Lobachevskii Journal of Mathematics Vol. 3, 1999, 197–200 \begin{matrix}c \\ ◯ \end{matrix}M.A. Malakhaltsev

M.A. Malakhaltsev

THE LIE DERIVATIVE AND COHOMOLOGY OF G-STRUCTURES

(submitted by B.N.Shapukov)

Abstract. In [1], J.F. Pommaret constructed the so-called Spencer P -complex for a differential operator. Applying this construction to the Lie derivative associated with a general pseudogroup structure on a smooth manifold, he defined the deformation cohomology of a pseudogroup structure. The aim of this paper is to specify this complex for a particular case of pseudogroup structure, namely, for a first-order G-structure, and to express this complex in differential geometric form, i. e., in terms of tensor fields and the covariant derivative. We show that the Pommaret construction provides a powerful tool for associating a differential complex to a G-structure. In a unified way one can obtain the Dolbeaut complex for the complex structure, the Vaisman complex for the foliation structure [2], and the Vaisman–Molino cohomology for the structure of manifold over an algebra [3].

1. Preliminaries

Let M be a smooth n-dimensional manifold, L(M) → M the frame bundle of M, and 𝔛(M) the Lie algebra of vector fields on M.

For a Lie subgroup G ⊂ GL(n), consider the bundle E_{G}(M) = L(M)/G → M. Let s: M → E_{G}(M) be a section. The Lie derivative of s with respect to a vector field X ∈ 𝔛(M) is defined in the following way. For each X ∈ 𝔛(M), the flow ϕ_{t} of X induces the flow dϕ_{t} on L(M) whose projection onto E_{G}(M) gives a flow \overline{ϕ}_{t} on E_{G}(M). The tangent vector field d\overline{ϕ}_{t}/dt is the complete lift \overline{X} ∈ 𝔛(E_{G}(M)) of X. We denote by V E_{G} the vertical subbundle of T E_{G}(M); then the pullback bundle s^{∗}(V E_{G}) is a vector bundle over M, and for each p ∈ M there is defined an isomorphism Π_{p}: (V E_{G})_{s(p)} → (s^{∗}(V E_{G}))_{p}. Then

(L_{X}s)_{p} = Π(\overline{X}(s(p)) − ds_{p}(X(p))),

is the Lie derivative of s with respect to X at a point p ∈ M.

Supported by Grant of President of Russian Federation (support to leading scientific schools, N 96-15-96276).

From the definition it immediately follows that L_{X}s = 0 if and only if X is an infinitesimal automorphism of s, i.e., if the flow \overline{ϕ}_{t} preserves s.

Now let Q ⊂ L(M) be an integrable G-structure on M. Then Q determines a section s: M → E_{G}(M), and L_{X}s = 0 if and only if X ∈ 𝔛_{Q}(M), where 𝔛_{Q}(M) is the Lie algebra of infinitesimal automorphisms of Q. We consider the Lie derivative L_{X}s as a first-order differential operator D_{L}: T M → s^{∗}(V E).

In [1] for a differential operator D: Γ(ξ) → ξ′, where ξ, ξ′ are vector bundles and Γ(ξ) is the sheaf of sections of ξ, a differential complex was constructed (the Spencer P-complex)

0 → Θ → Γ(ξ) ⟶^{D} F^{0} → F^{1} → …,

where Θ is the kernel of D. This construction, applied to a Lie derivative associated with a general pseudogroup structure, gives the deformation complex of this structure. We shall consider a particular case of pseudogroup structure, a first-order G-structure, and express this complex in terms of tensor fields and covariant derivative.

2. P -complex for the Lie derivative

Let us recall the definition of the P -complex [1] associated to an involutive linear differential operator D = Φ ∘ j^{q}: ξ → ξ′, where ξ, ξ′ are vector bundles over M, and Φ: J^{q}ξ → ξ′ is a morphism of vector bundles.

Let ρ^{1}(Φ) be the first prolongation of Φ, R^{1} = im ρ^{1}(Φ), σ^{1}(Φ) the first prolongation of the symbol of Φ, G^{1} = im σ^{1}(Φ), and π^{1} : R^{1} → im(Φ) the canonical projection. Let us denote by Ω^{∗}(M) the algebra of differential forms on M. Then the groups of the P -complex are

F^{0} = im Φ, F^{q} = \frac{Ω^{q}(M) ⊗ F^{0}}{δ(Ω^{q−1}(M) ⊗ G^{1})},

where δ is induced by the Spencer algebraic operator. The differential D: F^{q} → F^{q+1} is defined as D([ω]) = [D_{S}ω^{1}], where ω ∈ Ω^{q} ⊗ F_{0}, ω^{1} ∈ Ω^{q} ⊗ R^{1} are such that π^{1}(ω^{1}) = ω, D_{S} is the Spencer differential operator (for details we refer the reader to [1]).

Now let us find the P -complex for the operator D_{L}.

Let us denote by E_{𝔤} the subbundle of the affinor bundle T _{1}^{1}(M) consisting of affinors whose matrices with respect to the frames from Q lie in the Lie algebra 𝔤 of G, and let F_{𝔤} = T _{1}^{1}(M)/E_{𝔤}.

Theorem 1. The P -complex of the Lie derivative is isomorphic to the complex (C^{q}(P), d), where

C^{q}(P) = \frac{Ω^{k}(M) ⊗ T M}{Alt(Ω^{k−1}(M) ⊗ E_{𝔤})},

and the differential d: C^{q}(P) → C^{q+1}(P) is induced by the differential operator D = Alt ∘∇, where Alt is the alternation and ∇ is the covariant derivative of a torsion-free connection adapted to Q (i. e. (Dω)_{i_{1}...}_{i_{q+1}}^{j} = ∇_{[i_{1}}ω_{i_{2}...}_{i_{q+1}]}^{j} with respect to local coordinates adapted to Q).

Proof. First we note that s^{∗}(V E) ≅ F_{𝔤}. Since Q is integrable, there exists an atlas A = {(U_{α}, x_{α}^{i})} adapted to Q. Then the section s: M → E(M) has the form s|_{U_{α}} = [{∂_{i}}], where {∂_{i}} is the natural frame field of (U_{α}, x_{α}^{i}) and brackets denote the frame equivalence class. For the Lie derivative, we have L_{X}s|_{U_{α}} = [∂_{i}X^{j}dx^{i} ⊗ ∂_{j}]. Therefore D_{L} = Φ ∘ j^{1}, where Φ: J^{1}(T M) → F_{𝔤}, Φ(X^{j}, X_{i}^{j}) = [X_{i}^{j}].

Let π: T _{1}^{1}(M) → F_{𝔤} be the projection, and we denote by the same letter π all the induced projections. From our calculation it follows that G^{1} = π(S^{2}(M) ⊗ T M) ⊂ T^{∗}(M)⊗F_{𝔤}. Therefore one can easily see that the projection π induces an isomorphism between the group C^{q} = Ω^{k}(M) ⊗ T M/Alt(Ω^{k−1}(M) ⊗ E_{𝔤}) and the group Ω^{k−1}(M) ⊗ F_{𝔤}/δ(Ω^{k−2}(M) ⊗ G^{1}) of the P -complex.

Since Q is integrable, there exists a torsion-free linear connection ∇ adapted to Q. By simple calculation, one can verify that the differential of the P - complex can be written in terms of ∇ in the following way: d[ω] = [Dω], where (Dω)_{i_{1}...}_{i_{q+1}}^{j} = ∇_{[i_{1}}ω_{i_{2}...}_{i_{q+1}]}^{j}. Note that this definition does not depend on the choice of ∇, because for two adapted connections ∇, ∇′ the deformation tensor T = ∇′ − ∇ is a section of the bundle S^{2}(M) ⊗ T (M) ∩ T^{∗}(M) ⊗ E_{𝔤}. Also using properties of the curvature tensor R and the fact that R is a section of the bundle Ω^{2}(M) ⊗ E_{𝔤}, one can directly verify that D^{2} = 0.

3. Examples

1. Let Q be a foliation structure on a smooth manifold M, and let Δ be the corresponding integrable distribution. Then E_{𝔤} = {A ∈ T _{1}^{1}(M) | A(Δ) ⊂ Δ}. Therefore C^{p} = Ω^{p}(Δ) ⊗ (T M/Δ). If (x^{i}, x^{α}) are adapted local coordinates, i.e., if Δ is given by the equations dx^{i} = 0, then d can be written locally as (dω)_{α_{1}...}_{α_{q+1}}^{i} = ∂_{[α_{1}}ω_{α_{2}...}_{α_{q+1}]}^{j}. Thus we arrive at Vaisman’s foliated cohomology [2].

2. Let Q be a symplectic structure given by a symplectic form θ. Then the subbundle E_{𝔤} consists of those affinors that are skew-symmetric with respect to θ, and using the isomorphism T _{1}^{1}(M) → T^{2}(M) determined by θ, we obtain

C^{q} = \frac{Ω^{q}(M) ⊗ T^{∗}(M)}{δ(Ω^{q−1} ⊗ S^{2}(M))},

where S^{2}(M) is the bundle of symmetric tensors of type (2, 0) and

(δω)_{i_{1}...i_{q−1}i_{q}i_{q+1}} = ω_{[i_{1}...i_{q−1}i_{q}]i_{q+1}}.

The adapted connection ∇ satisfies ∇ω = 0 (a symplectic connection), and the differential d: C^{q} → C^{q+1} is induced by the operator (D̃ω)_{i_{1}...i_{q}i_{q+1}i_{q+2}} = ∇_{[i_{1}}ω_{i_{2}...i_{q}i_{q+1}]i_{q+2}}. In particular, C^{0} ≅ Ω^{1}(M), C^{1}(M) ≅ Ω^{2}(M), and d: C^{0} → C^{1} is the exterior differential. Thus we find that the kernel of d is the Lie algebra of (locally) Hamiltonian vector fields.

3. For a complex structure, the complex (C^{q}, d) is the Dolbeaut complex [1]. And it was shown in [4] that for the structure of manifold over algebra, (C^{q}, d) is the Vaisman–Molino complex constructed in [3].

M.A. Malakhaltsev 200

References

[1] Pommaret, J.F. Systems of partial differential equations and Lie pseudogroups, Math. and Appl., 14 (1978).

[2] Vaisman I. d_{f} -cohomologies of Lagrangian foliations, Monatshefte für Math., 106 (1988), pp. 221–244.

[3] Shurygin, V.V. On the cohomology of manifolds over local algebras, Russian Mathematics (Izv. vuzov. Matematika), 9 (1996), pp. 71–85.

[4] Gaisin, T.I. Spencer complex for manifolds over an algebra, Proc. of Geom. Semin., No. 23, Kazan, 1997, pp. 33–41.

Department of Mathematics, Kazan State University, Kremlevskaya 18, Kazan 420008, RUSSIA E-mail address: Mikhail.Malakhaltsev@ksu.ru

Received July 27, 1999