CHAPTER 3

Foliations and connections

We now recall the definition of a and discuss some fundamen- tal examples. We then use affine connections to investigate , par- ticularly Lagrangian foliations on symplectic manifolds.

1. Foliations and flat bundles There are many equivalent ways to define foliations. One of them is the following.

DEFINITION 3.1. A k-dimensional foliation of an n-manifold M is a partition of M into connected injectively immersedF k-manifolds, called the leaves of , which is locally trivial in the following sense: around every point in MFthere is a chart (U, j) such that j sends the connected compo- nents of the intersections of the leaves of with U to of the form Rk c in Rk Rn k = Rn. F ⇥{ } ⇥ The difference n k is called the of . The special charts appearing in the definition are called foliation charts.F Another way to phrase the definition is through the existence of an k k n k k whose transition maps send R c R R to some R d(c) for every c. Note that within a⇥ foliation{ }⇢ chart⇥ there is always a⇥ comple-{ } n k k n k mentary foliation with leaves corresponding to e R R R . However, this is only defined locally. The transition{ }⇥ maps of⇢ an atlas⇥ of fo- k n k liation charts for preserve slices parallel to the first factor in R R , but not usually theF slices parallel to the second factor. ⇥ We have avoided the use of the word in the definition of a foliation because the leaves are not embedded, they are only injectively im- mersed. Each leaf has an intrinsic and manifold structure which is induced by the restrictions of foliation charts, but this intrinsic topology is not the of the leaf considered as a of M.

EXAMPLE 3.2. A flow of irrational slope on the square 2- defines a 1-dimensional foliation, in which all leaves are densely injectively im- mersed – but not embedded – copies of R. The case of surfaces is very special, because the only non-trivial folia- tions are those of 1. On a every 1-dimensional foliation has a complementary 1-dimensional foliation, and every foliation is La- grangian with respect to the symplectic structure given by an area form.

35 36 3. FOLIATIONS AND CONNECTIONS

Nevertheless, the dynamics of the foliation can be very complicated, and there can be a mix of closed and non-closed orbits, Reeb components, etc. For a foliation on M we denote by T the subbundle of TM consist- ing of vectors tangentF to the leaves. This subbundleF has the property that its of sections is closed under taking commutators of vector fields, since the commutator of vector fields to the leaves is again tangent to the leaves. The Frobenius theorem gives the converse to this statement: an arbitrary subbundle E TM is integrable, i.e. E = T for some folia- tion on M, if and only if⇢ the space of sections of E is closedF under taking commutators.F This condition of being closed under brackets can be inter- preted as the vanishing of curvature for E. The intrinsic curvature of E is the map

c : L2(E) TM/E ! X Y [X, Y] . ^ 7! This map vanishes identically if and only if the sections of E are closed under brackets. The following example provides another connection between flatness and integrability.

EXAMPLE 3.3. Let p : V B be a vector equipped with a con- nection . The tangent vectors! to curves in V with re- spect to r define a vector subbundle H TV which is complementary to the tangentr bundle along the fibers of p,⇢ denoted Tp. We call Tp, respec- tively H, the vertical, respectively the horizontal, subbundle of TV. The vertical subbundle is always integrable, and the corresponding foliation of V is the foliation by the fibers. It is a standard theorem in differential ge- ometry that the horizontal subbundle H is integrable if and only if is flat. In this case the pair (V, ) is called a flat (or foliated) bundle. Ther foliation tangent to H is called ther horizontal foliation of the flat bundle. Its leaves are covering spaces of B. Choosing a -point p B and lifting curves from B to horizontal curves in V, parallel transport2 with respect to defines a representation r hol : p1(B, p) GLk(R) , r ! where k is the rank of V. This is called the representation of the flat connection . Conversely,r given any representation r : p (B, p) GL (R) 1 ! k one can define V =(B Rk)/ , the quotient by the diagonal action, ⇥ p1(B,p) where the fundamental group acts on the universal covering B by deck transformations and actse on Rk via r. Then V is a rank k with a horizontal foliation whose leaves are the images of B xe in V. ⇥{ } e 1. FOLIATIONS AND FLAT BUNDLES 37

In this way one obtains a bijective correspondence between conjugacy classes of (holonomy) representations of p1(B, p) and suitable isomorphism classes of flat bundles.

EXAMPLE 3.4. In the previous example one can replace the vector bun- dle V with structure group GLk(R) by an arbitrary smooth fiber bundle over B with fiber F and structure group the group Diff(F). Then isomorphism classes of foliated F-bundles over B correspond to con- jugacy classes of representations r : p (B, p) Diff(F) , 1 ! although there is no linear connection defining the horizontal foliation. Now let L be a leaf of a k-dimensional foliation on Mn. If there exists a tubular neighbourhood of L which is saturated, meaningF that it is a union of leaves of , then the diffeomorphism between the tubular neighbour- hood and theF total space of the of L in M equips the normal bundle with the structure of a foliated bundle. However, in general this is n k a foliated bundle with structure group Diff(R ), rather than GLn k(R). In the special situation of Example 3.3, where a horizontal foliation on a vector bundle V is defined by a flat linear connection on p : V B, the normal bundle of this foliation can be identified with the pullback! bundle p⇤(V), which carries the flat pullback connection. This idea that the normal bundle to the leaf of a foliation should be foli- ated, can be abstracted into the notion of a so-called Bott connection on the normal bundle of a foliation. For this one linearizes the discussion to obtain a linear connection defined by the infinitesimal holonomy of an arbitrary foliation. This will be defined throughout, without any assumptions about the existence of saturated tubular neighbourhoods. For technical reasons we will give a formulation which, instead of the normal bundle, uses the annihilator of T . This will be useful when we consider Lagrangian foliations on symplecticF manifolds.

DEFINITION 3.5. For a foliation on M define the annihilator as F A = a T⇤ M a T = 0 . { 2 | | F } This is a vector subbundle of T⇤ M whose rank equals the codimension of . The Bott connection on A defined by is the map F F : G(T ) G(A) G(A) r F ⇥ ! (X, a) L a . 7! X The defining term LXa = iXda by the Cartan formula, since by defini- tion a annihilates all X T . The assumption that T is integrable, i.e. is closed under brackets, ensures2 F that for all Y T weF have 2 F (i da)(Y)= a([X, Y]) = 0, X so that L a = i da indeed takes values only in the annihilator A of T . X X F 38 3. FOLIATIONS AND CONNECTIONS

The Bott connection is clearly bilinear over R, and it satisfies ( f a)=L ( f ) a + f L a = L ( f ) a + f a rX X · · X X · ·rX for all smooth functions f . This analog of the Leibniz rule justifies calling it a connection, although it is not a connection on the vector bundle A M ! in the usual sense. The reason is that the covariant X is only defined if X is tangent to the foliation, so that is only a partial connectionr on A. However, it becomes an honest connectionr whenever A is restricted to a leaf L of . The name Bott connection is used both for the partial connection definedF above, and for its restrictions to the leaves. The following lemma shows that the Bott connection captures the intu- itive flatness of the normal bundle.

LEMMA 3.6. For every leaf L of the restriction of the Bott connection to A L is flat. F ! PROOF. Let X, Y (L)=G(T L). The curvature of evaluated on X and Y acts on sections2X of A as follows:F| r

Fr(X, Y)a = a a a rXrY rYrX r[X,Y] = L L a L L a L a = 0 X Y Y X [X,Y] by the definition of and the definition of the commutator [X, Y]. r ⇤ 2. Lagrangian foliations We now consider foliations on a symplectic manifold (M, w).

DEFINITION 3.7. A foliation on M is called Lagrangian, if T is a Lagrangian subbundle of the symplecticF vector bundle (TM, w). F If one thinks of the leaves of a foliation as (which they are not, in the strictest sense), then a Lagrangian foliation is a foliation by Lagrangian submanifolds. In any case, if dim(M)=2n, the leaves of a Lagrangian foliation are of dimension n. Recall from Lemma 2.2 that the symplectic form w determines an iso- morphism f : TM T M via the contraction f(X)=i w. ! ⇤ X LEMMA 3.8. For a Lagrangian foliation , the contraction f restricts to an isomorphism between T and the annihilatorF A. F PROOF. Since T is Lagrangian, f(X) vanishes on T for all X T , and so f(T ) AF. However, T and A have the sameF rank n, and2 fFis injective, soFf maps⇢ T isomorphicallyF onto A. F ⇤ Under this isomorphism, the Bott connection on A corresponds to a partial connection on T , again defined only along the leaves of . Explic- itly, for X, Y G(T ) itF is given by F 2 F 1 1 1 Y = f (L f(Y)) = f (L i w)=f (i di w) , rX X X Y X Y 2. LAGRANGIAN FOLIATIONS 39 where the last equality is from the Cartan formula and the assumption that T is Lagrangian. Since w is non-degenerate, this formula is equivalent to theF following identity for all Z (M): 2X w( Y, Z)=(i di w)(Z)=(di w)(X, Z) rX X Y Y (3.1) = L (w(Y, Z)) L (w(Y, X)) w(Y, [X, Z]) X X = LX(w(Y, Z)) + w([X, Z], Y) , where we have used that w(Y, X) vanishes because X, Y are in the same Lagrangian subbundle for w. We shall also use the name Bott connection for this partial connection on T . Formula (3.1) is the analog for the Bott connection of the Koszul formulaF for the Levi-Civita connection of a . The importance of the Bott connection for the study of Lagrangian foli- ations stems from the follwing result.

PROPOSITION 3.9. For a Lagrangian foliation on a symplectic manifold, the Bott connection on T is flat and torsion-free. F F PROOF. The Bott connection on A is flat by Lemma 3.6, and therefore it is flat on T as well. It is an instructive exercise to prove flatness directly from (3.1) usingF the Jacobi identity for the commutator of vector fields. If T denotes the of , then in order to prove that T r r r vanishes, we prove that w(Tr(X, Y), Z) vanishes for all X, Y tangent to and arbitrary Z (M). First we calculate using (3.1): F 2X w(Tr(X, Y), Z)=w( Y, Z) w( X, Z) w([X, Y], Z) rX rY = L (w(Y, Z)) + w([X, Z], Y) L (w(X, Z)) X Y w([Y, Z], X) w([X, Y], Z) . This compares favourably with the usual formula for the exterior deriva- tive of w: dw(X, Y, Z)=L (w(Y, Z)) L (w(X, Z)) + L (w(X, Y)) X Y Z w([X, Y], Z)+w([X, Z], Y) w([Y, Z], X) . The two sums are the same, except for the term LZ(w(X, Y)), which van- ishes because X and Y are in the same Lagrangian subbundle with respect to w. We conclude

w(Tr(X, Y), Z)=dw(X, Y, Z) , and this vanishes because w is closed. Thus the Bott connection on T r F is torsion-free. ⇤ COROLLARY 3.10. If L is a leaf of a Lagrangian foliation, then TL L admits a torsion-free flat connection. In other words, L is an affinely flat manifold.! This shows that the leaves of Lagrangian foliations are very special. 40 3. FOLIATIONS AND CONNECTIONS

EXAMPLE 3.11. A closed oriented surface is affinely flat if and only if it is a torus. The torus is flat even in the , Riemannian, sense. It is a theorem of Milnor and Benzecri that surfaces of genus different from one do not admit flat affine connections. Of course this does not follow from Gauss–Bonnet, since the connections in question are not assumed to be metric. Recall that if we consider just Lagrangian submanifolds, without asking that they be leaves of Lagrangian foliations, then there are no restrictions on the manifolds.

EXAMPLE 3.12. We saw in Subsection 3.2 of Chapter 2 that the cotan- gent bundle of any smooth manifold is symplectic, and that the zero- is a Lagrangian submanifold. Corollary 3.10 admits the following sharpening and converse, origi- nally due to Weinstein.

THEOREM 3.13. A smooth manifold is a leaf of a Lagrangian foliation if and only if it admits a torsion-free flat affine connection.

PROOF. One direction was already proved in Corollary 3.10. For the converse assume that M admits a torsion-free flat affine connection . r We consider the T⇤ M with its canonical symplectic form wcan. Without using , we know that the zero-section is a Lagrangian sub- manifold. However,r we now want to prove more, namely that this La- grangian submanifold is actually a leaf of a Lagrangian foliation on T⇤ M. Instead of working with on TM, we use the dual connection ⇤ on T M. This is flat because ris. Flatness means that there exist localr trivi- ⇤ r alizations by parallel sections. In the case of ⇤ on T⇤ M we can trivialize T M by -parallel one-forms a ,...,a . Ther horizontal subbundle for ⇤ r⇤ 1 n ⇤ is integrable, and locally the leaves of the corresponding foliations are justr the graphs of constant linear combinations of the parallel one-forms a . Now because is torsion-free, every -parallel one-form is closed, i r r⇤ and so constant linear combinations of the ai are closed. Thus the leaves of the horizontal foliation are locally graphs of closed one-forms, and are therefore Lagrangian by Proposition 2.63. ⇤ We will need the following Darboux theorem for Lagrangian foliations.

PROPOSITION 3.14. Let (M, w) be a symplectic manifold with a Lagrangian foliation . Then locally, around an arbitrary point in M, there are coordinates F (p1,...,pn, q1,...,qn) such that w = Âi dpi dqi and, at the same time, the leaves of are given by setting q ,...,q equal^ to constants. F 1 n In other words, any Lagrangian foliation is locally standard, in that there is a foliation-preserving symplectomorphism with a suitable in a cotangent bundle foliated by its fibers. 3. INTEGRABILITY AND TORSION 41

3. Integrability and torsion We have already related the integrability of subbundles in the to the absence of curvature. One can also relate integrability to the absence of torsion, and this is what we do now. The basic observation in this direction is the following. As usual, we say that preserves E if XY G(E) for all Y G(E) and arbitrary X. In this caser one also says thatrE is 2 -parallel. 2 r LEMMA 3.15. If a subbundle E TM is preserved by a torsion-free affine connection on M, then E is integrable.⇢

PROOF. Suppose X and Y are sections of E. The torsion-freeness of means r [X, Y]= Y X . rX rY The two summands on the right hand side are in E because we assumed that E is preserved by . Thus we conclude that G(E) is closed under taking r commutators. ⇤ We can elaborate on this to obtain a characterization of integrability through the existence of certain torsion-free connections.

PROPOSITION 3.16. For a subbundle E TM the following are equivalent: ⇢ (1) the subbundle E TM is integrable, (2) the bundle E ⇢M has a torsion-free connection, and (3) the manifold M! admits a torsion-free affine connection preserving E.

PROOF. The third condition implies the second one by restricting the connection, and the second condition implies the first one by the argument in the proof of the Lemma above. That proof required only a torsion-free connection on E, not on all of TM. The proof will be complete once we show that (1) implies (3). So assume that E is integrable, i.e. E = T for some foliation . We can M by foliation charts for , so that inF each of these chartsF the leaves of corre- spond to the parallelF copies of the first factor in the product decompositionF k n k n R R = R . Locally in these charts we define torsion-free flat con- nections⇥ be declaring the coordinate vector fields to be parallel. Then these locally defined connections preserve E, since it just the span of the first k co- ordinate vector fields, which are by definition parallel. Now patch together these locally defined connections using a partition of unity. The resulting connection is still torsion-free and preserves E, since these properties sur- vive the formation of convex combinations1. Thus we have proved that (1) implies (3). ⇤

1The resulting connection is not usually flat, since flatness does not survive under con- vex combinations. 42 3. FOLIATIONS AND CONNECTIONS

We want to adapt this result to the case of Lagrangian foliations on sym- plectic manifolds. In this case we consider symplectic, or w-compatible, connections defined as follows.

DEFINITION 3.17. Let (M, w) be a symplectic manifold. An affine con- nection on M is compatible with w if r (3.2) L (w(X, Y)) = w( Y, Z)+w(Y, Z) X rX rX for all X, Y, Z (M). 2X PROPOSITION 3.18. Let (M, w) be a symplectic manifold. For a Lagrangian subbundle E TM the following are equivalent: ⇢ (1) the subbundle E TM is integrable, (2) the bundle E ⇢M has a torsion-free connection, and (3) the manifold! M admits a torsion-free affine connection compatible with w which preserves E.

PROOF. The implications from (3) to (2) and from (2) to (1) are as in the proof of Proposition 3.16. For the implication from (1) to (3) we can also consult that proof, and we see that the novelty is in ensuring that the connection is compatible with w. For this we can use the Darboux theo- rem for Lagrangian foliations proved in Proposition 3.14. By that result we may cover M by Darboux charts for w which have the additional property that the Lagrangian foliation with T = E is standard, given by the F F first factor in Rn Rn = R2n. Then the torsion-free connection defined on the domain of such⇥ a chart by the requirement that the coordinate vector fields ∂ and ∂ are parallel preserves E and is compatible with w. Patch- ∂pi ∂qi ing together with a partition of unity we obtain a globally defined affine connection. This is still torsion-free, compatible with w, and preserves the subbundle E. ⇤

Note that within each of these special Darboux charts there is a La- grangian foliation complementary to , however, the transition maps be- tween these charts only preserve , andF usually do not preserve the La- grangian complements singled outF by the Darboux coordinates. The special connections for Lagrangian foliations that we just constructed have the additional property that they restrict to the Bott connection along the foliation. This may be suprising since the Bott connection is unique, and the connections we have just constructed are certainly not unique. The use of partitions of unity to construct them indicates that they are quite flexible and plentiful.

PROPOSITION 3.19. Let (M, w) be a symplectic manifold with a Lagrangian foliation . Any w-compatible torsion-free connection which preserves T restricts toF T as the Bott connection. r F F 3. INTEGRABILITY AND TORSION 43

PROOF. Recall from (3.1) that for X, Y T the Bott connection B is given implicitly by 2 F r w( B Y, Z)=L (w(Y, Z)) + w([X, Z], Y) . rX X We can compare this with the w-compatibility of , which by (3.2) is equiv- alent to r w( Y, Z)=L (w(Y, Z)) + w( Z, Y) . rX X rX The torsion-freeness of gives r w( Z, Y)=w( X, Y)+w([X, Z], Y) , rX rZ where the first term on the right hand side vanishes because X and Y are tangent to , preserves T , and T is Lagrangian. Thus we conclude F r F F that the implicit formulas above for B Y and for Y agree. rX rX ⇤ Finally we would like to work out the special properties of differences of torsion-free symplectic connections preserving a given Lagrangian foli- ation . If and are two affine connections on M, then their difference F r r0 A = 0 is a one-form with values in the endomorphisms of the tangent bundle.r r We shall write this as

A(X, Y)= 0 Y Y . rX rX The endomorphism-valued one-form A determines a trilinear map w : (M) (M) (M) C•(M) A X ⇥X ⇥X ! (X,Y, Z) w(A(X, Y), Z) . 7! Conversely, because w is non-degenerate, wA determines A. Using this notation we prove the following characterization of symplec- tic torsion-free connections preserving . F THEOREM 3.20. Let (M, w) be a symplectic manifold with a Lagrangian foli- ation . There exist w-compatible torsion-free connections which preserve T . The setF of all such connections is naturally an affine spacer whose vector spaceF of translations is the space of endomorphism-valued one-forms A W1(End(TM)) which have the property that 2

(1) wA is symmetric under permutations of all three arguments, and (2) w vanishes on triples for which at least two of the vectors are in T . A F PROOF. Existence of such connections was proved in the proof of Propo- sition 3.18. Let A be the difference of two such connections. Because the two con- nections have the same torsion, we conclude that A is symmetric, equiv- alently wA is symmetric in its first two arguments. Writing out the w- compatibility of each of the two connections as in (3.2) and taking the dif- ference of the two identities, we see that w(A(X, Y), Z)=w(A(X, Z), Y), in other words wA is symmetric in the second and third arguments. Thus, combining the two symmetries, wA satisfies (1). 44 3. FOLIATIONS AND CONNECTIONS

Since both connections preserve T , we know that A(X, Y) is in T as soon as Y is. As we have already provedF that (1) holds, if two of the threeF arguments are in T , we may assume that they are Y and Z. Then A(X, Y) F and Z are both in T and wA(X, Y, Y)=0 because T is Lagrangian with respect to w. This provesF (2). F We have shown that the difference of two torsion-free w-compatible connections preserving T satisfies (1) and (2). These conditions define a F in W1(End(TM)). Conversely, of we modify a given con- nection with the desired properties by the addition of an element from this subspace, then one can check directly that all the desired properties sur- vive. ⇤