Infinitesimal Einstein Deformations of Nearly K\"ahler Metrics

It is well-known that every 6-dimensional strictly nearly K\"{a}hler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under the Laplace operator $\Delta$. Let $E(a)$ denote the $a$-eigenspace of the restriction of $\Delta$ to $E$. If $M$ is compact, we prove that the moduli space of infinitesimal Einstein deformations of the nearly K\"{a}hler metric $g$ is naturally isomorphic to the direct sum $E(scal/15)\oplus E(scal/5)\oplus E(2scal/5)$. It is known that the last summand is itself isomorphic with the moduli space of infinitesimal nearly K\"{a}hler deformations.


Introduction
Nearly Kähler manifolds, introduced by Alfred Gray in the 70s in the framework of weak holonomy, are defined as almost Hermitian manifolds (M, g, J) which are not far from being Kähler in the sense that the covariant derivative of J with respect to the Levi-Civita connection of g is totally skew-symmetric.
The class of nearly Kähler manifolds is clearly stable under Riemannian products. Using the generalization by Richard Cleyton and Andrew Swann of the Berger-Simons holonomy theorem to the case of connections with torsion [3], Paul-Andi Nagy showed in [7] that every nearly Kähler manifold is locally a Riemannian product of Kähler manifolds, 3-symmetric spaces, twistor spaces over positive quaternion-Kähler manifolds and 6-dimensional nearly Kähler manifolds. This result shows, in particular, that genuine nearly Kähler geometry only occurs in dimension 6. It turns out that in this dimension, strict (i.e. non-Kähler) nearly Kähler manifolds have several other remarkable features: They carry a real Killing spinor -so they are in particular Einstein manifolds with positive scalar curvature -and they have a SU 3 structure whose intrinsic torsion is parallel with respect to the minimal connection (cf. [3]). A strict nearly Kähler structure on a compact 6-dimensional manifold with normalized scalar curvature scal = 30 is called a Gray structure.
In [5] we have studied the moduli space G of infinitesimal deformations of Gray structures on compact 6-dimensional manifolds, and showed that this space is isomorphic to the space E(12), where E(λ) denotes the intersection of the λ-eigenspace of the Laplace operator and the space of co-closed primitive (1, 1)-forms.
In the present paper we consider the related problem of describing the moduli space E of Einstein deformations of a Gray structure. Since every Gray structure is in particular Einstein, one has a priori E ⊃ G. Our main result (Theorem 5.1) gives a canonical isomorphism between E and the direct sum E(2) ⊕ E(6) ⊕ E(12).
The main idea is the following. The space of infinitesimal Einstein deformation on every compact manifold consists of trace-free symmetric bilinear tensors in a certain eigenspace of a second order elliptic operator called the Lichnerowicz Laplacian ∆ L . On a 6-dimensional nearly Kähler manifold, one can decompose every infinitesimal Einstein deformation H (viewed as symmetric endomorphism) into its parts h and S commuting resp. anti-commuting with J. Under the SU 3 representation, the space of symmetric endomorphisms commuting with J is isomorphic to the space of (1, 1)-forms and that of symmetric endomorphisms anti-commuting with J is isomorphic to the space of primitive (2, 1) + (1, 2)-forms and one may interpret the eigenvalue equation for ∆ L in terms of the forms ϕ and σ corresponding to h and S. The problem is that ∆ L does not commute with the isomorphisms above, because J is not parallel with respect to the Levi-Civita connection. It is thus natural to introduce a modified Lichnerowicz operator ∆ L , corresponding to the canonical Hermitian connection, better adapted to the nearly Kähler setting. It turns out that the eigenvalue equation for ∆ L translates, via∆ L , into a differential system for ϕ and σ involving the usual form Laplacian, which eventually yields the claimed result.

Preliminaries
2.1. Notation. In this section we introduce our objects of study and derive several lemmas which will be needed later. Here and henceforth, (M 2m , g, J) will denote an almost Hermitian manifold with tangent bundle T M, cotangent bundle T * M and tensor bundle T M. We denote as usual by Λ (p,q)+(q,p) M the projection of the complex bundle Λ (p,q) M onto the real bundle Λ p+q M. The bundle of g-symmetric endomorphisms SymM splits in a direct sum SymM = Sym + M ⊕ Sym − M, of symmetric endomorphisms commuting resp. anti-commuting with J. The trace of every element in Sym − M is automatically 0, and Sym + M decomposes further Sym + M = Sym + 0 M ⊕ id into its trace-free part and multiples of the identity.
2.2. Nearly Kähler manifolds. An almost Hermitian manifold (M 2m , g, J) is called nearly Kähler if where ∇ denotes the Levi-Civita connection of g. The canonical Hermitian connection ∇, defined by∇ is a U m connection on M (i.e.∇g = 0 and∇J = 0) with torsionT X Y = −J(∇ X J)Y . A fundamental observation, which -although not explicitly stated -goes back to Gray, is the fact that∇T = 0 on every nearly Kähler manifold (see [1]).
We denote as usual the Kähler form of M by ω := g(J., .). The tensor ψ + := ∇ω is totally skew-symmetric by (1). Moreover, since J 2 = −id, it is easy to check that ψ + (X, JY, JZ) = −ψ + (X, Y, Z). In other words, ψ + is a form of type (3, 0) + (0, 3). Let us now assume that the dimension of M is 2m = 6 and that the nearly Kähler structure is strict, i.e. (M, g, J) is not Kähler. The form ψ + can be seen as the real part of ā ∇-parallel complex volume form on M, so M carries an SU 3 structure whose minimal connection (cf. [3]) is exactly∇.
JX . We will sometimes identify the endomorphism A X with the corresponding form in Λ (2.0)+(0,2) M, e.g. in formula (8) below. By definition, we have ∇ X =∇ X + 1 2 A X on T M. In fact this relation can be extended on the whole tensor bundle, provided we use the right extension for A X .
where A * is the adjoint of A and {e i } is a local orthonormal basis of T M. Here, as well as in the remaining part of this paper, we adopt the Einstein convention of summation on the repeated subscripts.
Notice that by (2), the extensions of ∇ and∇ to the tensor bundle T M are related by∇ 2.4. Algebraic results on nearly Kähler manifolds. Assume that (M 6 , g, J) is a strict nearly Kähler manifold and that the metric on M is normalized such that scal = 30. From [4,Theorem 5.2] it follows that for every unit vector X, the endomorphism ∇ X J (which vanishes on the 2-plane spanned by X and JX) defines a complex structure on the orthogonal complement of that 2-plane. Then the same holds for A X (because The exterior bundle Λ 2 M decomposes into irreducible SU 3 components as follows: The map X → X ψ + identifies the first summand with T M, and h → g(Jh., .) defines an isomorphism between Sym + 0 M and the second summand. Similarly, one can decompose Λ 3 M into irreducible SU 3 components The first summand is a rank 2 trivial bundle spanned by ψ + and its Hodge dual * ψ + , and the isomorphism S → S ⋆ ψ + identifies Sym − M with the second summand.
If {e i } denotes a local orthonormal basis of T M, it is straightforward to check the following formulas: (8) (ii) If S is a section of Sym − M, then the following formula holds for every X ∈ T M: Proof. (i) An easy computation shows: (ii) The symmetric endomorphism A X⋆ S = A X • S − S • A X commutes with J and is trace-free. Consequently, by Schur's Lemma (cf. [5] for a more detailed argument) Notice that if X ♭ is the 1-form corresponding to X (which we usually identify with X), then f ⋆ X ♭ = −(f X) ♭ for every symmetric endomorphism f . We then compute:

The curvature operator
Let (M n , g) be a Riemannian manifold. The curvature operator R : , for any vector fields X, Y, U, V on M, identified with the corresponding 1-forms via the metric. In a local orthonormal frame {e i } it can be written as Using the identification of 2-vectors and (skew-symmetric) endomorphisms given by Notice that a manifold with curvature operator R = cid has Ricci curvature c(n−1) and in particular the curvature operator of the sphere is a positive multiple of the identity.
Let EM be the vector bundle associated to the bundle of orthonormal frames via some representation π : SO(n) → Aut(E). Every orthogonal automorphism f of T M defines in a canonical way an automorphism of EM, denoted, by a slight abuse of notation, π(f ). The differential of π maps skew-symmetric endomorphisms of T M (or equivalently elements of Λ 2 M) to endomorphisms of EM. The Levi-Civita connection of M induces a connection on EM whose curvature R E satisfies R E (X, Y ) = π * (R(X, Y )) = −π * (R(X ∧ Y )). Notice that π * (A) is exactly A ⋆ in the notation of Section 2.
We now define the curvature endomorphism q(R) ∈ End(EM) as For example the curvature endomorphism q(R) on the form bundle EM = Λ p M satisfies In particular we have q(R) = Ric on 1-forms, and q(R) = −Ric ⋆ − 2R on 2-forms.
It is easy to check that the action of q(R) is compatible with the identification of Λ 2 M with the space of skew-symmetric endomorphisms (and, more generally, with all SO n equivariant isomorphisms): Lemma 3.1. Let ϕ ∈ Λ 2 M be a 2-form with associated skew-symmetric endomorphism A, i.e. ϕ(Y, Z) = g(AY, Z) for any vector fields Y, Z. Then We now return to the case of a 6-dimensional strict nearly Kähler manifold (M 6 , g, J) with scalar curvature scal = 30.
Let R be the curvature of the Levi-Civita connection and letR be the curvature of the canonical Hermitian connection∇. The following relation between R andR is implicitly contained in [4].
Proof. The stated formula follows using equation (3.1) and the polarization of equation (5.1) from [4]. Note that there is a different sign convention for the curvature tensors in [4].
The Ricci curvature ofR satisfies Ric = 4g. This follows from the formula above and the fact that (M 6 , g, J) is Einstein with Ric = 5g.
Replacing R byR in formula (14) yields a curvature endomorphism q(R). It is easy to check that the curvature operator with respect to∇, denoted byR, is a section of Sym(Λ 3 , we see that q(R) preserves all tensor bundles associated to SU 3 representations. Moreover, a straightforward computation using the fact that α ⋆ J = 0 and α ⋆ ψ + = 0 for every α ∈ su 3 yields: for every sections h ∈ Sym + M and S ∈ Sym − M, where ϕ denotes the (1, 1)-form defined by ϕ = g(hJ., .).
The following lemma describes the difference q(R) − q(R). It is an immediate consequence of the curvature formula in Lemma 3.2. We will denote with Cas = 1 2 (e i ∧ e j ) ⋆ (e i ∧ e j ) ⋆ the Casimir operator of so(n) acting on the representation E and at the same time the corresponding endomorphism of EM.
In the remaining part of this section we will apply Lemma 3.3 in order to compute q(R) − q(R) on certain spaces of endomorphisms and forms.
Finally we may substitute the Casimir eigenvalues and our explicit expressions into the formula of Lemma 3.3. Recall that in the normalization with {e i ∧ e j } as orthonormal basis of Λ 2 M ∼ = so(T M), the Casimir operator acts as −p(n − p)id on Λ p M, and as −2nid on SymM. Hence we obtain for n = 6:

Comparing rough Laplacians
Let (M, g, J) be a strict nearly Kähler manifold with Levi-Civita connection ∇ and canonical Hermitian connection∇. In this section we compare the actions of the rough Laplacians ∇ * ∇ and∇ * ∇ on several tensor bundles.
We will perform all calculations below at some fixed point x ∈ M using a local orthonormal frame {e i } which is ∇-parallel at x. On any tensor bundle on M we can write ∇ * ∇ = −∇ e i ∇ e i and because∇ e i e i = ∇ e i e i − 1 2 J(∇ e i J)e i = 0, we also havē ∇ * ∇ = −∇ e i∇ e i . We are interested in the operator P := ∇ * ∇ −∇ * ∇ . Using (6) and the fact that the tensor A := J∇J is∇-parallel, we have We now compute the action of the two operators occurring in the previous formula on several tensor bundles which are of interest in the deformation problem. (17) Proof. Let X ∈ T M be a tangent vector. Since A X and ϕ are 2-forms of type (2, 0) + (0, 2) and (1, 1) respectively, we get We then compute = −8ϕ − e j e k (ω 2 ϕ(e k , e j )) = −8ϕ − 2e j (Je k ∧ ωϕ(e k , e j )) = −8ϕ − 2ωϕ(e k , Je k ) + 2Je k ∧ Je j ϕ(e k , e j )) = −8ϕ + 2e k ∧ e j ϕ(e k , e j )) = −8ϕ + 4ϕ = −4ϕ.
Lemma 4.2. The following relations hold: Proof. Simple application of the Schur Lemma, taking into account the decomposition of the exterior bundles into irreducible components with respect to the SU 3 action.
Here δ denotes the co-differential on exterior forms and the divergence operator whenever applied to symmetric endomorphisms.
Proof. Since A e i anti-commutes with J and∇ e i ϕ is of type (1, 1), it follows that A e i ⋆∇e i ϕ is a form of type (2, 0) + (0, 2), so there exists a vector field α such that A e i ⋆∇e i ϕ = α ψ + . In order to find α, we use the relation (α ψ + ) ∧ ψ + = α ∧ ω 2 (see [5]) and compute: so α = −Jδϕ, thus proving (26). Using the fact that ψ + is∇-parallel, we get: This proves (27). We next use (23) to write In order to check (29) we first compute Let B denote the endomorphism of T M corresponding to the 2-form δ(S ⋆ ψ + ) + δS ψ + . By the calculation above we get Replacing X by JX yields (29).
The last term is a section of Λ (1,1) 0 M since both∇ and q(R) preserve this space.

The moduli space of Einstein deformations
We now have all the ingredients for the main result of this paper: Theorem 5.1. Let (M 6 , g, J) be a Gray manifold. Then the moduli space of infinitesimal Einstein deformations of g is isomorphic to the direct sum of the spaces of primitive co-closed (1, 1)-eigenforms of the Laplace operator for the eigenvalues 2, 6 and 12.
Proof. Let g be an Einstein metric with Ric = Eg. From [2], Theorem 12.30, the space of infinitesimal Einstein deformations of g is isomorphic to the set of symmetric trace-free endomorphisms H of T M such that δH = 0 and such that ∆ L H = 2EH, where ∆ L = ∇ * ∇ + q(R) is the so-called Lichnerowicz Laplacian ∆ L . Remark that q(R) = 2R • + 2Eid in the notation of [2].
In our present situation the Einstein constant equals E = 5, so the space of infinitesimal Einstein deformations of g is isomorphic to the set of H ∈ SymM with δH = 0 = trH such that Let h := pr + H and S := pr − H denote the projections of H onto Sym ± M. We define the primitive (1, 1)-form ϕ(., .) := g(Jh., .) and the 3-form σ := S ⋆ ψ + . The key idea is to express (35) in terms of an exterior differential system for ϕ and σ. Using Corollary 3.5 and Corollary 4.4, (35) becomes where s is the section of Sym − M defined in the second part of (33). Since the operator (∇ * ∇ + q(R)) preserves the decomposition SymM = Sym + M ⊕ Sym − M, the previous equation is equivalent to the system Taking the composition with J and using (15), the first equation of (36) becomes Similarly, taking the action on ψ + and using (15) and the definition of s, the second equation of (36) becomes Notice that δh = δH − δS = 0, which can also be seen by examining the algebraic types in equation (38). From (30) we get so finally the system (36) is equivalent to Using Corollary 3.5 together with the equations (31) and (32) (keeping in mind that δS = 0 and δϕ = 0) we get the two identities (∇ * ∇ + q(R))ϕ = (∇ * ∇ + q(R))ϕ and (∇ * ∇ + q(R))σ = (∇ * ∇ + q(R))σ. Hence the classical Weitzenböck formula for the Laplace operator on forms implies that (39) is equivalent to Lemma 5.2. Let E(λ) be the λ-eigenspace of ∆ restricted to the space of co-closed primitive (1, 1)-forms. Then the space of solutions of the system (40) is isomorphic to the direct sum E(2) ⊕ E(6) ⊕ E(12). The isomorphism can be written explicitly as Proof. The first thing to check is the fact that Φ and Ψ take values in the right spaces.
Finally, it is straightforward to check that Φ and Ψ are inverse to each other. This proves the lemma and the theorem.
Computations of the Laplace spectrum using the Peter-Weyl theorem show that E(2) and E(6) vanish on each of these spaces. Moreover, E(12) vanishes on CP 3 and on SU 2 × SU 2 , and is 8-dimensional on F (1, 2) (cf. [6]). As a consequence of these facts, we deduce: (1) every infinitesimal Einstein deformation of the 6-dimensional 3-symmetric spaces is an infinitesimal Gray deformation (cf. [5]); (2) the nearly Kähler structure on CP 3 and on SU 2 × SU 2 is rigid; (3) there is an 8-dimensional space of infinitesimal deformations of the nearly Kähler structure on F (1, 2).