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Geometry of coisotropic submanifolds in symplectic and K\"ahler manifolds

The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropic Grassmanians and their indices. Then we introduce the notions of {\it transverse Maslov bundle} of coisotropic submanifolds, and {\it gradable coisotropic submanifolds}. We then define {\it graded coisotropic submanifolds} and the {\it coisotropic Maslov index} of the maps with boundary lying on such graded coisotropic submanifolds, which reduces to the standard Maslov index of disc maps for the case of Lagrangian submanifolds. The second purpose is to study the geometry of coisotropic submanifolds in K\"ahler manifolds. We introduce the notion of the leafwise mean curvature form and transverse canonical bundle of coisotropic submanifolds and study various geometric properties thereof. Finally we combine all these to define the notion of {\it special coisotropic submanifolds} for the case of Calabi-Yau manifolds, and prove various consequences on their properties of the coisotropic Maslov indices.


§1. Introduction
A triple (X, ω, J) is called an almost Kähler manifold where (X, ω) is symplectic and J is compatible to ω in that linear form defined by g(X, Y ) := ω(X, JY ) is positive definite. The mean curvature one form α L of the Lagrangian submanifold L ⊂ (X, g) is then defined by α L := ( H⌋ω)| T L (1.1) Morvan [Mo] and Dazord [Da] proved that the mean curvature one form, of the Lagrangian submanifold L ⊂ (X, ω, J) becomes closed when (X, ω, J) is Einstein-Kähler, i.e., J is integrable and its Ricci form ρ satisfies ρ = λω, for λ ∈ R. (1.2) More precisely, Dazord [Da] proved the identity Typeset by A M S-T E X on general Kähler manifolds from which together with (1.2) follows closedness of α L in the Einstein-Kähler case. We will call the mean curvature class the real one dimensional cohomology class [α L ] ∈ H 1 (L; R) of the Lagrangian submanifold L ⊂ (X, ω, J). In fact, Morvan (in the case of C n ) and Dazord (in the Calabi-Yau, i.e., in the Ricci-flat case) proved that the mean curvature one form 1 π α L is the one form whose de Rham cohomology class is an integral class which represents the well-known Maslov class [Ar] of Lagrangian submanifolds in symplectic geometry. The latter measures the rotation index of the angle of the tangent plane of the Lagrangian submanifolds which illustrates an interesting interplay between symplectic and Riemannian geometry of Lagrangian submanifolds in Einstein-Kähler manifolds. In this respect, the author [Oh2] also proved that the mean curvature class α L is invariant under the Hamiltonian isotopy of (X, ω) (see [Oh2] for its precise meaning).
One purpose of the present paper is to generalize all of the above facts on Lagrangian submanifolds on Kähler manifolds to coisotropic submanifolds Y ⊂ (X, ω): (1.5) Coisotropic submanifolds have played some important role in symplectic geometry in relation to generalizing calculus of Lagrangian submanifolds to some corresponding calculus in Poisson manifolds in relation to the geometric quantization (see e.g., [We]). Recently they attracted some physicist's attention [KaOr] in an attempt to correct and complete Kontsevich's homological mirror symmetry proposal in which inclusion of coisotropic D-branes plays an important role. For the case of Kähler manifolds, we introduce the notion of leafwise mean curvature vector of a coisotropic submanifold Y ⊂ (X, ω, J) and attempt to prove all the analogs in this coisotropic case to the above mentioned relationships between the Maslov index and the mean curvature class that are valid for Lagrangian submanifolds. It turns out that in a suitable foliated or leafwise context, all the coisotropic analogs to the above mentioned geometric properties of Lagrangian submanifolds can be proved. We also introduce the notion of Maslov loops of coisotropic Grassmanians Γ k (R 2n , ω 0 ) = {C ∈ Gr n+k (R 2n ) | C ω0 ⊂ C, dim ker ω 0 | C = n − k} (1.6) and their indices. Note that when k = 0, C become Lagrangian and k = n, it becomes the full space R 2n . This class is not an homotopy invariant unlike the Lagrangian case but enjoys certain symplectic invariance property (see section 2 and 3). Then we introduce the notions of gradable coisotropic submanifolds and graded coisotropic submanifolds in general symplectic manifolds, for which we also define an index for any (disc) map w : (D, ∂D) → (X, Y ). In section 7, we will also introduce the notion of special coisotropic submanifolds in the Calabi-Yau case which generalize the notion of special Lagrangian submanifolds. And we will derive various consequence on the coisotropic Maslov index of disc maps with boundary lying on special coisotropic submanifolds.
Finally in section 8, we analyze the symplectic Π-transverse curvature F Π introduced in [OP1] with respect to the orthogonal splitting Π : T Y = T F ⊕ N J F in the case of Kähler manifolds (X, ω, J) and relate the curvature with the classical Levi form for the case of hypersurfaces Y ⊂ (X, ω, J).
The present work is a by-product of the joint works [OP1,2] with Jae-Suk Park. We would like to thank Jae-Suk Park for exciting collaboration on the coisotropic D-branes. We also thank Wei-Dong Ruan for his interest on this work and for pointing out a couple of imprecise points in our calculations in the precious version of this paper. §2. Maslov loops of coisotropic Grassmanians and their indices In this sub-section, we introduce the notion of Maslov loops of coisotropic subspaces and their associated indices.
Denote the set of coisotropic subspace of rank 2k in the symplectic vector space From the definition, for any coisotropic subspace we have the canonical flag, Combining this with the standard complex structure j on R 2n ∼ = C n , we have the splitting where H C is the complex subspace of C. The following proposition is not difficult check, whose proof we omit and leave to [OP1].
Proposition 2.1. Let 0 ≤ k ≤ n be fixed. The unitary group U (n) acts transitively on Γ k . The corresponding homogeneous space is given by Now we have the following commutative diagram of short exact sequences , ω 0 ) be a continuous loop and consider the flag

Considering
S γ := C ω γ ⊕ jC ω γ , C ω γ (t) := γ(θ(t)) ω we associate to the each coisotropic loop γ the pair (S γ , L γ ) of loops of symplectic vector space S γ (θ) of dimension 2(n − k) and its Lagrangian subspace L γ (θ) = C ω γ for θ ∈ S 1 . We denote by (C ω ) ⊥ ⊂ C * the set of annihilators of C ω in the dual space C * of C. We will need the following general discussion on the coisotropic subbundle of symplectic vector bundles. Let (E, σ, J) → N be a symplectic vector bundle with a compatible complex structure J on the fiber. Let C ⊂ E be a coisotropic subbundle and C σ the kernel bundle of C. Denote by C σ = C/C σ the quotient symplectic vector bundle and J σ the induced complex structure. We then identify C σ with the orthogonal complement of C σ in C. We have the splitting with respect the complex structure J. We recall that as a complex vector bundle (C σ , J σ ) and C 0,1 σ = C 1,0 σ,J are canonically isomorphic. We suppress the J σ from notations and write the splitting Definition 2.2.
(1) We denote the determinant line bundle of Λ 1,0 (C * σ ) by and call it the transverse canonical bundle of C. We also denote by Ω k,0 (C * σ ) the set of smooth sections thereof. We call any section ζ ∈ Ω k,0 (C * σ ) of unit norm a transverse complex volume form of C.
(2) We call the square bundle K ⊗2 C the transverse Maslov line bundle of C ⊂ (E, Ω, J).
We note that since the real line bundle (det R (C σ )) ⊗2 is trivial and det R E is always trivial. In general, the transverse Maslov bundle K ⊗2 C is not trivial as a complex vector bundle. Definition 2.3. Let C ⊂ (E, Ω) be a coisotropic subbundle of nullity n − k or of rank 2k.
(1) We call the coisotropic subbundle C ⊂ (E, σ) gradable if K ⊗2 C is trivial, for a (and so any) compatible almost complex structure.
(2) For a given compatible almost complex structure J on (E, Ω), we call a section ζ ∈ Γ(K ⊗2 C ) a transverse Maslov section of C. We call the pair (C, σ) a graded coisotropic subbundle of E.

Examples 2.4.
(1) Any Lagrangian subbundle is gradable in this sense since in that case the transverse Maslov bundle is just a scalar as the transverse space is trivial.
(2) Any coisotropic subbundle over the base N = S 1 is gradable since any complex line bundle over S 1 is trivial. (3) Consider (X, ω, J) is Calabi-Yau, or more generally any symplectic manifold (X, ω) with 2c 1 (X, ω) = 0. Then for any coisotropic submanifold Y ⊂ (X, ω), the coisotropic subbundle T Y ⊂ (T X| Y , ω) is gradable (see section 7). Now we go back to the case E = S 1 × (R 2n , ω 0 ) the trivial symplectic vector bundle over S 1 with the standard complex structure j, with a map γ : S 1 → Γ k (R 2n , ω 0 ) which corresponds to a coisotropic subbundle of E. This is a special case where N = S 1 . In this case, the real bundle (C σ ) ⊗2 → S 1 is always trivial. Since any symplectic vector bundle is trivial over S 1 , both C * σ and (det C C * σ ) ⊗2 are also trivial (C n is Calabi-Yau!).
(1) We denote and call the transverse canonical bundle of γ (with respect to the complex structure induced from the standard complex structure of C n ). We call any section ζ of K ⊗2 γ a transverse Maslov section of γ.

⊗2
. (2.4) The following is easy to check whose proof we omit.
(1) Note that in the Lagrangian case there is no transverse direction and so the bundle K γ just becomes a scalar. Furthermore ζ (γ;dz) becomes the standard angle function for the Lagrangian loops in C n , and hence Definition 2.6 reduces to the standard Maslov index of the loop of Lagrangian Grassmanians.
(2) In fact, the above discussion does not depend on the particular choice of the standard complex structure j on R 2n but can be repeated verbatim for the complex structure on E γ = C n induced from any almost complex structure J compatible to the standard symplectic structure ω 0 . Using the fact that the set of compatible almost complex structure is contractible and the degree is an homotopy invariant, the coisotropic Maslov index is an invariant depending only on the homotopy class of the complex line bundle K (J;γ) . We note that this homotopy class is an invariant depending only on the coisotropic loop γ. In this sense, the index µ(γ; ζ) defined above is a symplectic invariant. (3) A complete parallel discussion can be carried out in the case of Calabi-Yau manifolds as we will discuss in section 7 later.
From the definition, it is clear that to discuss invariance property of the index µ(γ; ζ), one needs more than just a homotopy of the loop γ but also to dictate a proper condition for the section ζ as well. This deviates from the homotopy invariance of the classical Maslov index of Lagrangian loops [Ar].
(2.7) §3. Graded coisotropic submanifolds, and the coisotropic Maslov index In this section, we will introduce the notion of grading on coisotropic submanifolds and define an index of maps carrying a transverse Maslov section with it.
First we denote by F the null foliation of the coisotropic submanifold Y ⊂ (X, ω) and consider the leafwise normal bundle N F and its dual N * F respectively.
We note that J preserves both N F ⊂ T X| Y and N * F ∼ = (T F ⊕ N Y ) ⊥ ⊂ T * X| Y and so induces decomposition of the complexifications By definition, the transverse canonical bundle is defined by In general K ⊗2 Y → Y is not trivial. However it is trivial if Y is either Lagrangian or if the total space X satisfies 2c 1 (X, ω 0 ) = 0.
This leads us to the following notion.
Y ) and [∆] its homotopy class. Then we call the ∆ transverse Maslov charge and its homotopy class [∆] a grading of Y .
Here we note that our definition of graded coisotropic submanifolds is manufactured so that an index for a map with boundary lying on the given coisotropic submanifold can be defined which is always the case for Lagrangian submanifolds. We would like to emphasize that it is not a generalization of the graded Lagrangian submanifold used in [Ko] or [Se]. For example, according to our definition, any Lagrangian submanifold is gradable and canonically graded.
Obviously for given compatible almost complex structure J on (X, ω), the space of transverse Maslov charges of Y is a principal homogeneous space of C ∞ (Y, S 1 ). In other words, we have We are now ready to define the coisotropic Maslov index of a map w : (Σ, ∂Σ) → (X, Y ) in the presence of transverse Maslov charge ∆ of Y which we denote by µ (Y,∆) (w). We will suppress ∆ from the notation, whenever there is no danger of confusion.
Consider the pull-back w * T X and its symplectic trivialization Such a trivialization, even the unitary one, always exists as long as ∂Σ = ∅. Furthermore it is unique up to homotopy when Σ is a disc. We will restrict to the case of discs in this paper and postpone more general discussions of the higher genus cases elsewhere. From now on, we assume Σ is a disc D = D 2 . We will denote by {u 1 , · · · , u n } the corresponding unitary frame such that In the presence of transverse Maslov charge ∆ on Y , we require that the pushforward grading on the loop α Φ ∈ Γ k (R 2n , ω 0 ) coincides with the canonical grading defined in section 2. In terms of the above unitary frame, this means that (1) We can always choose a unitary frame of γ * (T X) {u 1 , · · · , u n } over ∂D that satisfies (3.2) and (3.3).
(2) We can then extend the above frame over ∂D to that of w * T X over D.
We will call such a trivialization Φ an admissible trivialization of w * (T X) with respect to (Y, ∆). We will sometimes denote by Φ ∆ if necessary to explicitly state this choice. Now we are ready to define the coisotropic Maslov index for the map w : Definition 3.3. Let w be as above, and let Φ : w * T X → Σ × C n be an admissible trivialization. Then we define the index and call it the coisotropic Maslov index of the map w with respect to ∆.
It is easy to check the above definition does not depend on the choice of admissible trivializations Φ ∆ .
In general, the coisotropic Maslov index will not be invariant under the homotopy of maps w t : (D, ∂D) → (X, Y ) for t ∈ [0, 1]. However this is so when the transverse Maslov charge ∆ is also parallel. We will prove the following theorem in the next section.
Theorem 3.4. Suppose that the transverse Maslov charge ∆ on Y is parallel with respect to the connection induced from the canonical connection from (X, ω, J). Let {(w t )} 0≤t≤1 be a smooth family of maps with boundary lying on Y . Then we have An obvious necessary condition for the existence of such a parallel section is vanishing of the curvature of K Y . This theorem leads us to the following definition.
We will study various geometric properties of transversally Ricci-flat coisotropic submanifolds in Kähler manifolds later in section 5-7. §4. Coisotropic submanifolds in almost Kähler manifolds In this section we recall the basic facts on the connection on the almost Kähler manifolds (X, ω, J) following [Kb].
We now consider an almost complex structure J that is compatible to ω, i.e, the triple (X, ω, J) defines an almost Kähler manifold with the standard convention g = ω(·, J·).
We will choose a connection ∇ on T X which preserves both J and ω. Such a connection always exist and in addition becomes unique if we impose the condition that ∇ω = 0 = ∇J and the corresponding torsion form Θ is of type (0,2).
Definition 4.1. A connection ∇ of the almost Kähler manifold (X, ω, J) is called the canonical connection (see [Kb]), if its torsion form is of type (0,2).
When we say that (X, ω, J) is an almost Kähler manifold, we will always assume that it carries the canonical connection. Now we need to do some basic calculations involving the moving frame adapted to the given coisotropic submanifold Y ⊂ X.
We choose an orthonormal frame of X {e 1 , e 2 , · · · , e n , f 1 , · · · , f n } (4.2) such that form a unitary frame of T (1,0) X and the vectors be the dual frame of (4.2). The complex valued one forms form a unitary frame of T X which are dual to the unitary frame {u 1 , · · · , u n }.
In terms of the above mentioned canonical connection ∇, we have where ω i j is the connection one form The first structure equation with respect to the frame {θ 1 , · · · , θ n } becomes where the torsion form Θ j is of the type (0,2) (see [Kb]), or equivalently The 2-nd structure equation of {θ 1 , · · · , θ n } is where K is the curvature 2-form.
The following proposition will play an important role in the proof Theorem 3.4 for the maps w : (D 2 , ∂D 2 ) → (X, Y ) and also in our calculation of covariant derivative of the transverse Maslov section later.
Together we have a system of linear first order ODE for u i , ξ and u i , ξ for 1 ≤ i ≤ k with the initial condition and in particular at t = 1. This finishes the proof.
Proof of Theorem 3.4. Because of the symplectic invariance of the index µ(γ; ζ), we may choose any trivialization of w * t T X for the definition of µ (Y,∆) (w t ). We denote the parameterized map and fix a trivialization of W * (T X) Under this trivialization, (∂w t ) * T Y gives rise to a loop of coisotropic subspaces for each θ ∈ ∂D. The pairs (S αw t , L αw t ) corresponding to the loop t → Π 0 t · α wt ∈ Γ k (C n ) are mapped to a t-parameter family of the pairs in C n and Proposition 4.3 implies that for all t ∈ [0, 1]. Furthermore since we assume that ∆ is parallel, ∆ is invariant under the parallel translation Π 0 t , i.e., we have where ∆ t = (∂w t ) * ∆. Then µ (Y,∆) (w 0 ) = µ (Y,∆) (w 1 ) follows from (4.11), (4.12), Proposition 4.3 and homotopy invariance of the usual Maslov index of the Lagrangian Grassmanians. §5. Leafwise mean curvature vector From now on, we use indices i, j, . . . from 1 to n, a, b, c, · · · from 1 to k and α, β, · · · from k + 1, · · · , n and use the summation convention for the repeated indices. We restrict the 1-st structure equation (4.4) to Y . Since the distribution E on Y is integrable and f * α = 0 on Y , both dθ a and dθ a should be contained in the ideal generated by {θ 1 , · · · , θ k , θ 1 , · · · , θ k }. The 1-st structure equation provides We will come back to the non-integrable case in the appendix and restrict our attention to the integrable case for the rest of the paper. From now on, we will assume (X, ω, J) is Kähler, i.e., Θ ≡ 0. In this case, the above first structure equation becomes Since this is contained in the ideal generated by {θ 1 , · · · , θ k , θ 1 , · · · , θ k }, and ω a α ∧ θ α = ω a αj θ j ∧ θ α + ω a αjθ j ∧ θ α we derive ω a αβ θ β ∧ θ α + ω a αβθ β ∧ θ α = 0 and so ω a αβ = ω a βα , ω a αβ = 0. (5.2) Now we need to incorporate some real aspect of the submanifold Y ⊂ (X, ω). For this, we write the connection one form ω i j as where α i j , β i j are real one forms. The unitarity of connection, i.e, the skew-Hermitian property of (ω i j ) implies 3) The first structure equation where the coefficients satisfy Therefore the second fundamental form S = α S α f α of Y is given by the symmetric matrix Now we are ready to define the leafwise mean curvature vector.
Definition 5.1. Let Y ⊂ X be a coisotropic submanifold and let S = S α f α be the second fundamental form of Y ⊂ (X, g). The leafwise mean curvature vector of Y is the partial trace of S H := S(e β , e β ) = S α (e β , e β )f α . (5.8) The leafwise mean curvature one form of Y is defined by It is easy to check that the right hand side of (5.8) is independent of the choice of the frame adapted to Y and so the leafwise mean curvature vector and so the leafwise mean curvature one form are well-defined.
In the above moving frame, the leafwise mean curvature vector is given by H = A α ββ f α Recalling the Kähler form is given by ω = i 2 θ j ∧θ j = e * k ∧ f * k , we prove that the mean curvature one form becomes We summarize the above calculations into the following proposition Proposition 5.2. Let Y ⊂ (X, ω) be coisotropic and (X, ω, J) be Kähler. Then the leafwise mean curvature one form has the formula (5.11) Now we will derive leafwise derivatives of various leafwise differential forms, including α Y . To carry out this derivation in a coherent manner, we need to provide a brief review the concept of E-de Rham complex [NT] associated to the structure of Lie algebroid [Mac]. is a Lie algebra homomorphism and, for any sections σ and τ of E and a smooth function f on M , the identity (5.14) , σ 1 , · · · ,σ i , · · · ,σ j , · · · , σ k+1 ). (5.15) The cohomology of this complex will be denoted by E H * (M ) and called the E-de Rham cohomology of M . An E-connection on a vector bundle F on M is a linear map satisfying the Leibnitz rule: Similarly we can define the notion of E-curvature E K ∇ ∈ Γ(Λ 2 (E * ) ⊗ End(F )) in an obvious way.
We would like to emphasize that E H * (M ) is not a topological invariant of Y but an invariant of the Lie algebroid E = T Y ω or of the null foliation F of Y . For example, the class is not invariant under the general homotopy but invariant only under the homotopy tangential to the foliation in an obvious sense. We can also denote T Y ω = T F the tangent bundle of the null foliation F and by E * = T * F the cotangent bundle of F .
In our case, M = Y and E := T Y ω and the anchor map ρ : E → T Y is nothing but the inclusion map i : T Y ω → T Y . The integrability of T Y ω implies that the restriction of the Lie bracket on Γ(T Y ) to Γ(T Y ω ) defines the Lie bracket [ , ] on Γ(E). Therefore the triple defines the structure of Lie algebroid and hence the E-differential and E-connections. In our case, the corresponding E-differential is nothing but d F the exterior derivative along the null foliation F . and its cohomology, denoted by H * (Y, ω), the cohomology H * (F ) of the foliation F . We denote the E-connection of a vector bundle F over Y of this Lie algebroid by ∇ ω in general. Now we go back to further study the leafwise mean curvature vector. The the Ricci form iρ of the metric g = ω(·, J·) represents the curvature of the canonical line bundle K := Λ n,0 (T * X ⊗ C).
We first recall one geometric fact about Lagrangian submanifolds on the general Kähler manifolds. For a given Lagrangian embedding i : L ֒→ (X, ω), we consider the pull-back bundle F = i * K → L. If L is orientable, the line bundle i * K is always trivial. In general, the square i * K 2⊗ is trivial. In fact, for the orientable Lagrangian submanifold, there exists a canonical section which we denote by Ω L : we choose a positively oriented Darboux frame {e 1 , · · · , e n , f 1 , · · · , f n } so that {e 1 , · · · , e n } spans, T L and the associated holomorphic frame {θ 1 , · · · , θ n } of Λ n,0 (T X ⊗ C) adapted to L. In particular, the n-form e * 1 ∧ · · · ∧ e * n provides the volume form of Y with respect to the induced metric. Then the complex n-form Ω L = θ 1 ∧ · · · ∧ θ n where θ j = e * i + if * i provides the canonical section. Now we study the general coisotropic cases. The discussion below will be parallel with the case in (R 2n , ω 0 ) considered in section 2. Recalling that the direct sum where the splitting is with respect to the complex structure J : T F ⊕ N J Y → T F ⊕ N J Y . One can also consider the dual bundle (5.17) Then we consider the top exterior power of E 1,0 , det E 1,0 , as a complex line vector bundle.
Note that the complex line bundle det E 1,0 ) ⊗2 has a canonical section given by for any adapted frame {e 1 , · · · , e n , f 1 , · · · , f n } and its associated Hermitian frame. It is easy to check that this form does not depend on the choice of the adapted frame (4.9) and so globally well-defined.
We will now compute the covariant derivative ∇(u k+1 ∧ · · · ∧ u n ) with respect to the induced connection on (det E 1,0 ) ⊗2 from that of (T F ⊕J(T F ))⊗ C. More precisely we have the following

Lemma 5.5. The canonical connection on T X| Y naturally induces a connection on T F ⊗ J(T F ) ⊂ T X| Y by restriction.
Proof. This is an immediate consequence of Proposition 4.3.
We will also compute the E-covariant derivative where the E-connection ∇ ω is nothing but the restriction of the induced connection on F = det E 1,0 from the connection ∇ on T F ⊗ J(T F ) defined by Lemma 5.5. One can easily check that ∇ ω really satisfies the defining property E-connection in Definition 5.4.
Proposition 5.6. Let (X, ω, J) be Kähler and i : Y ֒→ (X, ω) be a coisotropic embedding. We denote by ∇ the connection on det E 1,0 → Y defined in Lemma 5.5. Then we have In particular, we have curv(det(E 1,0 )) = i * (dω α α ) (5.20) for the curvature of the line bundle det(E 1,0 ) is given by and its E-curvature is given by As in the Lagrangian case [Oh2], the following is an immediate corollary of Proposition 5.6 whose proof we omitted.
Corollary 5.7 (Compare with [Corollary 3.3,Oh2]). Let i : Y → X be any coisotropic embedding. Then the holonomy of the bundle det(E 1,0 ) over a loop γ ⊂ Y with respect to the induced connection is given by provided that γ is tangent to the null foliation of Y , i.e., γ ′ (t) ∈ T Y ω for all t.
We first state the following symmetry property of the second fundamental form B of coisotropic submanifolds Y in a Kähler manifolds, which is the analog of the symmetry property for the Lagrangian submanifolds [Lemma 3.10,Oh1]. Proof. This is a re-statement of the property that A α βγ is fully symmetric over α, β and γ which follows from (5.3) and (5.7).
(5.24) for X ∈ T F . On the other hand, we have already shown that β γ γ = A γ γα e * α = −α Y in (5.6), (5.11) and the symmetry property of A α βγ . This finishes the proof (5.21). §6. Pre-Kähler manifolds and its transverse canonical bundle The beginning discussion in this section is intrinsic in that it depends only on the corresponding pre-symplectic structure ω Y on Y , while the discussions in the previous sections are extrinsic in that it describes the property of the coisotropic embedding into (X, ω). In the end of the section, we will derive a compatibility condition between them.
For this purpose, it seems to useful to define another intrinsic notion Definition 6.1. Let (Y, ω) be a pre-symplectic manifold and fix a projection Π : T Y → T Y and its associated splitting T Y = T F ⊕ G Π Let J be an almost complex structure on the normal bundle N F ⊂ T Y compatible to the pre-symplectic structure ω Y in the sense that We call the triple (Y, ω Y , J) a pre-Kähler manifold. We denote by curv(K Y ) and E curv(K Y ) the curvature and E-curvature of the transverse canonical bundle K Y of Y , respectively.
Obviously any coisotropic submanifold Y in a Kähler manifold (X, ω, J) carries the induced pre-Kähler structure.
Considering θ 1 ∧ · · · ∧ θ k as a local frame of K Y , a straightforward computation shows that the covariant derivative for any X ∈ T Y . Here again we use Proposition 4.3 to define the natural connection on the bundle H E := (T F ) ⊥,J induced from T X| Y . The calculation is done with this natural connection. Therefore the curvature of K Y is given by On the other hand if we denote by the pull-back of the form ω a a to the leaves of the null-foliation F , The the Ecurvature of the bundle K Y is given by the leafwise differential These forms do not depend on the choice of frames adapted to Y but depends only on the triple (Y, ω Y , J Y ). We now derive the following which is the coisotropic analog of the result by Morvan [Mo] and Dazord [D], which relates the intrinsic curvatures (Y, ω Y , J Y and the extrinsic curvatures of Y ⊂ (X, ω, J) and the ambient Ricci-curvature of (X, ω, J).
Theorem 6.2. Let (X, ω, J) Kähler and i : Y ⊂ (X, ω, J) be any coisotropic submanifold. Let K = (K i j ) be the curvature two form and iρ = K j j the Ricci form of the metric. Then we have the formula Proof. We will follow the above used notations in section 4. We first note that dβ β γ = Im(dω β γ ). (6.8) Now we note that if we restrict β α γ in (5.6) to the leaves of E, we have On the other hand, taking the trace of the 2-nd structure equation we have We decompose the left hand side of (6.9) and rewrite dω α α + dω a a = iρ. This immediately proves (6.6). Restricting this to the leaves of F , we have On the other hand, we derive from (6.8) and ) from the definition (6.3) of the E-curvature of K Y . This finishes the proof. Corollary 6.3. If (X, ω, J) is Einstein-Kähler, i.e, ρ = λω, then we have curv(K Y ) − curv(det E 1,0 ) = 0. (6.12) This is the coisotropic analog to the well-known fact that the mean curvature one form is always closed for the Lagrangian submanifolds in Einstein-Kähler manifolds. Now we restrict to the case when Y carries a parallel transverse Maslov charge ∆. An obvious necessary condition for Y to carry such a parallel transverse Maslov charge is the vanishing of its ordinary curvature curv(K Y ) = 0 (6.14) i.e., Y must be transversally Ricci-flat. Therefore we have the following Proposition 6.4. Suppose that Y satisfies (6.14), i.e., is transversally Ricci-flat. Then we have −id F (α Y ) = i * F (ρ). In particular, for the Einstein-Kähler case, the mean leafwise mean curvature form is leafwise closed and so define an infinitesimal deformation of Y as a coisotropic submanifold. §7. Special coisotropic submanifolds in Calabi-Yau manifolds In this section, we restrict to the case when (X, ω, J) is Calabi-Yau and study special features of geometry of coisotropic submanifolds thereof. A coisotropic submanifold will have an induced pre-Kähler structure (Y, ω Y , J Y ) from X.
Let Ω be a holomorphic volume form of X with constant length one or equivalently a non zero holomorphic section of the canonical line bundle K with length one.
For any given coisotropic submanifold Y ⊂ (X, ω), taking the restriction of Ω to the leaves of F , we define a bundle map In particular, this pushes forward the global section (u k+1 ∧ · · · ∧ u n ) ⊗2 ∈ Γ(E 1,0 ) to a global section on K ⊗2 Y . Therefore we have Definition & Proposition 7.1. Any coisotropic submanifold Y in Calabi-Yau manifolds is canonically graded by the section induced by Ω. We denote this canonical transverse Maslov charge by ∆ (Ω;Y ) and call it the canonical transverse Maslov charge of Y and the corresponding grading the canonical grading.
This canonical grading induced by Ω enables us to define the coisotropic Maslov index for the maps w : (D, ∂D) → (X, Y ) for all coisotropic submanifolds Y in X simultaneously, which we denote by µ (Y ;Ω) .
Obviously one necessary condition for the embedding Y ⊂ (X, ω, J) is special coisotropic is that the pre-Kähler structure (Y, ω Y , J Y ) must be transversely Ricciflat.
We recall that for the Lagrangian case, N F = {0} and (i * Ω) ⊗2 = g(Ω L ) ⊗2 for g : L → S 1 which is a globally well-defined function and called the angle function of the Lagrangian submanifold. Therefore in this case, the above condition means g = (u 1 ∧ · · · ∧ u n ⌋Ω) ⊗2 satisfies dg = 0 on L, i.e., f is constant. Therefore Definition 7.2 reduces to the usual special Lagrangian condition for the Lagrangian case. Furthermore for the Lagrangian case, special Lagrangian condition implies minimality of Lagrangian submanifolds and vice versa (at least for orientable Lagrangian submanifolds).
The following theorem is the coisotropic analog to this fact. Proof. This immediately follows from (7.5).
Now we relate the above geometric study of coisotropic submanifolds in Calabi-Yau with the coisotropic Maslov index of the maps w : (D, ∂D) → (X, Y ).
For given w, we choose our unitary frame {v 1 , · · · , v n } on w * (T X) so that it is admissible to the transverse Maslov charge ∆ (Y ;Ω) . We denote by {φ 1 , · · · , φ n } its dual frame. The admissibility means (∂w) * (∆ (Y,Ω) ) = (φ 1 ∧ · · · ∧ φ k ) ⊗2 . (7.11) for g : ∂D → S 1 , where we recall T F = span R {e k+1 , · · · , e n }. The Maslov index µ (Y ;Ω) (w) is computed by the degree of the map where dθ is the canonical angular form on S 1 . On the other hand, we can choose our frame {v ′ 1 , · · · , v ′ n } of w * T X, which is not necessarily admissible, so that v ′ j = u j on (∂w) * T Y (7.13) where {u 1 , · · · , u n } the restriction of a frame of (∂w) * T X adapted to (∂w) * T Y over ∂D. We write In particular, recalling Definition 3.3 of µ (Y ;Ω) (w), (7.14) implies Noting that the calculation in the beginning of this section is purely on Y , we will have exactly the same equation as in (7.5) with the connection one form ω i j replaced by (ω ′ ) i j with respect to the frame {v ′ 1 , · · · , v ′ n }. However when restricted to Y , we have (ω ′ ) i j = ω i j . This proves the following local index formula Theorem 7.4. For any coisotropic submanifold Y and a frame {u 1 , · · · , u n } of (∂w) * T X adapted to (∂w) * T Y , we have for all w : (D, ∂D) → (X, Y ). In particular, coisotropic Maslov index depends only on the boundary map ∂w.
Corollary 7.5. Let Y ⊂ (X, ω) be a coisotropic embedding. If the unitary frame {u 1 , · · · , u n } of T X| Y adapted to Y extends over the neighborhood of the image of w, then µ (Y ;Ω) (w) = 0.
Proof. In this case, we note that the integral (7.15) becomes since ρ ≡ 0 for the Calabi-Yau metric.
Corollary 7.6. Suppose that Y is a special coisotropic submanifold. Then we have Note that (7.16) for special Lagrangian submanifolds reduces to the well-known fact that the Maslov index vanishes since in that case K ⊗2 Y is just a scalar C. §8. The transverse symplectic curvature in Kähler manifolds In [OP1], we have introduced the notion of Π-transverse symplectic curvature F Π ∈ Γ(Λ 2 (N * F ) ⊗ T F ), which measures non-integrablity of the complementary subbundle G Π of T Y in a given splitting Then the complementary subbunlde G Π is integrable, if and only if F Π = 0.
In the (almost)-Kähler case, we have the canonical Riemannian splitting We denote the corresponding curvature as F = F Y . We first recall the definition Definition 8.1 [Definition 4.1,OP1]. The transverse symplectic curvature of Y is a T F -valued two form on N J F or a section of ∧ 2 (N * F ) ⊗ T F defined as follows: Let π : T Y → N J F be the orthogonal projection. For any given v, u ∈ N J F | y , we define where X, Y be any vector field on Y with X(y) = v, Y (y) = u that is normal to the foliation and (·) the component of (·) tangential to the null-foliation.
It was proven in [OP1] that this is well-defined as an element We will compute the components of the intrinsic curvature F with respect to those of the second fundamental form of the embedding Y ⊂ (X, g) where g is the associated Kähler metric of X.
Let {e 1 , · · · , e n , f 1 , · · · , f n } be an orthonormal frame of T X adapted to Y and ω i j = α i j + iβ i j be the associated Hermitian frame. The first structure equation is This proves the following formula (8.1) In terms of the unitary frame, we can also write in addition to (5.7), in which case we have (2) We have F = 0 if and only if the second fundamental form S satisfies in addition to (5.7).
Now we consider the hypersurface case in detail.
Example 8.3. Let (X, ω, J) be any almost Kähler manifold and consider the hypersurface Y ⊂ X defined by for a smooth function ρ : X → R such that |∇ρ(x)| g ≡ 1 (8.10) for any x ∈ Y , where ∇ρ is the gradient of ρ with respect to the associated metric g = ω(·, J·). We denote by X ρ = J∇ρ is the Hamiltonian vector field of ρ. It follows that X ρ is tangent to Y and (8.10) and N J F ⊂ T Y is nothing but the distribution of the maximal complex or Jinvariant subspace of T Y . By the definition of F , we have for any X, Y ∈ N J F . If we denote for any X, Y ∈ N J F . Combining (8.11) and (8.14), we have proved (8.15) In general, the two forms dd c on the complex vector bundle N J F ⊗ C will not be of type (1, 1), unless the almost complex structure is integrable. On the other hand, in the Kähler case, the following idenity is well-known dd c ρ = 2i∂∂ρ whose restriction to N J F is the well-known Levi form on the hypersurface. Therefore in the integrable Kähler case, the transverse symplectic curvature F is automatically of type (1,1) and the two form reduces to the Levi form of the hypersurface. The case F = 0 corresponds to the hypersurface that is Levi-flat and the associated foliation of N J F in that case is nothing but the foliation by the maximally complex submanifolds in Y .
Motivated by this example, we now introduce the following definition Definition 8.4. We say that a pre-Kähler manifold (Y, ω Y , J Y ) is integrable, if the associated transverse curvature F is of type (1, 1), or equivalently the second fundamental S = S α f α in any adapted frame satisfies where all A α , C α and D α are symmetric k × k matrices.
We will further study geometry of such structures elsewhere in the future.

Appendix
A.1. Calculation for the almost Kähler case In this appendix, we generalize the formula in Theorem 4.3 in the (non-integrable) almost Kähler manifold as in the spirit of the calculation carried out by Schoen and Wolfson [SW] for the Lagrangian submanifolds in the almost Kähler case.
For this, we need to compare the canonical connection used in the main part of this paper and the Levi-Civita connection of the metric g = ω(·, J·).
The corresponding first structure equation with respect to the oriented orthonormal frame {e 1 , · · · , e n , f 1 , · · · , f n } can be obtained by decomposing (3.3) into the real and the imaginary parts: Writing The symmetry properties of ω i j and τ ī j ω i j = −ω j i , τ ī j = τ j i , coming from the unitarity of ω and from (3.6), immediately imply that (A.1-2) defines a torsion free Riemannian connection of g which is nothing but the unique Levi-Civita connection of g. Now we apply the same kind of analysis as in section 4 noting that the definitions of E-differential and E-connections depend only on the symplectic structure and the given coisotropic submanifold Y ⊂ (X, ω), but not on the almost complex structure J.
In the non-integrable case, there will be torsion terms appearing in various places. First, (4.4) is replaced by (A.1-2) and (4.5) by on Y . Again by Cartan's lemma, (4.6) is replaced by By the same reasoning as in section 4 (see [SW] for a similar calculation for the Lagrangian case), we still obtain the same formula α Y = A α βα e * β (A.5) as (4.11).
On the other hand, the formula (4.18) is replaced by when restricted to T F and so (4.21) replaced by Restricting this to T F and combining with (A.5-7), we have obtained the following coisotropic analog to [Proposition A.1,SW].
Theorem A.1. Let (X, ω, J) be almost Kähler with the canonical connection. Let K = (K i j ) be the curvature two form and iρ = K j j be the Ricci form of the connection. Then we have the formula id F α Y − E curv(K Y ) = i * F (ρ) + d F (τ ) where τ ∈ Γ(T * F ) is a one form canonically pulled-back via the embedding i : Y → X from the torsion form of the canonical connection of X.

A.2. Criterion for the minimality of the null foliation
In this section, we study the question when the null foliation becomes a minimal foliation in the sense that each leaf is a minimal submanifold of Y .
Note that the form ν Y := e * k+1 ∧ · · · ∧ e * n ∈ Λ n−k (E * ) = Ω n−k (F ) is independent choice of the frames {e 1 , · · · , e n , f 1 , · · · , f k } adapted to Y and so defines a globally well-defined n − k form on Y which restricts to the volume form on each leaf. We recall from [Ru,Su,Ha] that existence of such form is a necessary and sufficient condition for the manifold (Y, g Y ) is foliated by minimal leaves of dimension n − k. In our case, the following theorem shows that the form ν Y plays the role of such form.

Theorem A.2. A leaf Σ of the null foliation is a minimal submanifold if and only
if the from ν Y is relatively closed on Σ: namely, dν Y (X 1 , · · · , X n−k+1 ) = 0 (A.8) on Σ if the first n − k vector fields X i are tangent to the leaves.