fromEquation (2.5.11) that L is totally geodesic if and only if T vanishes along L. IfTUV = 0 for all U and V then all leaves of F are totally geodesic. In this case wesay that the foliation F is a totally geodesic foliation. Summarizing we haveProposition 2.5.13: A foliation F is totally geodesic if and only if T = 0.

Reversing the roles of the subbundles E and E⊥ in Equation (2.5.9) we obtain

(2.5.14) AE1E2 = π(∇π⊥(E1)π⊥(E2)) + π⊥(∇π⊥(E1)π(E2)).

Exercise 2.7: Show thatA defined by Equation (2.5.14) is a tensor field of type (2, 1).For O’Neill’s A tensor field, we have the following

AUV = 0, AUX = 0,(2.5.15)

AXU = π⊥(∇XU), AXY = π(∇XY ),(2.5.16)

AXY = −AY X,(2.5.17)

A is anti-symmetric, i.e., g(AXU, Y ) = −g(U,AXY ).(2.5.18)

Equations (2.5.15) and (2.5.16) are obvious from the definition. Equation (2.5.17)will be proved in Proposition 2.5.14 below, while again Equation (2.5.18) followssince ∇g = 0.Proposition 2.5.14: For any horizontal vector fields X,Y we have

AXY =12π([X,Y ]).

Proof. We compute

π([X,Y ]) = π(∇XY −∇Y X) = AXY −AY X.


66 2. FOLIATIONS

The result will now follow if we establish Equation (2.5.17). It suffices to show thatAXX = 0. Without loss of generality we may take the horizontal vector field X tobe foliate. We have for arbitrary vertical U

g(U,AXX) = g(U, π(∇XX)) = g(U,∇XX) = −g(∇XU,X)

= −g(∇UX,X)− g([X,U ], X) = −12Ug(X,X) = 0.

The last expression vanishes since g(X,X) is basic when X is foliate.

Notice that Equation (2.5.18) implies that A vanishes precisely when AXYvanishes for all horizontal vector fields X and Y, and Proposition 2.5.14 says thatAXY vanishes for all X,Y if and only if the subbundle E⊥ is integrable. Next letus see how the covariant derivative on M decomposes. We let ∇ be the inducedconnection along the leaves. Then we have Gauss’ formula for the leaves

(2.5.19) ∇UV = ∇UV + TUV.

We also have

∇UX = TUX + π⊥(∇UX),(2.5.20)

∇XU = π(∇XU) +AXU,(2.5.21)

∇XY = AXY +∇TXY.(2.5.22)

Concerning the covariant derivatives of A and T we have the following:Lemma 2.5.15: The following hold:

(i) (∇UA)V = −ATUV , (∇XT )Y = −TAXY ,

(ii) (∇XA)V = −AAXV , (∇V T )Y = −TTV Y ,

(iii) g((∇UA)XV,W ) = g(TUV,AXW )− g(TUW,AXV ),(iv) g((∇E1A)XY, V ) is anti-symmetric in X,Y,

(v) g((∇E1T )UV,X) is anti-symmetric in U, V,

(vi) Sg((∇ZA)XY, V ) = Sg(AXY, TV Z), where S denotes the cyclic sumtaken over X,Y, Z.

Proof. We establish the first identity of (ii), and leave the rest as an exercise.For any vector field F we have

(∇XA)V F = ∇X(AV F )−A∇XV F −AV (∇XF ) = −AAXV F.

Exercise 2.8: Finish the proof of Lemma 2.5.15.

Theorem 2.5.16: Let R, R, and RT denote the Riemann curvature associatedto the connections ∇, ∇, and ∇T , respectively. Then the following curvature



identities hold:

g(R(U, V )W,W ′) = g(R(U, V )W,W ′) + g(TUW,TV W ′)− g(TV W,TUW ′);

g(R(U, V )W,X) = g((∇UTV )W,X)− g((∇V TU )W,X);

g(R(X,U)Y, V ) = g(TUX,TV Y )− g((∇XTU )V, Y )

− g(AXU,AY V )− g((∇UA)XY, V );

g(R(U, V )X,Y ) = g((∇V A)XY,U)− g((∇UA)XY, V )

+ g(AXV,AY U)− g(AXU,AY V )

+ g(TUX,TV Y )− g(TV X,TUY );

g(R(X,Y )Z,U) = g(AY Z, TUX) + g(AZX,TUY )

+ g((∇ZA)XY,U)− g(AXY, TUZ);

g(R(X,Y )Z,Z ′) = g(RT (X,Y )Z,Z ′) + 2g(AXY,AZZ ′)

+ g(AXZ,AY Z ′)− g(AXZ ′, AY Z).

Proof. The first equation is just Gauss’ Equation along the leaves viewed asimmersed submanifolds, while the second equation is the Codazzi equation alongthe leaves. We leave the proof for the other equations to the reader.

We now have an easy corollary for the various sectional curvatures.Corollary 2.5.17: Let X,Y, U, V be linearly independent unit vector fields withrespect to the metric g, and let K,KT , K denote the sectional curvatures with respectto the metrics g, gT , g|L, respectively. Then one has

K(U, V ) = K(U, V ) + |TUV |2 − g(TUU, TV V ),

K(X,U) = g((∇XT )UU,X)− |TUX|2 + |AXU |2,K(X,Y ) = KT (X,Y )− 3|AXY |2.

We would like expressions for the Ricci and scalar curvatures similar to thoseof Theorem 2.5.16. First we need some preliminary development which we lift fromBesse [Bes87]. In what follows we let Xii and Uaa denote local orthonormalframes for the horizontal subspace E⊥ and vertical subspace E, respectively. Thefollowing identities hold:

g(AX , AY ) =∑i

g(AXXi, AY Xi) =∑a

g(AXUa, AY Ua),(2.5.23)

g(AX , TU ) =∑i

g(AXXi, TUXi) =∑a

g(AXUa, TUUa),(2.5.24)

g(AU,AV ) =∑i

g(AXiU,AXi

V ),(2.5.25)

g(TX, TY ) =∑a

g(TUaX,TUaY ).(2.5.26)

Next for any tensor field T we define the projected divergence operators,

δTT = −∑i

(∇XiT )Xi ,(2.5.27)

δT = −∑a

(∇UaT )Ua ,(2.5.28)


68 2. FOLIATIONS

which satisfy δ = δ + δT with respect to the ordinary divergence operator δ. Wedefine the mean curvature vector field N along the leaves of F by

(2.5.29) N =∑a

TUaUa

which vanishes if and only if each leaf is a minimal submanifold of M. We are nowready for the relations between the Ricci curvatures. We let Ric,RicT , and Ricdenote the Ricci curvatures of g, gT , and the family of metrics g|L, respectively.Theorem 2.5.18: The Ricci curvatures Ric,RicT , and Ric satisfy

Ric(U, V ) = Ric(U, V )− g(N,TUV ) + g(AU,AV ) +∑i

g((∇XiT )UV,Xi),

Ric(X,U) = g((δT )U,X) + g(∇UN,X)− g((δTA)X,U)− 2g(AX , TU ),

Ric(X,Y ) = RicT (X,Y )− 2g(AX , AY )− g(TX, TY ) +12(g(∇XN,Y ) + g(∇Y N,X)).

Defining the quantities

|A|2 =∑i

g(AXi, AXi

) =∑a

g(AUa, AUa),(2.5.30)

|T |2 =∑i

g(TXi, TXi) =∑a

g(TUa , TUa),(2.5.31)

we can give the relation between the various scalar curvatures, viz.Corollary 2.5.19: If s, sT , s denote the scalar curvatures of the metrics g, gT , g|L,respectively, then

s = sT + s− |A|2 − |T |2 − |N |2 − 2δTN.

2.5.2. Transverse Complex Geometry. Here we describe only the basicsof transverse complex and almost complex geometry. We present much more detailin the special case of interest to us, namely transverse almost Kahler and Kahlergeometry, in Sections 7.2 and 7.5 below. For related work on transverse Kahlerfoliations see [NT88]. Let (M,F) be a foliated manifold, thenDefinition 2.5.20: A transverse almost complex structure is a smooth sec-tion J of End ν(F) that satisfies J2 = −1lν(F).

This implies that the normal bundle ν(F) is even dimensional, say 2q. Thenalternatively, a transverse almost complex structure on a foliated manifold (M,F)is a transverse G-structure with G = GL(q,C). We are particularly interested inthe case when the transverse almost complex structure is integrable. This is donein terms of a Nijenhuis torsion tensor in the case of almost contact structures inSection 6.5.4 of Chapter 6. Here it seems both more conceptual and more convenientto describe the integrable case in terms of Haefliger structures. It is more convenientfrom the point of view of deformation theory adopted in Section 8.2. Let (M,F) bea foliated manifold of codimension 2q with local submersions fα : Uα−−→R2q = Cq

related by Equation (2.2.1) with transition functions τβα that satisfy Haefliger’scocycle conditions (2.2.2). ThenDefinition 2.5.21: The foliated manifold (M,F) is said to have a transverseholomorphic structure if the local diffeomorphisms τβα : fα(Uα∩Uβ)−−→fβ(Uα∩Uβ) are biholomorphisms for all α, β.


2.6. RIEMANNIAN FLOWS 69

It is easy to see that a foliated manifold (M,F) with a transverse holomorphicstructure defines a transverse almost complex structure J on the normal bundleν(F). We can then decompose the complexification ν(F)C = ν(F) ⊗ C into theeigenspaces of J ,

(2.5.32) ν(F)C = ν(F)1,0 ⊕ ν(F)0,1.Similarly, we get a splitting of the co-normal bundle

(2.5.33) ν∗(F)C = ν∗(F)1,0 ⊕ ν∗(F)0,1,and this induces a splitting of the exterior differential algebra over ν∗(F)C

Λrν∗(F)C =⊕p+q=r

Λp,qν∗(F)C,

where Λp,qν∗(F)C = Λpν∗(F)1,0 ⊗ Λqν∗(F)0,1. We let Ωp,qB denote the set of basicforms of type (p, q), that is the set of smooth basic sections of Λp,qν∗(F)C. As inthe usual case there is a splitting

(2.5.34) ΩrB(F)⊗ C =⊕p+q=r

Ωp,qB ,

as well as the basic Dolbeault complex

(2.5.35) 0−−→Ωp,0B

∂−−→Ωp,1B

∂−−→ · · ·−−→Ωp,nB −−→0,together with its basic Dolbeault cohomology groups Hp,q

B (Fξ). This will be devel-oped much further in the context of K-contact and Sasakian geometry inSection 7.2.

We end this section with a discussion of transformation groups that preserve thetransverse holomorphic structure. Notice that any φ ∈ Fol(M,F) induces a map φ∗ :ν(F)→ ν(F). So we define the group of transversely holomorphic transformationsHT (F , J) by

(2.5.36) HT (F , J) = φ ∈ Fol(M,Fξ) | φ∗ J = J φ∗.Since a 1-parameter subgroup of any element of XF (M) induces the identity onν(Fξ), the group HT (F , J) is infinite dimensional. We are mainly interested in theinfinitesimal version, that is in the “Lie algebra” of HT (F , J).Definition 2.5.22: Let (M,F) be a foliated manifold with a transverse holomor-phic structure J . We say that a vector field X ∈ fol(M,F) is transversally holo-morphic if given any section Y of ν(F), we have

[X, J Y ] = J [X,Y ] .

The set of all such vector fields will be denoted by hT (F , J).Exercise 2.9: Show that hT (F , J) is a Lie subalgebra of trans(M,F).

2.6. Riemannian Flows

We shall be interested in 1-dimensional Riemannian foliations.Definition 2.6.1: A 1-dimensional oriented Riemannian foliation is called aRiemannian flow.


70 2. FOLIATIONS

Before we begin our study of Riemannian flows, we briefly mention1-dimensional foliations that are never Riemannian, the so-called Anosov flows[Ano69]. Any oriented 1-dimensional foliation on a smooth manifold M is deter-mined by a nowhere vanishing vector field V. The flow φt determined by V is saidto be Anosov if the tangent bundle TM splits as TM = LV ⊕ Es ⊕ Eu, where LVis the trivial line bundle generated by V and Es(Eu) denote the stable (unstable)subbundles defined as follows: there exist constants c ≥ 1 and a > 0 such that forall t ≥ 0 the estimates hold:

(i) |(φt)∗X| ≤ ce−at|X| for every smooth section X of Es,(ii) |(φt)∗X| ≥ c−1eat|X| for every smooth section X of Eu.

Here the norm | · | is taken with respect to some Riemannian metric on M, butwhen M is compact the condition of being Anosov is actually independent of themetric. An important example of an Anosov flow is the geodesic flow on the unittangent bundle over a compact manifold of constant negative curvature [Ano69].See also [Ton97].

For a given Riemannian foliation (M,F) one is interested in whether the leavesare closed in the topology of M. This is of particular interest to us in the case ofRiemannian flows. In order to study this in more detail we consider certain typesof “singular foliations.” Here we follow Molino [Mol88].Definition 2.6.2: Let M be a smooth manifold. A singular Riemannianfoliation SF is a partition of M into connected immersed submanifolds, calledleaves that satisfy:

(i) The Lie subalgebra of vector fields tangent to the leaves of SF is transitiveon each leaf.

(ii) There exists a Riemannian metric g on M such that every geodesic thatis perpendicular at one point to a leaf remains perpendicular to every leafit meets.

It is not difficult to see that any ordinary Riemannian foliation satisfies theseconditions, and thus is a special case of a singular Riemannian foliation. Further-more, Molino shows a converse in the following sense. If F is a foliation on M (inthe usual sense) and the conditions of Definition 2.6.2 hold, then F is a Riemannianfoliation with respect to g. For us the importance of singular Riemannian foliationsis given byProposition 2.6.3: Let (M,F) be a Riemannian foliation on a compact connectedmanifold M. Then the closures F of the leaves of F form a singular Riemannianfoliation for which the leaves are embedded submanifolds of M.

Proof. See [Mol88] Proposition 6.2 and Theorem 5.1.

We are interested in the case when the Riemannian flow is an isometry, andthen many simplifications occur. Indeed the following theorem of Carriere [Car84b](see also Appendix A of [Mol88] by Carriere) becomes much easier to prove whenthe flow is an isometry.Theorem 2.6.4: Let F be Riemannian flow on a compact manifold M. Then theleaf closure L of a leaf L of F is diffeomorphic to a torus T k, and F restricted toL is conjugate to a linear flow.



Proof. Here we give the proof assuming that the flow is an isometry, but thetheorem holds more generally. See [Car84b] for the proof of the general case. Let gbe a bundle-like Riemannian metric for (M,F) and let φ denote the one parametersubgroup of the isometry group lsom(M, g) generated by F . Since M is compact,so is lsom(M, g), and the closure φ is a torus T ⊂ lsom(M, g). For any x ∈ M theclosure φx of the orbit φx coincides with the orbit φx of the closure φ. So if φxdenotes the isotropy subgroup at x, we see that φ/φx is a torus T k. Hence, the leafclosure L is a torus. Furthermore, φ is a subgroup of the torus, so it is conjugateto a linear flow.

Note that the dimension k = k(L) of the torus in general depends on the leaf Land is clearly bounded by the dimension of the manifold. This leads toDefinition 2.6.5: The toral rank or just rank rk(M,F) of a Riemannian flowF on a compact manifold M is defined by

rk(M,F) = maxL∈M

k(L).It is clear that 1 ≤ rk(M,F) ≤ dim (M). The rk(M,F) = 1 case will be of

great interest to us. In this case the leaves are all circles, and the space of leaveswill be a compact orbifold. See Chapter 4 for details.Definition 2.6.6: A 1-dimensional foliation F on M is said to be isometric ifthere is a Riemannian metric g on M such that the flow is an isometry of g.

We haveProposition 2.6.7: An isometric 1-dimensional foliation is Riemannian.

Proof. Let V be a nowhere vanishing vector field on M generating a1-dimensional foliation F that is isometric with respect to a Riemannian metricg, and let X,Y be any horizontal foliate vector fields on M. Then

(2.6.1) V g(X,Y ) = (£V g)(X,Y ) + g([V,X], Y ) + g(X, [V, Y ]).

The first term vanishes since V is an infinitesimal isometry (Killing vector field),and the last two terms vanish since X,Y are both horizontal and foliate. Thus, gis a bundle-like metric for (M,F). So F is Riemannian by Proposition 2.5.7.

The converse is not true in general; however, see Proposition 2.6.11 below.Carriere has related the isometry of a 1-dimensional foliation to the orbits beinggeodesics. The following result is well-known [Ton97].Lemma 2.6.8: Let F be a 1-dimensional foliation on a manifold M, and let V bea nowhere vanishing vector field generating F . Then the following are equivalent:

(i) There exists a Riemannian metric g on M such that V has unit lengthand the orbits of V are geodesics.

(ii) There exists a 1-form η on M such that η(V ) = 1 and £V η = 0.(iii) There exists a 1-form η on M such that η(V ) = 1 and V dη = 0.

Proof. Choose a metric g on M such that V has unit length and define ηto be the dual of V , i.e., η = g(V, ·). Then the equivalence of (ii) and (iii) followsimmediately from the well-known formula £X = ιX d+ d ιX . Now let ∇ denoteany affine connection on M. Then, the orbits of V are geodesics with respect tothe connection ∇ if and only if the mean curvature vector field (∇V V )⊥ vanishes,where ⊥ indicates the component perpendicular to V. Equivalently, this means that


72 2. FOLIATIONS

the dual mean curvature 1-form κ = g((∇V V )⊥, ·) vanishes. But for any vector fieldX perpendicular to V , we have

κ(X) = g(∇V V,X) = V g(V,X)−∇V g(V,X)− g(V,∇V X)

= (£V η)(X) + η([V,X])−∇V g(V,X)− η(∇V X)

= (£V η)(X) + η(∇V X)− η(∇XV ) + η(τ(V,X))−∇V g(V,X)− η(∇V X)

= (£V η)(X) +12(∇Xg)(V, V )− (∇V g)(V,X) + η(τ(V,X)),

where τ denotes the torsion tensor of ∇. In particular, choosing ∇ to be the Levi-Civita connection proves the result.

Now we have the following result due to Carriere.Proposition 2.6.9: A Riemannian flow whose orbits are geodesics is isometric.Conversely, the orbits of a 1-dimensional isometric foliation are geodesics.

Proof. Let F be a Riemannian flow generated by the nowhere vanishing vectorfield V whose orbits are geodesics, and let g be a bundle-like metric such that Vhas unit length. Consider Equation (2.6.1) for horizontal vector fields X,Y. Sincethis equation is tensorial we can take X,Y to be foliate as well. So the last twoterms on the right hand side vanish as does the left hand side since g is bundle-like.Thus, (£V g)(X,Y ) = 0 for X,Y horizontal. On the other hand if η is the 1-formdual to V , then by (iii) of Lemma 2.6.8 we have for X horizontal

0 = dη(V,X) = V η(X)−Xη(V )− η([V,X]) = −η([V,X]),

which implies that (£V g)(V,X) = 0. Also (£V g)(V, V ) = 0 since V has constantlength. Thus, V is a Killing vector field and the flow is isometric.

For the converse let F be a 1-dimensional isometric foliation. Then F is gen-erated by a nowhere vanishing vector field V, and there is Riemannian metric gon M such that V is a Killing vector field. Let η be the 1-form dual to V withrespect to g. Then condition (ii) of Lemma 2.6.8 is satisfied. Thus, the orbits of Vare geodesics.

The following is an immediate consequence of the proof of Theorem 2.6.4.Proposition 2.6.10: Let (M,F) be an isometric Riemannian flow with toral rankrk(M,F) = k. Then the isometry group lsom(M, g) contains a torus subgroup Tk ofdimension k. Moreover, the closure of any leaf is a subtorus Tj ⊂ Tk of dimension1 ≤ j ≤ k.

Molino (See [Mol88] and references therein) has studied the leaf closures forgeneral Riemannian foliations F and has shown the existence of a sheaf of localtransverse Killing vector fields C(M,F) that he calls the commuting sheaf of thefoliation. Proposition 2.6.10 says that for isometric Riemannian flows the sheafC(M,F) admits precisely k global sections. In fact on a compact manifold M aresult of Molino and Sergiescu [MS85] says that the commuting sheaf admits aglobal trivialization if and only if F is isometric. Furthermore, Proposition 5.5of [Mol88] says that for a Riemannian foliation on a compact simply connectedmanifold, C(M,F) admits a global trivialization. Thus,Proposition 2.6.11: On a simply connected compact manifold every Riemannianflow is isometric.



We are now ready for a Theorem of Wadsley [Wad75] that will be of importanceto us in the sequel.Theorem 2.6.12: Let F be a 1-dimensional foliation on M such that all the leavesof F are circles. Then there is a smooth action A : S1 ×M−−→M whose orbits areprecisely the leaves of F if and only if there exists a Riemannian metric on M suchthat the leaves of F are geodesics.

Next we discuss an obstruction to the existence of Riemannian flows noticed byCarriere. This comes from a seminal paper of Gromov [Gro81] in which he definestwo important invariants of smooth manifolds. The first, called the minimal volumeis defined by

(2.6.2) MinVol(M) = inf|K(g)|≤1

Vol(M, g),

where Vol(M, g) denotes the volume ofM with respect to the Riemannian metric g,and the infimum is taken over all complete metrics whose sectional curvatures K(g)are bounded in absolute value by 1. One easily sees that, for example, for compactoriented 2-manifolds M2 one has MinVol(M2) = 2π|χ(M2)|, where χ denotes theEuler characteristic. On the other hand Gromov noticed that if a smooth manifoldM admits a locally free S1 action, then by scaling the metrics along the orbits ofthe S1, one sees that MinVol(M) = 0, and Carriere [Car84b] noticed that thisargument generalizes to arbitrary Riemannian flows.

The second invariant of Gromov is intuitively less geometrical, but computa-tionally more useful. It is the so-called simplicial volume of M defined3 as follows:consider the singular chain complex C∗ of all finite combinations c =

∑i riσi of

singular simplices σi in M with real coefficients ri. Define the simplicial norm ofc by ||c|| =∑i |ri|. This gives a pseudo-norm on the singular homology H∗(M,R)by ||[z]|| = infz ||[z]||, where the infimum is taken over all singular homology cyclesrepresenting [z]. If M is an orientable manifold, its simplicial volume ||M || (alsoknown as the Gromov invariant of M) is defined to be the simplicial norm of thefundamental class ofM. IfM is not orientable, then ||M || = 1

2 ||M ||, where M is the2-fold cover of M. The simplicial volume enjoys the following important properties:

(i) If f : M−→M ′ is a proper map of degree d, then ||M || ≥ d||M ′||, and iff is a d-sheeted covering map, ||M || = d||M ′||.

(ii) If M1 is compact and M2 is arbitrary, then there is a positive constantC depending only on the dimension of M1 ×M2 such that

C||M1|| · ||M2|| ≥ ||M1 ×M2|| ≥ C−1||M1|| · ||M2||.(iii) If dim(M1) = dim(M2) ≥ 3, then ||M1#M2|| = ||M1||+ ||M2||.

Furthermore, Gromov proves the following estimate for the simplicial and minimalvolumes

(2.6.3) ||M || ≤ (n− 1)nn!MinVol(M).

We mention that if M is closed (compact without boundary), then ||M || is com-pletely determined by the fundamental group. In particular, compact simply con-nected manifolds have vanishing Gromov invariant. (More generally ||M || vanishesfor any compact manifold whose fundamental group is amenable.)

Summarizing we have the following theorem of Carriere.

3Here M can be any topological space.


74 2. FOLIATIONS

Theorem 2.6.13: If a compact manifold M admits a Riemannian flow, thenMinVol(M) = 0. In particular, ||M || = 0 and all the Pontrjagin numbers of Mvanish.

We are interested in seeing which manifolds M have non-vanishing Gromovinvariant ||M ||. In [Gro81] Gromov proves a theorem which he attributes toThurston which says that a complete Riemannian manifold of finite volume whosesectional curvatures are pinched between −k and −1 for some k > 1 has non-vanishing Gromov invariant ||M ||. At about the same time Inoue and Yano [IY82]proved that any compact Riemannian manifold whose sectional curvatures isbounded from above by −δ for δ > 0 has non-vanishing Gromov invariant. Westate the precise results as:Theorem 2.6.14: Let (M, g) be a n-dimensional Riemannian manifold.

(i) Suppose that (M, g) is complete with finite volume 0 < Vol(M, g) < ∞,and that the sectional curvatures of g are pinched between −k and −1 forsome real number k > 1. Then the estimate

Vol(M, g) ≤ Cn||M ||holds, where Cn is a positive constant depending only on n.

(ii) Let (M, g) be compact without boundary whose sectional curvatures arebounded from above by −δ for some δ > 0. Then there exists a positiveconstant Cn such that

||M || ≥ Cnδn/2Vol(M, g).

Hence, in both cases ||M || = 0.It follows immediately that no manifold satisfying either of the two conditions

of this theorem can admit a Riemannian flow.

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