GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 2, 1997, 185-200

THE HÖRMANDER AND MASLOV CLASSES AND FOMENKO’S CONJECTURE

Z. TEVDORADZE

Abstract. Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko’s conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.

Introduction

In this work we are interestied in the characteristic classes of sympletic bundles and Lagrangian subbundles. One of such classes was discovered when studying asymptotic solutions of linear partial differential equations [1] and called a Maslov class. In [2] Arnold gives a pure geometric interpretation of the Maslov class.

A generalization of the Maslov class to higher cohomological dimensions was defined by Arnold and studied in [3]. Another generalization of the above-mentioned classes is defined by Trofimov in [4].

In [5] and [6] Hörmander defined the cohomology classes σ(E; X, Y) ∈ H^{1}(M; ℤ) for arbitrary sections X and Y of Lagrange’s Grassmanian L(E) →^{π} M, where E →^{p} M is a sympletic fibre bundle. These classes are described in [7].

In [8] Fomenko formulates a conjecture that all Maslov–Arnold characteristic classes of minimal Lagrangian surfaces are equal to zero, and this conjecture is proved when Lagrangian surfaces are submanifolds in M^{2n} = ℝ^{2n} = ℂ^{n}.

The paper is organized as follows. In §1 we will describe the condition when the Maslov index ℓ(γ) is an even number for an arbitrary closed curve γ in the Lagrangian manifold (Theorem 1.1) and prove one functorial property for Hörmander classes σ(E; X, Y) (Theorem 1.3). In §2 a class of Lagrangian

185 manifolds LH(M) is considered in an almost Hermitian manifold M with the following property: J^{∗}α_{H} is an exact 1-form in each manifold from LH(M), where J is an almost complex structure and H is a mean curvature with respect to the inclusion in M. It is proved that c|_{N} is an exact 2-form for the Lagrangian manifold N in the Hermitian manifold M and c|_{N} = 0, if N ∈ LH(M), where c is the first Chern form. The extended variant of Fomenko’s conjecture on the class LH(M) was also formulated and studied. This conjecture is proved for the class LH(T^{∗}Q^{n}), where Q^{n} satisfies some additional condition (Theorem 2.8).

1991 Mathematics Subject Classification. 58F05, 57R20.

Key words and phrases. Symplectic manifold, connection, Maslov class, minimal surface, Hermitian manifold.

1072-947X/97/0300-0185$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

In conclusion we note that it remains unknown whether Fomenko’s conjecture is true or not in the general case and how much the class LH(M) extends the class of minimal Lagrangian submanifolds in M.

§ 1. Construction of Maslov and Hörmander Classes

Let (V^{2n}, ω) be a symplectic vector space over ℝ. A real n-plane ζ in V^{2n} is called a Lagrangian plane if the restriction of ω on ζ vanishes. By L(V) we denote Lagrange’s Grassmanian which consists of all Lagrangian n-planes. It is well known that if we have a fixed point X ∈ L(V), then L(V) can be represented as a homogeneous space U(n)/O(n), where the orthogonal group O(n) is cannonically imbedded in the unitary group U(n).

Let us consider the fibre bundle

SU(n)/SO(n) → U(n)/O(n) →^{det^{2}} S^{1},

where the map det^{2} is a fibre map and SU(n)/SO(n) is a fibre at the point 1 ∈ S^{1} ⊂ ℂ. The space SU(n)/SO(n) is simply connected; therefore the long exact homotopy sequence of fibre bundles implies that π_{1}(L(V)) = ℤ and hence H_{1}(L(V); ℤ) = ℤ. From the formula of universal coefficients

0 → Ext(H_{0}(L(V), ℤ)) → H^{1}(L(V); ℤ) →^{h} Hom(H_{1}(L(V); ℤ); ℤ) → 0

(where (h{f}){∑ c_{i} ⊗g_{i}} = ∑ f(c_{i})⊗g_{i}, {f} ∈ H^{1}(L(V); ℤ), {∑ c_{i} ⊗g_{i}} ∈ H_{1}(L(V); ℤ)), we obtain H^{1}(L(V); ℤ) = 0.

The differential 1-form (det^{2})^{∗} \frac{dz}{2πiz} is the generator of H^{1}(L(V); ℤ), where the form \frac{dz}{2πiz} is the 1-form on S^{1} = \{z ∈ ℂ : |z| = 1\} ⊂ ℂ.

Definition 1.1. A class of cohomology which is defined by the 1-form (det^{2})^{∗} \frac{dz}{2πiz} is called a Maslov class.

Definition 1.2. A submanifold N ↪ V^{2n} is called a Lagrangian manifold if the tangent plane at each point x ∈ N is a maximal isotropic plane with respect to ω_{x} = ω|_{TxV ^{2n}}, i.e., ω_{x}|_{TxN} = 0.

HÖRMANDER AND MASLOV CLASSES, FOMENKO’S CONJECTURE 187

For each closed curve γ in N there is a map

G : γ → L(V), G(x) = (x, T_{x}N), x ∈ γ,  (1.1)

which is called a Gaussian map. This map defines the integer

ℓ(γ) = ∫_{γ} (det^{2} ∘ G)^{∗} \frac{dz}{2πiz}  (1.2)

and thereby the class of cohomology from H^{1}(N; ℤ). This class does not depend on the choice of a Lagrangian plane X ∈ L(V) and is called the Maslov class of a Lagrangian manifold N.

Theorem 1.3. If the Gaussian map G : γ → L(V) can be covered by the continuous map G̃ : γ → U(n), i.e., p ∘ G̃ = G, where p : U(n) → L(V) is the natural projection, then ℓ(γ) is an even number.

Proof. First notice that the map det : U(n) → S^{1} induces an isomorphism between the fundamental groups π_{1}(U(n)) and π_{1}(S^{1}). Indeed, the long exact homotopy sequence of fibre bundles

SU(n) →^{j} U(n) →^{det} S^{1}

contains the sequence

π_{1}(SU(n)) →^{j_{∗}} π_{1}(U(n)) →^{(det)_{∗}} π_{1}(S^{1}) →^{∂} π_{0}(SU(n)).

The map (det)_{∗} is an isomorphism, since SU(n) is a simply connected space. It is clear that the map

π_{1}(S^{1}) ∋ [γ] → \frac{1}{2πi} ∫_{γ} \frac{dz}{z} ∈ ℤ

is an isomorphism. The composition of (det)_{∗} and the above-mentioned map give the natural isomorphism

π_{1}(U(n)) ∋ [γ] ⟼ \frac{1}{2πi} ∫_{γ} \frac{d(det U)}{det U} ∈ ℤ.

The projection p : U(n) → U(n)/O(n) defines the monomorpohism

p_{∗} : π_{1}(U(n)) → π_{1}(L(V)),

which is multiplication by 2. This statement immediately follows from the commutative diagram

\begin{matrix}U(n) & ⟶^{p} & U(n)/O(n) \\ ↓^{det} & & ↓_{det}^{2} \\ S^{1} & ⟶^{k} & S^{1} \end{matrix},

where k(z) = z^{2}, z ∈ S^{1} ⊂ ℂ. This monomorphism can be written explicitly as

ℤ ∋ \frac{1}{2πi} ∫_{γ} \frac{d(det U)}{det U} ⟼^{p_{∗}} \frac{1}{2πi} ∫_{γ} \frac{d(det^{2}U)}{det^{2}U} ∈ ℤ,

where γ is the closed curve is U(n).

Now from the commutative diagram p ∘ G̃ = G we have ℓ(γ) = [G(γ)] = p_{∗}[G̃(γ)], where [G(γ)] ∈ π_{1}(L(V)) and [G̃(γ)] ∈ π_{1}(U(n)). Clearly,

\frac{ℓ(γ)}{2} = \frac{1}{2πi} ∫_{_{G}(γ)} \frac{d(det U)}{det U} ∈ ℤ.

Remark 1.4. The experience accumulated in investigating various concrete mechanical systems and calculations of the Maslov class for the Liouville tori of such systems show that an overwhelming majority of systems even numbers as components of the Maslov class [9], [10]. Theorem 1.3 describes the mechanism of such occurrence.

Now we briefly recall the definition and construction of more general classes, namely, of Hörmander classes [7].

Let X, Y, Z, W be four Lagrangian n-planes in V^{2n} such that Z ∩ X = Z ∩ Y = {0} and W ∩ X = W ∩ Y = {0}. For them Hörmander defined the index at the intersection

(X, Y, Z, W) = \frac{1}{2}(sign Q_{Z} − sign Q_{Z}) = ind Q_{W} − ind Q_{W} , (1.3)

where Q_{Z} and Q_{W} are nondegenerate quadratic forms on Y/X ∩ Y defined by the formulas

Q_{Z}(y) = ω(p_{Z}^{X}y, y), Q_{W} (y) = ω(p_{W}^{X} (y), y)

(here p_{Z}^{X} and p_{W}^{X} are the projections of Y/Y ∩X onto Z and W, respectively, accross X). The following relations immediately follow from (1.3):

\begin{matrix}(X, Y, Z, W) = −(X, Y, W, Z), \\ (X, Y, Z, W) + (X, Y, W, V) + (X, Y, V, Z) = 0, \\ (X, Y, Z, W) = −(Z, W, X, Y). \end{matrix}  (1.4)

Let E →^{p} M be a symplectic vector fibre bundle, i.e., for each point m ∈ M there is a symplectic 2-form ω_{m} on E_{m} which smoothly depends on m. Then we can consider the fibre bundle L(E) →^{π} M, where L(E)_{m} = L(E_{m}). Now let X and Y be the sections of the bundle L(E), and U = {U_{α}}_{α∈I} be an open covering of M such that all nonempty finite intersections are diffeomorphic to ℝ^{n} (such a covering is called a good covering). Then for arbitrary neighborhoods U_{α} and U_{β} one can find sections Z_{α} and Z_{β} of L(E) on U_{α} and U_{β}, respectively, such that Z_{α}(m) and Z_{β}(m) are transversal to X(m) and Y (m) for each point m ∈ U_{α} ∩ U_{β}. In this situation the Čech 1-cocycle σ can be defined by the formula

σ(U_{α}, U_{β}) = (X, Y, Z_{α}, Z_{β}).  (1.5)

The cohomology class [σ] is called the Hörmander class for a symplectic vector fibre bundle E and sections X, Y and denoted by σ(E; X, Y) ∈ Ȟ^{1}(M; ℤ).

From (1.4) and (1.5) one can conclude that

σ(E; X, Y) = −σ(E; Y, X).  (1.6)

Let (E′, p′, L(E)) be a pullback bundle of the bundle E to L(E):

\begin{matrix}E′ & ⟶ & E \\ ↓^{p′} & & ↓^{p} \\ L(E) & ⟶^{π} & M \end{matrix} \begin{matrix}L(E′) & \overrightarrow{⟵}^{π′}_{s′} & L(E) \\ ^{T} ↑↓_{π′} & & ↓^{π} \\ L(E) & \overrightarrow{⟵}^{π}_{s} & M \end{matrix} .

Then we can consider the fibre bundle L(E) →^{π′} L(E) and the natural section T : L(E) → L(E′) which is defined by the formula T (x) = x, where x ∈ L(E)_{m} and the fibres E′_{x}, E_{m} are identified. An arbitrary section S : M → L(E) can be lifted to the section S′ : L(E) → L(E′) by the formula S′(x) = S(m), where x ∈ L(E)_{m}, m ∈ M, i.e., if x = (m, ξ) and S(m) = (m, η), then S′(x) = (m, ξ, η) (here, as before, L(E′)_{x} and L(E)_{m} are identified).

One can verify that Y^{∗}(S′) = S and Y^{∗}(T ) = Y for each section Y of the bundle L(E).

It is not difficult to see that class [σ] has the following functorial property: if (f^{∗}E, f^{∗}(p), M_{1}) is a pullback bundle of the bundle (E, p, M), where f : M_{1} → M is a smooth map, then

f^{∗}σ(E; X, Y) = σ(f^{∗}E; f^{∗}X, f^{∗}Y)  (1.7)

(f^{∗}X, f^{∗}Y are pullback sections of the sections X and Y by the map f).

The following equalities arise from (1.4) and (1.6):

\begin{matrix}σ(L(E′); X′, T ) − σ(L(E′); Y′, T ) = \\ = σ(L(E′); X′, Y′) ∈ Ȟ^{1}(L(E); ℤ), \\ σ(E; X, Y) = Y^{∗}σ(L(E′); X′, T ). \end{matrix}  (1.8)

For each section X of the bundle L(E) there is a characteristic class σ_{X} defined by the formula

σ_{X} ≡ σ(L(E); X′, T ) ∈ Ȟ^{1}(L(E); ℤ),  (1.9)

such that

(i) the restriction of σ_{X} on a fibre L(E)_{m}, m ∈ M, is the generator of Ȟ^{1}(L(E)_{m}; ℤ);

(ii) X^{∗}σ_{X} = 0.

The property (ii) immediately follows from (1.8) and (1.6). Indeed, X^{∗}σ_{X} = X^{∗}σ(L(E′); X′, T ) = σ(E; X^{∗}(X′), X^{∗}(T )) = σ(E; X, X) = 0. The first property is more difficult and can be found in [7].

To prove Theorem 1.6 we will use the following well-known theorem [11].

Theorem 1.5 (Dold–Thom–Gysin). Let h^{∗} be a multiplicative cohomology theory and let F →^{i} E →^{π} B be an h^{∗}-fibration. Suppose there are elements a_{1}, …, a_{r} in h^{∗}(E) such that (i^{∗}a_{1}, …, i^{∗}a_{r}) is an h^{∗}(point)-base for h^{∗}(F) as an h^{∗}(point)-module; then (a_{1}, …, a_{r}) is an h^{∗}(B)-base for h^{∗}(E) as an h^{∗}(B)-module.

Theorem 1.6. If X and Y are sections of the bundle L(E), and X′, Y′ are the lifted sections of the bundle L(E′), then

σ(L(E); X′, Y′) = π^{∗}σ(E; X, Y).  (1.10)

Proof. By virtue of the properties (i) and (ii) of the Hörmander class σ_{X} and Theorem 1.5 we can write

σ_{Y} = π^{∗}[β(X, Y)] + k(X, Y)σ_{X},  (1.11)

where β(X, Y) is the closed 1-form on M and k(X, Y) ∈ ℤ. Applying X^{∗} and Y^{∗} to both sides of (1.11) and taking into account (1.8) and (ii), we respectively have

σ(E; X, Y) = [β(X, Y)] + k(X, Y) · 0 = [β(X, Y)],  (1.12)
Y^{∗}σ_{Y} = [β(X, Y)] + k(X, Y)σ(E; X, Y) = 0.  (1.13)

Now from (1.6), (1.12), (1.13) we obtain k(X, Y) = 1, and formula (1.11) acquires the form

σ_{Y} = π^{∗}(E; Y, X) + σ_{X}.  (1.14)

It is clear that (1.14) and (1.10) are equivalent equalities (see (1.8)).

§ 2. Fomenko’s Conjecture

Let M^{2n} be an almost Hermitian manifold, i.e., there exist a tensor field J and a Riemannian metric g on M, where J is an endomorphism of the tangent bundle T M with the property J^{2} = −id and g is an invariant with respect to J, i.e., g(JX, JY) = g(X, Y) for all vector fields X, Y ∈ T (M) on M. The fundamental 2-form on M is defined by the formula

Φ(X, Y) = g(X, JY), X, Y ∈ T (M).  (2.1)

When Φ is a closed 2-form, M is called an almost Kaehlerian manifold and Φ is called an almost Kaehler form on M. In this case M is a symplectic manifold with a symplectic 2-form Φ. The converse is also true: for every symplectic manifold (M^{2n}, ω) there exists an almost Hermitian structure on M such that the fundamental 2-form of this structure coincides with the symplectic 2-form ω. Thus every symplectic manifold is an almost Kaehlerian manifold [12].

Now let N^{n} be a Lagrangian submanifold in an almost Kaehlerian manifold M^{2n}, i.e., i^{∗}Φ = 0, where i : N → M is the imbedding map. By V N we denote the normal fibre bundle on N with respect to the metric g on M (i.e., V_{x}N is a n-plane in T_{x}M, which is orthogonal to T_{x}N, x ∈ N), and by ∇ we denote the Riemannian connection on M. We recall that the trace of the following morphism of the fibrations

B : T N ⊕ T N → V N, B(x, y) = (∇_{x}Y)^{V}

(where x, y ∈ T_{n}N, n ∈ N, and Y is the locally defined vector field on M which extends the vector y, Y |_{N} is the section of T N, and (∇_{x}Y)^{V} is the normal component of vector ∇_{x}Y), which can also be considered as a section of the fibration V N, is called the mean curvative of the submanifold N and denoted by H. More precisely, if e_{1}, …, e_{n} is an orthobasis of the plane T_{n}N and V is a locally defined vector field on M whose restriction on N is a section of V N and V (n) = v, then the mean curvature H is defined by the relation

g(H(n), v) = − ∑_{i=1}^{n} g(∇_{e_{i}}V, e_{i}).  (2.2)

We recall that N ↪^{i} M is called a locally minimal submanifold if the mean curvature H is zero at every point of N. From (2.1) we can conclude that the diagram

[figure: V N; T N; N; J; π_{N}; π_{N}]

is commutative. By α_{H} we denote a section of the fibration A^{1}(V N) (the fibration of the exterior 1-forms on V N) defined as

α_{H} (n)(X) = g(H(n), X), n ∈ N, X ∈ V_{n}N.  (2.3)

Definition 2.1. We say that the Lagrangian manifold N in M is locally minimal from the cohomological point of view if the 1-form β = J^{∗}α_{H} defines the trivial class of cohomology H^{1}(N; ℝ).

The class of the above-defined submanifolds is denoted by LH(M).

In [13] Le Hong Van, relying on the calibrated geometry methods developed in the fundamental work [14], shows that there exists a 1-form ψ on L(T M) such that the Lagrangian submanifold N in M is locally minimal if and only if

G^{∗}(ψ) = 0,

where G : N → L(T M) is the Gaussian map defined as G(x) = (x, T_{x}N) ∈ L(T M)_{x}, x ∈ N.

Now we will briefly describe the 1-form ψ.

Let T_{x}^{c}(M^{2n}) be the complexification of the tangent space of M at point x, x ∈ M^{2n}. By S_{x}^{c} we denote a complex subspace of T_{x}^{c}(M^{2n}) which contains all eigenvectors of the operator J with the eigenvalue i. The unitary bases R^{c}(x) = {ε_{1}(x), …, ε_{n}(x)} of the space S_{n}^{c}(x) construct the principal bundle U(M) over M, with the structure group U(n). By ε\overline{1}, …, ε\overline{n} we denote the complex conjugate vectors of the vectors ε_{1}(x), …, ε_{n}(x). Then the vectors

e_{i}(x) = \frac{1}{\sqrt{2}}(ε_{i}(x) + ε\overline{i}(x)), i = \overline{1, n},
e\overline{i}(x) = \frac{i}{\sqrt{2}}(ε_{i}(x) − ε\overline{i}(x)) = Je_{i}, i = \overline{1, n},

make up families of orthonormal real vectors.

An infinitesimal connection on the principal bundle U(M) is called an almost Hermitian connection on M.

Let U = {U_{α}}_{α∈I} be the open covering of the manifold M, which is equipped with local sections S_{α} : U_{α} → U(M); then an almost Hermitian connection π can be defined by the given 1-forms π_{α} on U_{α}, α ∈ I, with values in the Lie algebra u(n). We can express π_{α} in terms of the (π_{j}^{i})_{i,j=\overline{1,n}} matrix, where π_{j}^{i} are the 1-forms on U_{α} and π_{j}^{i} + \overline{π}_{i}^{j} = 0. If we define S_{α} as S_{α}(x) = R_{α}^{c}(x), x ∈ U_{α}, then there exist matrices B_{β}^{α}(x) ∈ U(n) such that

S_{α}(x)B_{β}^{α}(x) = S_{β}(x),
π_{β} = (B_{β}^{α})^{−1}π_{α}B_{β}^{α} + (B_{β}^{α})^{−1}dB_{β}^{α}, α, β ∈ I, x ∈ U_{α} ∩ U_{β}.

Frames of the type R^{c}(x) and \overline{R}^{c}(x) = \{ε\overline{1}(x), …, ε\overline{n}(x)\} construct the principal bundle U′(M) on M with a structure group U(n) which is a sub-

group in U(2n) by means of the imbedding A → (\begin{matrix}A & 0 \\ 0 & \overline{A} \end{matrix}), A ∈ U(n). Then

the above-mentioned Riemannian connection ∇ with respect to the metric g can be expressed on each open set U_{α} by the matrix of 1-forms

(\begin{matrix}π_{j}^{i} & πi ; \overline{j} \\ π\overline{i} ; j & π\overline{i} ; \overline{j} \end{matrix}) ,

where π\overline{i} ; \overline{j} = \overline{π_{j}^{i}}, π\overline{i} ; j = \overline{πi ; \overline{j}} (since ∇ is a real connection) and π\overline{i} ; j + πi ; \overline{j} = 0 (for ∇g = 0).

The matrices (π_{j}^{i}) (on each U_{α}) define an almost Hermitian connection which is called the first canonical connection for an almost Hermitian manifold M.

From the first structure equations of E. Cartan we have

dθ^{i} = θ^{k} ∧ π_{k}^{i} + θ\overline{k} ∧ πi ; \overline{k},  (2.4)

where θ^{j} = \frac{1}{\sqrt{2}}(e_{j})^{∗} + i(Je_{j})^{∗}, j = \overline{1, n}, k = \overline{1, n}. (2.4) is equivalent to

dθ^{i} = θ^{k} ∧ π_{k}^{i} + Γ_{\overline{k}q}^{i}θ\overline{k} ∧ θ^{q} + \frac{1}{2} T_{\overline{k}\overline{q}}^{i}θ\overline{k} ∧ θ\overline{q}, i = \overline{1, n}, k = \overline{1, n}, q = \overline{1, n},

where Γ_{\overline{k}q}^{i} are the coefficients of the Riemannian connection with respect to frames from the space U′(M), and T_{\overline{k}\overline{q}}^{i} = Γ_{\overline{k}\overline{q}}^{i} − Γ_{\overline{q}\overline{k}}^{i}. Now we note that

the forms T_{\overline{k}\overline{q}}^{i}θ\overline{k} ∧ θ\overline{q}, i = \overline{1, n}, construct the vector 2-form on U(M) which coincides with the torsion vector T of an almost complex structure. By direct calculation (using the fact that dΦ = 0) we can obtain Γ_{\overline{k}q}^{i} = 0. So (2.4) finally has the following form:

dθ^{i} = θ^{k} ∧ π_{k}^{i} + \frac{1}{2} T_{\overline{k}\overline{q}}^{i}θ\overline{k} ∧ θ\overline{q}, i = \overline{1, n}, k = \overline{1, n}, q = \overline{1, n}. (2.5)

By L(T M) we denote Lagrange’s Grassmanian associated with the symplectic fibration T M →^{p} M. The map q : U(M) → L(T M), defined as q((ε_{1}, …, ε_{n})) = e_{1} ∧ ⋯ ∧ e_{n} (here the multivector e_{1} ∧ ⋯ ∧ e_{n} is identified with the Lagrange n-plane span {e_{1}, …, e_{n}}), defines the principal bundle with the structure group O(n). On the space U(M) there exists a 1-form

\overline{ψ} = −( ∑_{k=1}^{n} iπ_{k}^{k} + 2 Im ( ∑_{i,k=1}^{n} T_{\overline{i}\overline{k}}^{i}θ\overline{k})),  (2.6)

which can be expressed as a pullback form by the map q. Indeed, the 1-form ∑_{k} π_{k}^{k} vanishes on the fibres of q : U(M) → L(T M) and is invariant under

the action of the group O(n) on U(M). Therefore the form ∑_{k} iπ_{k}^{k} can be

pulled down on the L(T M). It is easy to verify that

∑_{i}(ε\overline{i}⌟T^{i}) = ∑_{i,k} T_{\overline{i}\overline{k}}^{i}θ\overline{k} − ∑_{i,k} T_{\overline{k} \overline{i}}^{i}θ\overline{k} = 2 ∑_{i,k} T_{\overline{i} \overline{k}}^{i}θ\overline{k},

and if R_{α}(x) = R_{β}(x)A_{α}^{β}(x), where A_{α}^{β}(x) = (A_{j}^{i′})_{i′,j=\overline{1,n}} ∈ O(n), α, β ∈ I, then

∑_{i}(ε\overline{i}⌟T^{i}) = ∑_{i,i′} A_{i}^{i′}(ε\overline{i}′⌟T^{i}) = ∑_{i,i′,p} A_{i}^{i′}(ε\overline{i}′⌟T^{p′})A_{p′}^{i} = ∑_{i′}(ε\overline{i′}⌟T^{i′}).

This means that the form 2 Im(T_{\overline{i} \overline{k}}^{i}θ\overline{k}) can also be pulled down and so there exists a 1-form ψ on L(T M) such that

\overline{ψ} = q^{∗}ψ.  (2.7)

When M is a Hermitian manifold (T = 0), the 1-form ψ on L(T M) has a simpler form [8]

ψ = J df + dθ,  (2.8)

where f and θ are the locally defined functions on L(T M).

Lemma 2.2. The diagram is commutative:

[figure: A^{1}(T N); A^{1}(V N); A^{1}(L(T M)); M; N; ψ; α_{H}; J^{∗}; G^{∗}; i]

where i is the imbedding map and G is the Gaussian map.

Proof. Let x ∈ N, D be an open neighborhood of x in N and S_{D} be a local section S_{D} : D → U(M), S_{D}(x) = (ε_{1}(x), …, ε_{n}(x)). Let S_{D′} be the extension of S_{D} to the tubular neighborhood D′ of D in M; then a locally n-form on D′,

φ = Re(S_{D′}^{∗}(θ^{1} ∧ θ^{2} ∧ ⋯ ∧ θ^{n})),

is an n-form of comass 1 on D′ and φ|_{D} ≡ 1. If X ∈ V_{n}N, n ∈ N, we have

(X⌟dφ)(e_{1}, …, e_{n}) = ∑_{i=1}^{n}(−1)^{i}e_{i}(φ(X, e_{1}, …, ê_{i}, …, e_{n})) +

+ X(φ(e_{1}, …, e_{n})) +

+ ∑_{i<j}(−1)^{i+j}φ([e_{i}, e_{j}], X, …, ê_{i}, …, ê_{j}, …, e_{n}) +

+ ∑_{i=1}^{n}(−1)^{i}φ([X, e_{i}], e_{1}, …, ê_{i}, …, e_{n}).

Since φ has comass 1, X(φ(e_{1}, …, e_{n})) = 0, and X⌟φ ≡ 0 for all X⊥T_{n}N. Thus from the above equality we have

(X⌟dφ)(e_{1}, …, e_{n}) = ∑_{i=1}^{n}(−1)^{i}φ([X, e_{i}], e_{1}, …, ê_{i}, …, e_{n}) =

= − ∑_{i} g([X, e_{i}], e_{i}) = − ∑_{i} g(∇_{X}e′_{i} − ∇_{e_{i}}X, e_{i}) =

= ∑_{i} g(∇_{e_{i}}X, e_{i}) = −g(H, X).

In the fourth equality the fact that ∇_{X}⟨e′_{i}, e′_{i}⟩ = 0 is used, and e′_{i} denotes the locally defined vector fields on D′ which extend the vectors e_{i}, i = \overline{1, n}. By direct calculation we can obtain (S_{D})^{∗}(\overline{ψ})(JX) = (X⌟dφ)(e_{1}, …, e_{n}) = −α_{H} (X). Since q ∘ S_{D} = G, we have

α_{H} (X) = −G^{∗}ψ(JX) = −J^{∗}G^{∗}ψ(X).

Therefore

β = J^{∗}α_{H} = G^{∗}ψ.  (2.9)

The second structure equations of E. Cartan for the first canonical connection π = (π_{j}^{i})_{i,j=\overline{1,n}} have the form

Ω = dπ + π ∧ π,  (2.10)

where Ω is the curvature form for the connection π. Taking into account π_{j}^{i} = −\overline{π}_{i}^{j}, from (2.10) we have ∑_{k} Ω_{k}^{k} = − ∑_{k} \overline{Ω}_{k}^{k} and d(∑_{k} π_{k}^{k}) = ∑_{k} Ω_{k}^{k}. The form \frac{i}{2π} ∑_{k} Ω_{k}^{k} is real and can be pulled down on M. It is called the Chern

first form and denoted by c. By (2.6) we obtain

d\overline{ψ} = −2πp^{∗}c − 2d( Im ∑_{i,k} T_{\overline{i} \overline{k}}^{i}θ\overline{k}).  (2.11)

For an open covering U = {U_{α}}_{α∈I} of the manifold M there exists a commutative diagram

[figure: U(M); N; N_{α} = N ∩ U_{α}; L(T M); M; S_{U_{α}}; p; q; G; i; i_{α}; π]

Since p^{∗}c = q^{∗}π^{∗}c and G^{∗}π^{∗} = i^{∗}, we have

S_{U_{α}}^{∗}d\overline{ψ} = −2πc|_{N_{α}} − 2S_{U_{α}}^{∗}d( Im ∑ T_{\overline{i} \overline{k}}^{i}θ\overline{k}).

Now, taking into account equality (2.9) and S_{U_{α}}^{∗}q^{∗} = i_{α}^{∗}G^{∗}, we obtain

i_{α}^{∗}dβ = i_{α}^{∗}G^{∗}dψ = −2πc|_{N_{α}} − 2d(S_{U_{α}}^{∗} Im ( ∑_{i,k} T_{\overline{i} \overline{k}}^{i}θ\overline{k})). (2.12)

Thus β is the global 1-form on N, and the expression d(S_{U_{α}}^{∗} Im(∑_{i,k} T_{\overline{i} \overline{k}}^{i}θ\overline{k})

defines the global closed 2-form on the Lagrangian submanifold N and we denote it by γ. We have

Proposition 2.3. The Chern form c on the Lagrangian submanifold N in an almost Hermitian manifold M can be expressed by the sum

i^{∗}c = −\frac{1}{2π} dβ − \frac{1}{π} γ,  (2.13)

where i is the imbedding map i : N → M, and the forms β and γ depend on the mean curvature and torsion vector form, respectively.

Theorem 2.4. If M is a Hermitian manifold, then:

(i) the form π^{∗}c is an exact form on L(T M);

(ii) if N is a Lagrangian submanifold in M, then i^{∗}c = −\frac{1}{2π} dβ; (iii) the form ψ is a closed form exactly when c = 0; (iv) if β is closed on N (in particular, when N ∈ LH(M)), then i^{∗}c = 0;

(v) if M is the Kaehlerian manifold, then ψ is closed exactly when Ricci’s tensor is identicaly equal to zero.

Proof. From formulas (2.11), (2.9) we have

q^{∗}dψ = −2πp^{∗}c.

Taking into account π ∘ q = p, we obtain

q^{∗}(dψ + 2π · π^{∗}c) = 0.

As q : U(M) → L(T M) is a fibration, the above equation implies

π^{∗}c = −\frac{1}{2π} dψ.  (2.14)

By formula (2.14) it is clear that ψ is closed exactly when c = 0. Now we recall that the form c on the Kaehlerian manifold can be locally expressed by the formula (see [12])

c = \frac{1}{2π} ∑_{i,j} R_{i\overline{j}}dz^{i} ∧ dz\overline{j}.  (2.15)

Formulae (2.14), (2.15) prove assertions (i), (iii), (v). (2.13) immediately implies (ii) and (iv).

When M = ℂP^{n} and N ↪ ℂP^{n} is a Lagrangian manifold, the conditions of Theorem 2.4 are fulfilled for Chern forms of higher dimension on ℂP^{n}. This result follows from the well-known fact that the Chern forms on ℂP^{n} are calculated by the formula

c_{k} = m(k)c^{k}, k = \overline{1, n}, m(k) ∈ ℕ.

Now let X and Y be the sections of the bundle i^{∗}L(T M) on N, where N is a submanifold in M. By Theorem 1.5, for every closed 1-form φ on L(T M) there exists a 1-form δ on N such that

î^{∗}[φ] = π_{1}^{∗}[δ] + k([φ], X)σ_{X}, k([φ], X) ∈ ℝ;  (2.16)

here î is an imbedding map i^{∗}L(T M) \underrightarrow{_{i}} L(T M) and π_{1} is the projection of the bundle i^{∗}L(T M). Applying X^{∗} and Y^{∗} to (2.16), we obtain respectively

X^{∗}î^{∗}[φ] = [δ]; Y^{∗}î^{∗}[φ] = [δ] + k([φ], X)σ(i^{∗}(T M); X, Y). (2.17)

From the above equations we have

k([φ], X)σ(i^{∗}(T M); X, Y) = Y^{∗}î^{∗}[φ] − X^{∗}î^{∗}[φ].  (2.18)

In the particular case, when î^{∗}[φ] is not a pullback class from H^{1}(N; ℝ), we have k ≠ 0 and therefore (2.18) can be rewritten as

σ(i^{∗}(T M); X, Y) = \frac{1}{k([φ], X)} (Y^{∗}î^{∗}[φ] − X^{∗}î^{∗}[φ]).  (2.19)

Thus we have proved

Proposition 2.5. For an arbitrary closed 1-form φ on L(T M) (it is assumed that î^{∗}[φ] is not a pullback class from H^{1}(N; ℝ) by the map π_{1}), the Hörmander class σ(i^{∗}(T M); X, Y) can be expressed by formula (2.19).

Proposition 2.6. If N is a manifold from LH(M) and ψ is the closed 1-form on L(T M), then

î^{∗}[ψ] = k([ψ], G)σ_{G}.  (2.20)

Proof. For the 1-form ψ and the Gaussian section G (2.16) can be rewritten as

î^{∗}[ψ] = π_{1}^{∗}[δ] + k([ψ]; G)σ_{G}.

Applying G^{∗} to the formula and taking into account N ∈ LH(M) and Lemma 2.2, we conclude that [δ] = 0.

Now let M^{2n} = T^{∗}Q^{n} be a cotangent bundle and ω be a canonical symplectic structure on M^{2n}. M^{2n} can be considered as an almost Kaehlerian manifold with the fundamental 2-form ω. There is a fixed section A of the fibre bundle L(T M^{2n}) →^{π} M^{2n}, which is defined at every point m ∈ M^{2n} as a tangent plane at the point m of the fibre p^{−1}(p(m)), where p is a natural projection T^{∗}Q^{n} →^{p} Q^{n}.

For every Lagrangian submanifold N^{n} ↪^{i} M^{2n}, the Gaussian map G : N^{n} → i^{∗}L(T M^{2n}) is defined as previously. The following definition is equivalent to Definition 1.1.

Definition 2.7. The cohomology class σ(i^{∗}(T M); G, A) ∈ H^{1}(N; ℤ) is called the Maslov class of the Lagrangian submanifold N and we denote it by σ_{N}.

Theorem 2.8. Let N^{n} be a manifold from LH(M^{2n}). Then:

(i) if H^{1}(Q^{n}; ℝ) = 0 and ψ is a closed 1-form, then k([ψ]; A)σ_{N} = 0,

k([ψ], A) ∈ ℝ;

(ii) if H^{1}(Q^{n}; ℝ) = 0 and [ψ] ≠ 0, then σ_{N} = 0.

Proof. In condition (i) we have [ψ] = k([ψ]; A)σ_{A} so that k([ψ]; A)σ_{N} = −G^{∗}î^{∗}[ψ] = −[β] = 0.

If [ψ] ≠ 0, then k([ψ]; A) ≠ 0 and (ii) follows from (i).

Remark 2.9. When M = ℝ^{2n}, all conditions of Theorem 2.5 are satisfied. The form ψ is a closed 1-form; more precisely, ψ = dθ (see formula 2.8), where θ is the function from L(T ℝ^{2n}) = ℝ^{2n} × U(n)/O(n) to S^{1}. It is not difficult to calculate that in this case k([ψ]; A) = 2π. Therefore Theorem 3 in [8] (Fomenko’s conjecture) follows from Theorem 2.8 as a corollary not only for minimal Lagrangian submanifolds in ℝ^{2n} as in [8], but already for the manifolds from LH(M) as well.

By Remark 2.9 and Theorem 2.8 we are able to extend Fomanko’s conjecture for the manifolds from the class LH(M^{2n}). When the form ψ is a closed 1-form, this conjecture can be reduced equivalently to the exactness of the form A^{∗}[ψ]|_{N}.

Corollary 2.10. If M^{n} is a Kaehlerian manifold, with H^{1}(M^{n}; ℝ) = 0, and Ricci’s tensor is identically equal to zero, then for N ∈ LH(T M^{n}) we have k([ψ], A)σ_{N} = 0. In the particular case where [ψ] ≠ 0, we have σ_{N} = 0.

This corollary follows from Theorems 2.4 and 2.8.

Aknowledgement

The research described in this paper was made possible in part by Grant No. BJV000 from the International Science Foundation.

References

1. V. P. Maslov, Theory of perturbation and asymptotic methods. (Russian) Moscow University Press, Moscow, 1965.

2. V. I. Arnold, Characteristic class entering in quantization conditions. (Russian) Funkts. Anal. Prilozh. 1(1967), No. 1, 1–14.

3. D. B. Fuks, Maslov–Arnold characteristic classes. (Russian) Dokl. Akad. Nauk SSSR 178(1968), No. 2, 303–306.

4. V. V. Trofimov, Symplectic connections, index of Maslov and conjecture of Fomenko. (Russian) Dokl. Akad. Nauk SSSR 304(1989), No. 6, 1302–1305.

5. L. Hörmander, Fourier integral operators I. Acta Math. 127(1971), 79–183.

6. L. Hörmander, The calculus of Fourier integral operators. Prospects in Math., 33–57, Ann. of Math. Stud., 70, Princeton Univ. Press., Princeton, N. Y., 1971.

7. V. Guilleman and S. Sternberg, Geometric Asymptotics. Amer. Math. Soc., Providence, Rhode Island, 1977.

8. Le Hong Van and A. T. Fomenko, The criteria of minimality of Lagrangian submanifolds in a Kaehlerian manifold. (Russian) Mat. Zametki 4(1987), 559–571.

9. Z. A. Tevdoradze, One note on the calculation of Arnold–Maslov class of Liouville tori in cotangent bundles. (Russian) Uspekhi Mat. Nauk 43(1988), No. 5, 219–220.

10. Z. A. Tevdoradze, Calculation of Maslov class in the Lagrange and Kovalevskaya problem for the motion of a rigid body about a fixed point. (Russian) Bull. Acad. Sci. Georgia 141(1991), No. 1, 45–49.

11. E. Dyer, Cohomology theories. W. A. Benjamin Inc., New York, 1969.

12. A. Lichnerowicz, Théorie globale des connexions et des groups d’holonomie. Ed. Cremonese, Roma, 1955.

13. Le Hong Van, Minimal Φ-Lagrangian surfaces in almost Hermitian manifolds. (Russian) Mat. Sb. 180(1989), 924–936.

14. R. Harvey and H. B. Lawson, Calibrated geometries. Acta Math. 148(1982), 47–157.

(Received 28.11.1994; revised version 15.03.1995)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia