CHAPTER 3 Foliations and connections We now recall the definition of a foliation and discuss some fundamen- tal examples. We then use affine connections to investigate foliations, 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 subsets of the form Rk c in Rk Rn k = Rn. F ⇥{ } ⇥ − The difference n k is called the codimension 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 atlas 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 submanifold in the definition of a foliation because the leaves are not embedded, they are only injectively im- mersed. Each leaf has an intrinsic topology and manifold structure which is induced by the restrictions of foliation charts, but this intrinsic topology is not the subspace topology of the leaf considered as a subset of M. EXAMPLE 3.2. A flow of irrational slope on the square 2-torus 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 dimension 1. On a surface 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 space of sections is closed under taking commutators of vector fields, since the commutator of vector fields tangent 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 bundle equipped with a con- nection . The tangent vectors! to parallel transport 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 base-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 holonomy 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 vector bundle 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 diffeomorphism 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 normal bundle 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 derivative 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 submanifolds (which they are not, in the strictest sense), then a Lagrangian foliation is a foliation by Lagrangian submanifolds.