J. London Math. Soc. (2) 104 (2021) 1795–1811 doi:10.1112/jlms.12475 Deformations of nearly G2 structures Paul-Andi Nagy and Uwe Semmelmann Abstract We describe the second order obstruction to deformation for nearly G2 structures on compact manifolds. Building on work of Alexandrov and Semmelmann, this allows proving rigidity under deformation for the proper nearly G2 structure on the Aloff–Wallach space N(1, 1). 1. Introduction Consider a compact oriented manifold (M7, vol). A G2 structure on M is a 3-form ϕ on M which is stable in the sense of [13] and compatible with the orientation choice. Such a structure induces in an unique way a Riemannian metric gϕ on M , with respect to which we consider ψ := �gϕϕ. The G2 structure is called (strictly) nearly G2 provided that dϕ = τ0ψ (1.1) for some τ0 ∈ R×. It is a well established result [2] that nearly G2 structures are in 1 : 1 correspondence with Riemannian metrics in dimension 7 admitting Killing spinors. In particular gϕ is an Einstein metric of positive scalar curvature, a fact which further drives the research in this area. The nearly G2 structure is called proper if aut(M, gϕ) ⊆ aut(M,ϕ); equivalently gϕ is required to admit a one-dimensional space of Killing spinors. The main classes of examples known are • homogeneous, as classified in [11], including the Aloff–Wallach spaces N(k, l) • obtained from a canonical variation [11, 12] of a 3-Sasaki metric in dimension 7. A distinguished rôle is played by the Aloff–Wallach space N(1, 1) which supports a 3-Sasaki metric and therefore a nearly G2 structure of proper type in the canonical variation. See also [5] for a classification of invariant G2 structures (with torsion) on Aloff–Wallach spaces. The deformation theory of (proper) nearly G2 structures, which is a potential tool for producing new examples is the main focus in this paper. Some evidence in this direction is supported by the fact that 3-Sasaki metrics in dimension 7 containing T3 in their automorphism group can be deformed to Sasaki-Einstein structures [7]. According to [1], infinitesimal deformations of nearly G2 structures correspond to the kernel F4 of Δgϕ − τ2 0 acting on Ω4 27(M, gϕ) ∩ ker d. Those are actually deformations which are normalised to lie, up to the action of the diffeomorphism group, in the Ebin slice for Riemannian metrics on M . We consider the cubic polynomial K : F4 → F∗ 4 , K(α)γ := ∫ M Q2(α) ∧ γ. (1.2) Here Q2 is the quadratic form associated to an explicit bilinear form b2 : Λ4 27 × Λ4 27 → Λ3 27 between G2-representation spaces. Received 24 July 2020; revised 7 April 2021; published online 22 June 2021. 2020 Mathematics Subject Classification 53C10, 53C25 (primary). This research has been financially supported by the Special Priority Program SPP 2026 ‘Geometry at Infinity’ funded by the DFG. C� 2021 The Authors. Journal of the London Mathematical Society is copyright C�London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. http://creativecommons.org/licenses/by/4.0/ 1796 PAUL-ANDI NAGY AND UWE SEMMELMANN The main result of this paper is the following Theorem 1.1. Let (M7, ϕ) be a compact manifold equipped with a proper nearly G2 structure. The set of infinitesimal deformations which are unobstructed to second order is parametrised by K−1(0). Here elements α ∈ F4 are called unobstructed to second order provided they arise from a second order Taylor series of nearly G2 structures. This notion models, at order two, instances of deformation by smooth curves of nearly G2 structures. We only work with G2 deformations of constant total volume which does not restrict generality by Moser’s theorem. In this situation, as it is well-known, those deformations can be parametrised algebraically by using the linearisation α �→ α̂ of Hitchin’s duality map [13]. Differentiating the structure equations (1.1) reveals that deformation theory is governed by the first order differential operator D : Ω4(M) → Ω4(M), Dα = d α̂− τ0α. This is overdetermined elliptic, thus has a well-defined Hodge theory which shows that (unnormalised) infinitesimal deformations lie in ker(D) and also that α ∈ ker(D) is unobstructed to second order provided that Q2(α) is L2-orthogonal to d� ker(D�). With these preliminary observations in hand, the proof of Theorem 1.1 consists in examining how D interacts with the type decomposition of Ω4(M). Here, a crucial observation we make is that a small modification of the Ebin slice for Riemannian metrics produces invariant subspaces for D and D�. This allows breaking those operators into blocks with very simple Hodge theory. For the Aloff–Wallach space N(1, 1) equipped with the canonical variation of the 3-Sasaki metric, we show that Theorem 1.1 provides a computationally efficient way to describe second order deformations. Theorem 1.2. All infinitesimal deformations of N(1, 1) are obstructed to second order. The first ingredient used for the proof is the representation theoretic description of F4 which turns out to be isomorphic to su(3) as an SU(3)-representation (cf. [1]). The second is the explicit computation of the bilinear form b2 restricted to F4. This is performed in detail in the last section of the paper. Related results have been proved by Foscolo in [10] for nearly Kähler manifolds in dimension 6 based on our earlier work [14, 15]. In particular he was able to show that the nearly Kähler metric on the flag manifold F (1, 2) has no non-trivial deformations. As it is the case in this paper, his work relies on the explicit parametrisation of curves of SU(3)-structures given by the linearisation of the duality map. Finally we would like to note that the same results as in our work were obtained by Dwivedi and Singhal in [9]. However, the details of the proofs in our work are quite different. The arguments in [9] follow very closely [10]. 2. Preliminaries 2.1. Linear algebra Consider a nearly G2 manifold (M,ϕ, vol) as above; the induced metric will be simply denoted by g in what follows. The metric g is recovered from the pair of G2-data (ϕ, vol) according to (v�ϕ) ∧ (w�ϕ) ∧ ϕ = −6g(v, w) vol . The Hodge star operator constructed from (g, vol) will be denoted by �. In many algebraic computations it is useful to express ϕ and ψ := �ϕ in an adapted frame. This is a local DEFORMATIONS OF NEARLY G2 STRUCTURES 1797 orthonormal basis {ek, 1 � k � 7} of one forms with respect to which ϕ = e123 + e145 − e167 + e246 + e257 + e347 − e356 ψ = e4567 − e1247 + e1256 − e2345 + e2367 − e3146 − e3157. As G2-representations the spaces of k-forms Λk with k = 3, 4 split into irreducible compo- nents as Λk = Λk 1 ⊕ Λk 7 ⊕ Λk 27, where the subscript indicates the dimension of the subspace. Accordingly, we write α = α1 + α7 + α27 whenever α ∈ Λk. Note that Λ4 1 = Rψ and Λ3 1 = Rϕ. Similarly the space of 2-forms splits as Λ2 = Λ2 7 ⊕ Λ2 14 as a G2-representation. The representation of G2 on R7, defining the tangent bundle will be denoted by T. Throughout this paper we use the metric to identify tangent vectors and 1-forms, as well as endomorphisms and (2,0)-tensors via A ∈ End T �→ g(A·, ·) ∈ Λ1. Consider the action (A,α) ∈ End T × Λ� given by A�α = Aei ∧ (ei�α), where {ei} is some orthonormal basis of T. Note that we use here and in the following the Einstein summation convention and sum over repeated indices. This action allows to define a G2-invariant linear isomorphism i : Sym2 0T → Λ3 27 via S �→ S�ϕ. A few well-known algebraic identities to be repeatedly used in this paper are �(S�ψ) = −S�ϕ and |i(S)|2 = 2|S|2 (2.1) i(S) ∧ (v1�ψ) ∧ v2 = 2g(Sv1, v2) vol (2.2) ϕ ∧ (v1�ψ) ∧ v2 = −4g(v1, v2) vol (2.3) (v�ϕ)�ψ = −3v ∧ ϕ (2.4) whenever S ∈ Sym2 0T and v, v1, v2 ∈ T. All these facts can either be proved by direct computation in an adapted frame or looked up in [1, 4 6]. Note that Bryant’s orientation convention in [4] is opposite to ours and that the isomorphism i differs from his by a factor of 1 2 . The inner product on Sym2T we work with here is 〈S1, S2〉 = tr(S1S2). Next we state several algebraic facts which will be used later on in the paper. Lemma 2.1. The following hold (i) If β ∈ Λ3 satisfies β ∧ (v�ψ) = 0 for all vectors v ∈ T, then β has to vanish. (ii) If α ∈ Λ4 satisfies α ∧ (v�ϕ) = 0 for all vectors v ∈ T, then α has to vanish. (iii) If α = λψ + V ∧ ϕ + α27 ∈ Λ4, then α ∧ (v�ψ) = −4g(V, v) vol for v ∈ T. Proof. (i) By a suitable contraction β ∧A�ψ = 0 for all A ∈ End T. Since ψ is a stable 4-form, that is, its GL(4,R)-orbit is open, the action of End T on ψ spans Λ4 and we get β ∧ Λ4 = 0 thus β = 0. Part (ii) follows similarly using that ϕ is a stable 3-form. (iii) We have (α27 + λψ) ∧ (v�ψ) = −(α27 + λψ) ∧ �(v ∧ ϕ) = −g(α27 + λψ, v ∧ ϕ) vol = 0. The claim follows from the algebraic identity ϕ ∧ (v�ψ) = −4v� vol. � Lemma 2.2. We have a linear map Λ4 �→ Λ3, α �→ α̂ uniquely determined from having α̂ ∧ (v�ψ) + ϕ ∧ (v�α) = 0 (2.5) for all v ∈ T. Moreover α̂ = − � α1 + �α7 − �α27. (2.6) This is checked by direct computation in an adapted frame, using sample elements. The uniqueness in the definition of α̂ follows by Lemma 2.1,(ii); if ϕ ∧ (v�α) = 0 for all v ∈ T then a contraction shows that α ∧ (v�ϕ) = 0 for all vectors v ∈ T and α has to vanish. As we will see by explicit calculation in Section 3.1 the algebraic map α �→ α̂ enters the parametrisation at order two of G2 deformations of constant total volume (see (3.1)). Finally, we need to deal with the natural G2-invariant polynomial introduced below. 1798 PAUL-ANDI NAGY AND UWE SEMMELMANN Proposition 2.3. There exists a symmetric bilinear form b2 : Λ4 × Λ4 → Λ3 uniquely determined from having b2(α1, α2) ∧ (v�ψ) + α̂1 ∧ (v�α2) + α̂2 ∧ (v�α1) = 0 (2.7) for all v ∈ T. Letting Q2(α) = b2(α, α) be the quadratic form associated to b2, then (i) Q2(Λ4 27) ⊆ Λ3 1 ⊕ Λ3 27 and (ii) the cubic polynomial Q : Λ4 27 → R given by Q(α) vol := Q2(α) ∧ α satisfies Q(α) = −2〈q(α, α), i−1(�α)〉 where the symmetric bilinear form q : Λ4 27 × Λ4 27 → Sym2T is defined by the equation q(α, α)(v1, v2) := 〈v1�α, v2�α〉, with respect to the form inner product. Proof. All statements follow from the following simple observation. Pick α ∈ Λ4 27 and compute α̂ ∧ (v1�α) ∧ v2 = − ∗ α ∧ (v1�α) ∧ v2 = −(v1�α) ∧ (v2 ∧ ∗α) = (v1�α) ∧ ∗(v2�α) = q(α, α)(v1, v2) vol . Since tr(q(α, α)) = 4〈α, α〉 we can split q = q0 + 4 7g ⊗ id according to Sym2T = Sym2 0T ⊕ Rid. Then (2.2) together with (2.3) imply that (i(q0(α, α)) − 2 7 |α|2ϕ) ∧ (v1�ψ) ∧ v2 = 2q(α, α)(v1, v2) vol . Comparing the last two displayed equations we see that Q2(α) := −i(q0(α, α)) + 2 7 |α|2ϕ satisfies the requirement in (2.7). As q is symmetric Q2 extends to a symmetric bilinear form on Λ4 27. Part (i) of the claim is thus proved. To prove (ii) we compute Q(α) = 〈Q2(α), �α〉 = −〈i(q0(α, α)), �α〉 = −2〈(q0(α, α)), i−1(�α)〉 = −2〈(q(α, α)), i−1(�α)〉 by taking into account (2.1) and that pure trace components can be ignored by type considerations. � At this stage a few remarks are in order. Remark 2.4. (i) Proposition 2.3 proves directly the existence for b2; an a priori proof of this fact stems from having b2 arising as the second derivative, in a suitable sense, of Hitchin’s duality map. (ii) In the last section of the paper it will be convenient to work with the cubic polynomial P : Λ3 27 → R given by P (β) = Q(�β). By part (ii) in Proposition 2.3 and some linear algebra this is computed from P (β) = 2〈p(β, β), i−1(β)〉 where the symmetric bilinear form p : Λ3 27 × Λ3 27 → Sym2T is determined from the equation p(β, β)(v1, v2) = 〈v1�β, v1�β〉. (iii) Another way of thinking about the bilinear form b2 is based on the following observation. After suitably contracting (2.7) and using (2.1) we obtain 〈b2(�β1, �β2), i(S3)〉 = 〈(S3)�β1, β2〉 + 〈(S3)�β2, β1〉 = 〈(S3)�(S1)�ϕ, (S2)�ϕ〉 + 〈(S3)�(S2)�ϕ, (S1)�ϕ〉 with �βk = i(Sk) ∈ Λ3 27, k = 1, 2 and S3 ∈ Sym2 0T. Using repeatedly relations of the type (S3)�(S1)�ϕ = (S1)�(S2)�ϕ + [S3, S1]�ϕ together with {F�ϕ : F ∈ Λ2} ⊆ Λ3 7 we see this is DEFORMATIONS OF NEARLY G2 STRUCTURES 1799 symmetric in S1, S2, S3. In other words P is induced by a G2-invariant tensor in Sym3(Λ3 27). Since G2 acts without fixed vectors on Λ3 27, this third order symmetric tensor actually belongs to Sym3 0(Λ 3 27). Classical invariant could then be used to recover P from octonian multiplication. 2.2. The Lie derivative Throughout this paper we systematically denote the various algebraic components in the codifferential with d� 7 α := (d� α)7 etc. for α ∈ Ω�(M), with respect to the type decompositions Ωk(M) = Ωk 1 ⊕ Ωk 7 ⊕ Ωk 27, k = 3, 4. The first objective is to render explicit the structure of Lie derivatives LXψ, X ∈ Γ(TM) according to the splitting Ω4(M) = Ω4 1 ⊕ Ω4 7 ⊕ Ω4 27. Let L : Γ(TM) → Γ(TM) denote the first order differential operator determined from d7 X = 1 3L(X)�ϕ. A second first order differential operator of relevance is the trace free part of the Lie derivative of the metric X ∈ Γ(TM) �→ SX = 1 2LXg + 1 7 (d� X) g ∈ Γ(Sym2 0TM) (2.8) as showed below. Lemma 2.5. We have LXψ = − 4 7 (d� X)ψ + (1 2L(X) − τ0 4 X) ∧ ϕ + (SX)�ψ (2.9) and d�(X ∧ ψ) = 3 7 (d� X)ψ + (− 1 2L(X) − 3τ0 4 X) ∧ ϕ + (SX)�ψ (2.10) whenever X ∈ Γ(TM). Proof. Recall the local expressions d = ei ∧∇ei and d� = −ei�∇ei where ∇ denotes the Levi-Civita connection of g. At the same time ∇Uψ = − τ0 4 U ∧ ϕ with U ∈ TM (see Proposition 2.4 in [1]). Direct computation based on these facts leads to d(X�ψ) = (∇tX)�ψ − τ0 4 X ∧ ϕ d�(X ∧ ψ) = (d�X)ψ + (∇X)�ψ − 3τ0 4 X ∧ ϕ where ∇tX denotes the transpose of ∇X ∈ End(T) with respect to g, that is, the difference of the symmetric and the anti-symmetric part of ∇X. All claims follow now by decomposing ∇X ∈ End T according to End T = Λ2 ⊕ Sym2T = Λ2 14 ⊕ Λ2 7 ⊕ Rid ⊕ Sym2 0T. Explicitly ∇X = 1 2 d14X + 1 2 d7X − 1 7 (d�X) 1TM + SX . Here we also use the identity (2.4) for computing the action of d7 X = 1 3L(X)�ϕ on ψ. � Recall that the divergence operator δ : Γ(Sym2TM) → Γ(TM) is defined according to the convention δS := −(∇eiS)ei. Below we work out how d� and δ relate via the algebraic isomorphism Sym2 0T → Λ4 27. In the rest of this paper we indicate with (·, ·) the L2-inner product on tensor fields. Lemma 2.6. For any symmetric tensor in S ∈ Γ(Sym2 0TM) we have d� 7(S�ψ) = 1 2 (δS)�ψ. (2.11) 1800 PAUL-ANDI NAGY AND UWE SEMMELMANN Proof. Pick X ∈ Γ(TM) and record that LXg = δ∗X where δ∗ is the formal adjoint of δ. Writing d� 7(S�ψ) = Z�ψ ensures that (d� 7(S�ψ), X�ψ) = 4(Z,X). At the same time (d� 7(S�ψ), X�ψ) = (S�ψ,d(X�ψ)) = −(S�ψ,d �(X ∧ ϕ)) = −(�S�ψ,d�(X ∧ ϕ)) = (S�ϕ, (SX)�ϕ) = 2(S, SX) = 2(S, δ∗X) = 2(δS,X) where we used (2.1) and that S is trace free. Hence Z = 1 2δS. � 3. Deformation theory 3.1. Curves of G2-structures Consider a compact manifold M7. Assume that it is equipped with a G2-structure (ϕ,ψ) ∈ Ω3(M) ⊕ Ω4(M) such that dϕ = τ0ψ. (3.1) Assume that (ϕt, ψt) ∈ Ω3(M) × Ω4(M) is a small time deformation of (ϕ,ψ) satisfying (3.1) and having constant volume vol ∈ Ω7(M). This can be assumed without loss of generality by Moser’s theorem. Consider the truncated Taylor series ψt = ψ + tψ1 + t2 2 ψ2 + O(t3). From here we obtain the truncated Taylor series for ϕt as follows. First differentiate the algebraic identity ϕt ∧ (X�ψt) = −4X� vol (3.2) for some vector field X. At t = 0 we obtain 0 = ϕ̇ ∧ (X�ψ) + ϕ ∧ (X�ψ1) = ϕ̇ ∧ (X�ψ) − ψ̂1 ∧ (X�ψ) by taking into account ψ̇ = ψ1 and (2.5). Since the wedge product with X�ψ is injective in the sense of Lemma 2.1,(i) we find ϕ̇ = ψ̂1. Differentiating at second order in (3.2) yields at t = 0 0 = ϕ̈ ∧ (X�ψ) + 2 ϕ̇ ∧ (X�ψ̇) + ϕ ∧ (X�ψ̈) = ϕ̈ ∧ (X�ψ) + 2 ψ̂1 ∧ (X�ψ1) + ϕ ∧ (X�ψ2) = (ϕ̈−Q2(ψ1) − ψ̂2) ∧ (X�ψ) after successive use of ψ̈ = ψ2 combined with (2.5) and the definition of the quadratic form Q2 from (2.7). Thus we find that ϕ̈ = Q2(ψ1) + ψ̂2 again by Lemma 2.1,(i). Summarising we obtain the following well-known parametrisation for ϕt essentially contained in [4, Proposition. 5]. Lemma 3.1. The truncated Taylor series for ϕt reads ϕt = ϕ + tψ̂1 + t2 2 (ψ̂2 + Q2(ψ1)) + O(t3). (3.3) This makes it straightforward to determine the differential operator which governs the deformation theory of nearly G2 structures by differentiating the structure equations. Let D : Ω4(M) → Ω4(M) be defined by Dα := dα̂− τ0α. In fact, differentiating at order two in dϕt = τ0ψt whilst using (3.3) yields at t = 0 Dψ1 = 0 and Dψ2 = − dQ2(ψ1). (3.4) This second equation prompts out the following DEFORMATIONS OF NEARLY G2 STRUCTURES 1801 Definition 3.2. An element ψ1 ∈ Ker(D) is unobstructed to second order provided there exists ψ2 ∈ Ω4(M) solving Dψ2 = − dQ2(ψ1). A first key step in the study of such objects is to observe that the first order differential operator D is overdetermined elliptic in the sense of [3, p. 462]. Indeed, its principal symbol which is given by ξ ∈ Λ1M �→ σ(ξ)α = ξ ∧ α̂ is injective. Thus, Hodge theory for overdetermined elliptic operators (see [3], Corollary 32,(b), p.464) makes that Ω4(M) = (ker(D�) ∩ Ω4(M)) ⊕ Im(D) orthogonally with respect to the L2-inner product. In particular the equation Dψ2 = − dQ2(ψ1) can be solved for ψ2 if and only if dQ2(ψ1) ⊥ ker(D�), that is Q2(ψ1) ⊥ d� ker(D�) (3.5) with respect to the L2-inner product. Rendering this constraint more transparent relies on the explicit computation of d� ker(D�) which is the main objective in the next section. 3.2. Structure of the operators D,D� The following general observation will be repeatedly used in this section. It stems from a first order version of Lemma 3.3 where the volume is allowed to vary with the deformation. Lemma 3.3. We have D(LXψ) = d(d�Xϕ) for X ∈ Γ(TM). Proof. Consider the flow (Φt)t∈R of X together with the nearly G2-structure defined by ϕt = Φ� tϕ and ψt = Φ� tψ, with volume form volt = Φ� t vol. Differentiating at t = 0 in the algebraic identity ϕt ∧ (Y �ψt) = −4Y � volt yields ϕ̇ ∧ (Y �ψ) + ϕ ∧ (Y �LXψ) = 4(d∗X)Y � vol = −(d∗X)ϕ ∧ (Y �ψ) where we used ˙volt = LXvol = −(d�X) vol, ψ̇t = LXψ at t = 0 and once again the algebraic identity. Then Lemma 2.2 leads to [ϕ̇− L̂Xψ + (d∗X)ϕ] ∧ (Y �ψ) = 0. By Lemma 2.1,(i) we obtain L̂Xψ = ϕ̇ + (d∗X)ϕ. This gives D(LXψ) = d L̂Xψ − τ0LXψ = d ϕ̇ + d(d∗X ϕ) − τ0ψ̇ and the claim follows by differentiating in dϕt = τ0ψt. � Remark 3.4. During the proof of Lemma 3.3 we have seen that the linearisation of Hitchin’s duality map satisfies L̂Xψ = LXϕ + (d�X)ϕ. This can be used to derive in a different way the relation between LXψ and d�(X ∧ ψ) in Lemma 2.5. Indeed d�(X ∧ ψ) = − � d(X�ϕ) = − � (LXϕ−X�dϕ) = − � LXϕ− τ0X ∧ ϕ. Then (2.10) follows from (2.9) while using the expression for the map α �→ α̂ from (2.6). Let K := aut(M, g) be the space of Killing vector fields and denote with K⊥ its L2-orthogonal within Γ(TM). We assume that the structure is proper in the sense of [11], that is, (M, g) admits a one-dimensional space of Killing spinors and in particular we have K ⊆ aut(M,ϕ). (3.6) Thus the Lie derivative of ϕ and ψ in the direction of Killing vector fields vanishes. This condition leads to an important simplification in subsequent calculations. A few more background facts we need are related to the L2-decomposition Γ(Sym2TM) = Im δ∗ ⊕ ker δ. 1802 PAUL-ANDI NAGY AND UWE SEMMELMANN This plays a key role in proving the Ebin slice theorem. In presence of an Einstein metric with positive scalar curvature it can be refined as follows, mainly due to Obata’s theorem. Proposition 3.5. Assume that the compact Einstein manifold (M7, g) with scal > 0 is not isometric to the standard sphere. Then K⊥ embeds in Γ(Sym2 0M) via X �→ SX , with SX defined in (2.8), and Γ(Sym2 0M) = {S ∈ Γ(Sym2 0TM) : δS ∈ K} ⊕ K⊥ (3.7) orthogonally with respect to L2-inner product. Proof. This can be extracted from [3, Theorem 4.60] and related material in that reference. In fact the statement is true in arbitrary dimension. We outline some of the arguments required mainly for the convenience of the reader. Computation using Bochner’s formula on 1-forms yields δ(LXg) = (Δ − 2Ric)X + d d�X with X ∈ Γ(TM). Thus δSX = δ( 1 2LXg + 1 7 d�X g) = 1 2ΔX − RicX + 1 2 dd�X − 1 7 dd�X =: AX (3.8) where the operator A : Γ(TM) → Γ(TM) is given by A = 1 2Δ − Ric + 5 14 dd�. Because g is an Einstein metric A preserves the splitting Ω1(M) = ker(d�) ⊕ Im d. On ker(d�) it is clear that ker(A) = K. At the same time we have A ◦ d = 6 7 d(Δ − scal 6 ) on the space C∞(M). As scal > 0 and g does not have constant sectional curvature Obata’s Theorem ensures that the operator Δ − scal 6 is invertible on C∞(M) thus A is invertible on K⊥. In particular the map X ∈ K⊥ �→ SX injective. To conclude, pick S ∈ Γ(Sym2 0TM) and decompose the vector field δS as δS = K + Y with K ∈ K and Y ∈ K⊥ and choose Z ∈ K⊥ such that AZ = Y . By (3.8) we have δ(S − SZ) = K + Y −AZ = K ∈ K and the claim is proved. � It turns out that the splitting (3.7) replicates at the level of 4-forms, in a way which is consistent with both the algebraic isomorphism Sym2 0T → Λ4 27 and the Hodge decomposition for D. To describe how this works consider the spaces E := {(LXψ)27 : X ∈ K⊥} and F := {α ∈ Ω4 27 : d� 7 α ∈ K�ψ}. For ease of reference we write K�ψ = {X�ψ : X ∈ K} and Ω4 7′ := K⊥ ∧ ϕ, as well as Ω4 1⊕7′ = Ω4 1 ⊕ Ω4 7′ . Lemma 3.6. We have an L2-orthogonal splitting Ω4(M) = (K ∧ ϕ) ⊕ (Ω4 1⊕7′ ⊕ E) ⊕F . (3.9) Proof. It is enough to check that Ω4 27 = E ⊕ F . Clearly E and F are L2-orthogonal. Indeed for α ∈ F and X ∈ K⊥, that is, (LXψ)27 ∈ E we compute (α, (LXψ)27) = (α,LXψ) = (α,d(X�ψ)) = (d� 7ψ,X�ψ) = 0 according to the definition of F . That E ⊕ F spans Ω4 27 is a consequence of Proposition 3.5 as outlined below. Let α = S∗ψ belong to Ω4 27. Decomposing the tensor field S = S̃ + SX according to (3.7) we find α = S̃�ψ + (SX)�ψ = S̃�ψ + (LXψ)27 after also taking into account (2.9). The last summand belongs, by definition, to E . To conclude we use (2.11) to check that d� 7(S̃�ψ) = 1 2δS̃�ψ ∈ K�ψ. Thus S̃�ψ ∈ F and the claim is proved. � DEFORMATIONS OF NEARLY G2 STRUCTURES 1803 The next objective is to determine how the decomposition above is acted on by the operators D and D�. This is based on the following Proposition 3.7. We have (i) the splitting (3.9) is preserved by the operators D and D� (ii) D is self-adjoint on F , that is D� |F = D|F . Proof. (i) First we check that D preserves Ω4 1 ⊕ (K⊥ ∧ ϕ) ⊕ E , the second summand of the splitting (3.9). By direct computation we obtain D(fψ) = − df ∧ ϕ− 2τ0fψ and D(X ∧ ϕ) = −LXψ − τ0X ∧ ϕ (3.10) with (f,X) ∈ C∞(M) × Γ(TM). Note that df ∈ K⊥ since Killing vector fields are co-closed. Clearly LXψ ∈ Ω4 1⊕7 ⊕ E . Moreover we see (LXψ,K ∧ ϕ) = (X�ψ,d�(K ∧ ϕ)) = −(X�ψ, �LKψ) = 0 for all K ∈ K by using (3.6). Thus LXψ ∈ Ω4 1⊕7′ ⊕ E . (3.11) Combining this observation with (3.10) leads to D(Ω4 1⊕7′) ⊆ Ω4 1⊕7′ ⊕ E . (3.12) It remains to consider the action of D on E . Here we find that D((LY ψ)27) = D(LY ψ) −D((LXψ)1⊕7′) ∈ Ω4 1⊕7′ ⊕ E , by using Lemma 3.3 and (3.12) above. The first summand in (3.9) is preserved by D because of the second equation in (3.10) and LXψ = 0 for Killing vector fields X. Finally we have to show that the operator D also preserves F , that is, the third summand of the decomposition (3.9). Here we take (α,X) ∈ F ×K⊥ and compute the L2-product (Dα, (LXψ)27) = −(d �α, (LXψ)27) = −(α, �d�((LXψ)27)) = (α,D((LXψ)27)). where we have taken into account that α ⊥ (LXψ)27, which is true since F ⊥ E and α ∈ F , (LXψ)27 ∈ E by assumption. As showed above DE ⊆ Ω4 1⊕7′ ⊕ E which is orthogonal to F . Thus the equation above shows DF ⊥ E , that is, we already have DF ⊆ Ω4 1⊕7 ⊕F . Next we will show that DF is orthogonal to Ω4 7′ , that is, to forms X ∧ ϕ with X ∈ K⊥. For a α ∈ F ⊂ Ω4 27 we have Dα = − d�α− τ0α. Hence, with X ∈ K⊥ we obtain (Dα,X ∧ ϕ) = −(d�α− τ0α,X ∧ ϕ) = −(�d�α, �(X ∧ ϕ)) = (d∗ 7α,X�ψ) = 0 by the defining condition of F . But Dα is also orthogonal to any 4-form X ∧ ϕ for X ∈ K. Indeed if X ∈ K we have LXψ = 0, since the structure is proper, and we obtain for any α ∈ Ω4 27 that (Dα,X ∧ ϕ) = −(d�α,X ∧ ϕ) = (d�α,X�ψ) = (α,d(X�ψ)) = (α,LXψ) = 0 . There remains to prove that DF is orthogonal to Ω4 1. Letting f be some function on M and α ∈ F we similarly compute (Dα, fψ) = −(d �α, fψ) = −(d�α, fϕ) = −(α,d(fϕ)) = −(α,df ∧ ϕ + fτ0ψ) = 0 . Thus DF ⊆ F as claimed. Since the splitting (3.9) is L2-orthogonal and preserved by D it must also be preserved by D�. Statement (ii) follows from �d� = d � on Ω4(M) and DF ⊆ F . � 1804 PAUL-ANDI NAGY AND UWE SEMMELMANN The infinitesimal deformation space of the nearly G2 structure is defined according to F4 := ker(D) ∩ Ω4 27. Rewriting F4 = {α ∈ Ω4 27 : � d �α = −τ0 � α} = {α ∈ Ω4 27 : d∗α = −τ0 � α} has several consequences. Firstly d� 7 vanishes automatically on F4, thus F4 ⊆ F ; secondly F3 := �F4 = {β ∈ Ω3 27 : �dβ = −τ0β} is exactly the space of infinitesimal deformations considered in [1]. Lastly, F4 is a subspace of the eigenspace of the Laplace operator acting on 4-forms for the eigenvalue τ2 0 . In particular F4 is finite dimensional. An important first consequence of Proposition 3.7 is the following Corollary 3.8. We have ker(D) = ker(D|Ω4 1⊕7′⊕E) ⊕F4 as well as ker(D�) = ker(D� |Ω4 1⊕7′⊕E) ⊕F4. Proof. We have already seen that D = −τ0 id on K ∧ ϕ. Thus D has no kernel on the first summand of (3.9) and the statement follows since D preserves the decomposition (3.9). The second part of the claim follows from having the restriction of D to F self-adjoint. � It is now straightforward to determine the action of D on Ω4 1⊕7′ ⊕ E based on a specific parametrisation of the latter space. The main observation here is that Proposition 3.9. We have an identification map (f,X, Y ) ∈ C∞(M) ⊕K⊥ ⊕K⊥ �→ fψ + X ∧ ϕ + LYψ ∈ Ω4 1⊕7′ ⊕ E (3.13) with respect to which D(f,X, Y ) = (τ0 d� Y − 2τ0f,d(d� Y − f) − τ0X,−X). (3.14) Proof. For ease of reference indicate with ι the map in (3.13). Clearly ι(f,X, Y ) = 0 forces (LY ψ)27 = 0 thus SY = 0 by (2.9). By Proposition 3.5 it follows that Y = 0 which ensures that f and X vanish as well. That ι is surjective follows, via the definition of E , from (LXψ)27 = LXψ − (LXψ)1⊕7′ whenever X ∈ K⊥, which allows absorbing the 27-component of LXψ into E . Note that we also use (3.11) to see that LXψ has no component on K ∧ ϕ. The claim in (3.14) is now granted by Lemma 3.3 and (3.10). � An easy argument based on (3.14) shows that ker(D) = {LXψ : X ∈ K⊥,d� X = 0} ⊕ F4. (3.15) Indeed by Corollary 3.8 we only have to study D on Ω4 1⊕7′ ⊕ E . Here we use the identification given in (3.13). Then (3.14) implies that D(f,X, Y ) = 0 if and only if X and f vanish and Y ∈ K⊥ is coclosed. Moreover under the identification (3.13) the element (0, 0, Y ) is then mapped to LY ψ. In particular, this decomposition shows that infinitesimal deformations can be normalised, up to the action of volume preserving diffeomorphisms, to lie in F4. This is consistent with the approach in [1] where the first variation of nearly G2 metrics has been normalised to lie in the DEFORMATIONS OF NEARLY G2 STRUCTURES 1805 Ebin slice for Einstein metrics, thus leading to the identification of the space of infinitesimal deformations with F3 = �F4. 3.3. Computation of ker(D�) Computing the kernel of D� on Ω4 1⊕7 ⊕ E requires a bit more work as the map in (3.13) is not an isometry with respect to the canonical L2-inner product on C∞(M) ×K⊥ ×K⊥. The restriction of D� to Ω4 1⊕7′ ⊕ E can be understood similarly to the restriction of D, using a slightly different parametrisation, compared to (3.13), of the space Ω4 1⊕7′ ⊕ E . Lemma 3.10. The spaces spanned by d(K ∧ ϕ) and K⊥ ∧ ψ are L2-orthogonal. Proof. Take K ∈ K; since LKψ = 0 we have d7 K = τ0 6 K�ϕ as guaranteed by (2.9). Thus d(K ∧ ϕ) = dK ∧ ϕ − τ0K ∧ ψ = (d14 K) ∧ ϕ + (d7 K) ∧ ϕ− τ0K ∧ ψ = �d14 K + τ0 6 (K�ϕ) ∧ ϕ − τ0K ∧ ψ by taking into account that Λ2 14 = {α ∈ Λ2 : α ∧ ϕ = �α}. Hence the scalar product reads 〈Y ∧ ψ,d(K ∧ ϕ)〉 = 〈�(Y ıϕ),d(K ∧ ϕ)〉 = τ0 6 〈�(Y ıϕ), (K�ϕ) ∧ ϕ〉 − τ0〈Y �ϕ,K�ϕ〉 = − 4τ0 3 〈Y ıϕ,Kıϕ〉 = −4τ0g(K,Y ) with Y ∈ Γ(TM) and the claim follows by integration, whilst taking Y ∈ K⊥. � Proposition 3.11. We have ker(D�) = {τ0Y ∧ ϕ + d�(Y ∧ ψ) : Y ∈ K⊥,d� Y = 0} ⊕ F4. Proof. Choose α ∈ Ω4 1⊕7′ ⊕ E . Since we are ultimately interested in the space spanned by d�α it is convenient to parametrise α = fψ + X ∧ ϕ + d�(Y ∧ ψ) (3.16) with (f,X, Y ) ∈ C∞(M) ×K⊥ ×K⊥. The existence proof relies on (2.10) which ensures that d� 27(Y ∧ ψ) = (LYψ)27 and on having d�(Y ∧ ψ) orthogonal to K ∧ ϕ, as granted by Lemma 3.10. Uniqueness is entirely similar to the argument used to establish (3.13). Because d ◦D = −τ0 d we have D� ◦ d� = −τ0 d� on Ω5(M). On the other hand, using (2.6) of Lemma 2.2 and an easy L2-orthogonality argument we compute the three components of D∗(X ∧ ϕ) in Ω4(M) as D� 1(X ∧ ϕ) = (LXψ)1, D� 7(X ∧ ϕ) = −(LXψ)7 − τ0 X ∧ ϕ, D� 27(X ∧ ϕ) = (LXψ)27. This leads to D�(X ∧ ϕ) = d�(X ∧ ψ) + 1 7 (d�X)ψ after comparing the type components in LXψ and d�(X ∧ ψ) according to (2.9) and (2.10). At the same time D�(fψ) = df ∧ ϕ− 2τ0fψ by direct computation. Thus having D�α = 0 for a form α ∈ Ω4 1⊕7′ ⊕ E reads, in terms of the parametrisation in (3.16), d�((X − τ0Y ) ∧ ψ) + df ∧ ϕ + (1 7 d� X − 2τ0f)ψ = 0. 1806 PAUL-ANDI NAGY AND UWE SEMMELMANN Since X − τ0Y and d f ∈ K⊥ we get X = τ0Y,d f = 0 as well as d∗ X = 14τ0f . Clearly this entails f = d� X = 0 and the claim is proved. � In particular, we obtain Corollary 3.12. d� ker(D�) = {d�(Y ∧ ϕ) : Y ∈ K⊥,d� Y = 0} ⊕ F3. (3.17) Proof. Because of Proposition 3.11 we only need to check that d� F4 = F3 which follows from having d� α = −τ0 � α whenever α belongs to F4. � Remark 3.13. For the non-proper nearly G2 structures supported by 3-Sasaki and Sasaki Einstein metrics parts of the infinitesimal deformation space F4 have been explicitly computed in cohomological terms by van Coevering [7]. 4. Proof of Theorem 1.1 To capture explicitly the properties of the subset of ker(D) consisting of second order unob- structed deformations, we consider again the Kuranishi-type map K : F4 → F∗ 4 as introduced in (1.2). Recall that the map K depends quadratically on its first argument. We see that a 4-form α belongs to the zero locus K−1(0) if and only if the corresponding 3-form Q2(α) is orthogonal to the space F3. We first establish the following preliminary Lemma 4.1. Whenever α ∈ F4 we have d7(Q2(α)) = 1 4 d |α|2 ∧ ϕ. Proof. X ∈ Γ(TM). Differentiate the defining equation Q2(α) ∧ (X�ψ) + 2α̂ ∧ (X�α) = 0 (4.1) in direction of ei to obtain ∇ei(Q2(α)) ∧ (X�ψ) + Q2(α) ∧ (X�∇eiψ) + 2∇ei α̂ ∧ (X�α) + 2α̂ ∧ (X�∇eiα) = 0. Note that ei ∧ (X�∇eiα) = ∇Xα−X�dα = ∇Xα since d α̂ = τ0α forces dα = 0. Similarly we have ei ∧ (X�∇eiψ) = ∇Xψ since ψ is closed. Taking the exterior product with ei in the displayed equation above whilst taking into account that ∇Xψ = − τ0 4 X ∧ ϕ we arrive at d(Q2(α)) ∧ (X�ψ) + τ0 4 Q2(α) ∧X ∧ ϕ + 2d α̂ ∧ (X�α) − 2α̂ ∧∇Xα = 0. As Dα = 0 we have d α̂ ∧ (X�α) = τ0α ∧ (X�α) = 0 since α ∧ α ∈ Ω8(M) = 0 and α has even degree. Having α ∈ Ω4 27 ensures that α̂ = − � α thus α̂ ∧∇Xα = −g(α,∇Xα) vol = − 1 2 d |α|2(X) vol . Summarising we obtain d(Q2(α)) ∧ (X�ψ) + τ0 4 Q2(α) ∧X ∧ ϕ + (d |α|2)(X) vol = 0. (4.2) The second (algebraic) summand vanishes since Q2(α) is orthogonal to Λ3 7 due to Proposi- tion 2.3,(i). The claim finally follows from (4.2) and Lemma 2.1,(iii). � The full description of normalised infinitesimal deformations which are unobstructed to second order is contained below. DEFORMATIONS OF NEARLY G2 STRUCTURES 1807 Theorem 4.2. An element ψ1 ∈ F4 is unobstructed to second order if and only if ψ1 ∈ K−1(0). (4.3) Proof. As we already have remarked in (3.5) an infinitesimal deformation described by ψ1 is unobstructed to second order if and only if Q2(ψ1) ⊥ d� ker(D�). The computation of the latter space in (3.17) leads to Q2(ψ1) ⊥ {d�(X ∧ ϕ) : X ∈ K⊥,d� X = 0} ⊕ F3, that is, to Q2(ψ1) ⊥ {d�(X ∧ ϕ) : X ∈ K⊥,d� X = 0} and Q2(ψ1) ⊥ F3. The second requirement is by the definition of K equivalent to ψ1 ∈ K−1(0), whereas the first is trivially satisfied by Lemma 4.1. Indeed, since d∗ X = 0 we have (Q2(ψ1),d∗(X ∧ ϕ)) = (d7(Q2(ψ1)), X ∧ ϕ) = 1 4 (d |α|2 ∧ ϕ,X ∧ ϕ) = (d |α|2, X) = 0. � This proves Theorem 1.2 in the Introduction. 5. The Aloff–Wallach space N(1, 1) The Aloff–Wallach space N(1, 1) can be described as the homogeneous space G/H = (SU(3) × SU(2))/(U(1) × SU(2)), where the subgroup U(1) is embedded into SU(3) × SU(2) as diag(eit, eit, e−2it, 1) for t ∈ R. The subgroup SU(2) is first embedded canonically into SU(3) and then diagonally into SU(3) × SU(2), see [1, p. 738] for details. On N(1, 1) we choose the Riemannian metric induced by − 1 24B, where B is the Killing form of the Lie algebra g = su(3) ⊕ su(2). In other words we consider G/H as a normal homogeneous space. The Lie algebra g can be written as g = h⊕m. Here h is the Lie algebra of H and m = h⊥ is its orthogonal complement which can be identified with the tangent space at the origin. The space m splits further as m = m3 ⊕m4 with m3 = span{e1, e2, e3} and m4 = span{e4, e5, e6, e7} where e1, . . . , e7 is an orthonormal basis of m explicitly given in matrix form in [1]. Note that m3 = su(2)o in the notation of that paper. With respect to this basis the nearly G2 structure on N(1, 1) is induced by ϕ ∈ Λ3m in the standard form of Section 2.1. By our previous work in [1] the space F3 = �F4 of infinitesimal deformations of this nearly G2 structure is canonically identified to the Lie algebra su(3) by means of an explicit homomorphism A ∈ Homh(su(3),Λ3 27m) as follows. Map ξ ∈ su(3) to a Λ3 27m valued function on G via g �→ A(g−1ξg). Since the SU(2)-part of G acts trivially on su(3) this actually lives on SU(3). The well-known identification Ω3 27(G/H) = C∞(G,Λ3 27m)H produces then an isomorphism ξ ∈ su(3) �→ βξ ∈ F3. With respect to this parametrisation the obstruction map K : F4 → F� 4 from Theorem 4.2 reads K(�βξ) � βξ = 1 vol(U(1)) ∫ SU(3) P (gξg−1) vol (5.1) where the polynomial P : su(3) → R is given by P (ξ) = 〈p(A(ξ), A(ξ)), i−1(A(ξ))〉 and the symmetric bilinear map p : Λ3 27m× Λ3 27m → Sym2(m) is defined according to p(γ1, γ2)(v1, v2) := g(v1�γ1, v2�γ2). This follows in an essentially algebraic way from Proposition 2.3 and Remark 2.4,(ii). The aim here is to show that all infinitesimal deformations in F4 are obstructed to second order. By an elementary harmonic analysis observation [10] only the SU(3)-invariant piece in the polynomial P can contribute to the integral (5.1). In turn this is determined by the scalar product of P with the unique invariant cubic polynomial on su(3) given by ξ �→ idet ξ. 1808 PAUL-ANDI NAGY AND UWE SEMMELMANN In the next section we will show that this scalar product is non-vanishing by first computing the polynomial P . Equivalently we obtain K−1(0) = {0} thus completing the proof of Theorem 1.2. 5.1. Computation of the obstruction polynomial In this section all computations are performed on m. It is convenient to write the G2 form ϕ and its Hodge dual ψ on m as ϕ = vol3 + ea ∧ ωa, ψ = vol4 − (e12 ∧ ω3 + e23 ∧ ω1 + e31 ∧ ω2) where we use the notation vol = e1234567, vol3 = e123, vol4 = e4567, as well as ω1 = e45 − e67, ω2 = e46 + e57, ω3 = e47 − e56 . For a = 1, 2, 3 we denote with Ia the skew-symmetric endomorphisms on m4 associated to the 2-form ωa via ωa(·, ·) = g(Ia·, ·). On basis elements these are determined from I1e4 = e5, I1e6 = −e7, I2e4 = e6, I2e5 = e7, I3e4 = e7 and I3e5 = −e6 and satisfy the relations I1I2 = −I2I1 = −I3. Moreover the forms ωa are anti-selfdual with ωa ∧ ωb = −2δab vol4. To outline how the map A : su(3) → Λ3 27m is explicitly build pick ξ ∈ su(3) and write ξ = ⎛ ⎜⎝ iv1 x1 + ix2 x3 + ix4 −x1 + ix2 iv2 x5 + ix6 −x3 + ix4 −x5 + ix6 iv3 ⎞ ⎟⎠ = ⎛ ⎜⎝ iv1 −z̄3 z2 z3 iv2 −z̄1 −z̄2 z1 iv3 ⎞ ⎟⎠ with v1 + v2 + v3 = 0 and z1 = −x5 + ix6, z2 = x3 + ix4, z3 = −x1 + ix2. Following the detailed description in [1] and using the matrix form of the basis elements {ek, 1 � i � 7} therein it follows that the map A then breaks into three pieces A(ξ) = sϕ̃ − 5 3y ∧ Ω + √ 5 3 √ 2 C(x). Here ϕ̃ := ϕ− 7 vol3 ∈ (Λ1m3 ⊗ Λ2m4) ⊕ Λ3m3, Ω := e45 + e67 ∈ Λ2m4 and y := v1−v2 2 e1 − x1e2 + x2e3 ∈ m3 x := x3e5 − x4e4 + x5e7 − x6e6 ∈ m4 s := v1+v2 2 . The last component in A(ξ) belongs to Λ3m4 ⊕ (Λ2m3 ⊗ Λ2m4) and reads C(x) = 3x� vol4 + e12 ∧ (x�ω3) + e23 ∧ (x�ω1) + e31 ∧ (x�ω2) = x�(4 vol4 −ψ). (5.2) In order to streamline the computations below we will write Iy = yaIa whenever y = yaea ∈ m3. The symmetric tensor product on vectors is defined according to the convention v � w = 1 2 (v ⊗ w + w ⊗ v). Finally we denote with J the complex structure on m4 defined via g(J ·, ·) = Ω(·, ·). The restriction of the quadratic map p to the subspace span{A(ξ) : ξ ∈ su(3)} ⊆ Λ3 27m is fully determined as follows. Lemma 5.1. The components of p(A(ξ), A(ξ)) are given by p(ϕ̃, C(x)) = −4Iax� ea ∈ m3 �m4 p(C(x), C(x)) = 2|x|2id3 + 10( |x|2id4 − x⊗ x) ∈ Rid3 ⊕ Sym2(m4) p(y ∧ Ω, C(x)) = 6y � Jx ∈ m3 �m4 p(ϕ̃, y ∧ Ω) = −JIy ∈ Sym2 0m4 p(ϕ̃, ϕ̃) = 38id3 + 3id4. DEFORMATIONS OF NEARLY G2 STRUCTURES 1809 The proof is by direct calculation using only the behaviour of the 3-forms of type ϕ̃, y ∧ Ω and C(x) with respect to m = m3 ⊕m4. The last piece of information needed in order to conclude is Lemma 5.2. The three endomorphisms corresponding to the summands of A(ξ) are i−1(C(x)) = − 1 2ea � Iax ∈ m3 �m4 i−1(ϕ̃) = −2id3 + 3 2 id4 i−1(y ∧ Ω) = − 1 2JIy ∈ Sym2 0m4. The routine, though lengthy proof is based on the transformation formula (2.2) which allows computing i−1. Alternatively these facts can be checked by writing down the action of i on decomposable elements in Sym2 0m. Further on, we write A(ξ) = sϕ̃ + y ∧ Ω + C(x) and compute all terms present in the polynomial P (ξ) by taking the scalar product of the quantities computed in Lemma 5.1 and 5.2 . The scalar product on symmetric tensors in Sym2(m) is defined here by 〈S1, S2〉 = tr(S1S2). Pure type considerations with respect to the splitting Sym2m = Sym2m3 ⊕ Sym2m4 ⊕ (m3 � m4) show that 〈p(y ∧ Ω, y ∧ Ω), i−1(A(ξ))〉 = 2s|y|2 〈p(y ∧ Ω, C(x)), i−1(A(ξ))〉 = − 3 2g(Jx, Iyx) 〈p(C(x), C(x)), i−1(A(ξ))〉 = 33s|x|2 − 5g(Jx, Iyx) 〈p(ϕ̃, ϕ̃), i−1(A(ξ))〉 = −210s 〈p(ϕ̃, y ∧ Ω), i−1(A(ξ))〉 = 2|y|2 〈p(ϕ̃, C(x)), i−1(A(ξ))〉 = 3|x|2. Plugging these relations into the expression for P (ξ) leads to 〈p(A(ξ), A(ξ)), i−1(A(ξ))〉 = −210s3 + s(39|x|2 + 6|y|2) − 8R(ξ) where R(ξ) := g(Jx, Iyx). Reverting to the original parametrisation of ξ via the transformations y �→ − 5y 3 and x �→ √ 5 3 √ 2 x yields P (ξ) = 210s3 + 65 6 s|x|2 + 50 3 s|y|2 + 100 27 R(ξ). This expression can be refined as follows in order to facilitate the computation of 〈P, idet〉. In the current complex notation further calculation shows that R(ξ) reads R(ξ) = v1−v2 2 (x2 3 + x2 4 − x2 5 − x2 6) − 2x1(−x3x6 + x4x5) + 2x2(x3x5 + x4x6) = v1−v2 2 (|z2|2 − |z1|2) + 2iRe (z1z2z3). Similarly, we also have |y|2 = (v1−v2) 2 4 + |z3|2 and |x|2 = |z1|2 + |z2|2 as well as idet(ξ) = v1v2v3 + 2iRe (z1z2z3) − (v1|z1|2 + v2|z2|2 + v3|z3|2). As explained above there remains to compute the scalar product of the two cubic polynomials P and idet considered as elements in Sym3(su(3)). We interpret the coordinates v1, v2, 1810 PAUL-ANDI NAGY AND UWE SEMMELMANN v3 = −v1 − v2, z1, z2, z3 as linear forms on su(3). The scalar product in use on su(3)∗ is induced by the trace form b(ξ, ξ) = − 1 2 tr(ξ2). It is completely determined from 〈va, va〉 = 4 3 , 〈va, vb〉 = − 2 3 for a �= b, 〈zj , z̄k〉 = 2δjk and 〈va, zj〉 = 〈vb, z̄k〉 = 0. Then the scalar product on monomials, that is, elements of Sym3(su(3)∗), is computed with the help of permanents. Recall that the permanent of a matrix A = (aij) is defined similarly to the determinant by the formula permA = ∑ σ∈Sn ∏ i ai,σ(i). Permanents help make explicit the scalar product on Symk(su(3)∗) induced from the scalar product 〈·, ·〉 on su(3)∗ via 〈a1 . . . ak, b1 . . . bk〉 = perm (〈ai, bj〉), where a1 . . . ak and b1 . . . bk are monomials in the coordinate functions. A straightforward calculations then gives 〈s3, idet〉 = − 4 9 , 〈s|x|2, idet〉 = − 8 3 , 〈s|y|2, idet〉 = 4, 〈R, idet〉 = 24 . Collecting all summands we finally obtain 〈P, idet〉 = 210(− 4 9 ) + 65 6 (− 8 3 ) + 50 3 4 + 100 27 24 = 100 3 �= 0. This proves Theorem 1.2. Acknowledgements. It is a pleasure to thank G. Weingart for his input on some of the computations in the last section of the paper. 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Moroianu, P.-A. Nagy and U. Semmelmann, ‘Deformations of nearly Kähler structures’, Pacific J. Math. 235 (2008) 57–72. 15. A. Moroianu and U. Semmelmann, ‘Infinitesimal Einstein deformations of nearly Kähler metrics’, Trans. Amer. Math. Soc. 363 (2011) 3057–3069. DEFORMATIONS OF NEARLY G2 STRUCTURES 1811 Paul-Andi Nagy and Uwe Semmelmann Institut für Geometrie und Topologie Fachbereich Mathematik Universität Stuttgart Pfaffenwaldring 57 Stuttgart 70569 Germany paul-andi.nagy@mathematik.uni-stuttgart.de Uwe.Semmelmann@mathematik.uni-stuttgart.de The Journal of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics. mailto:paul-andi.nagy@mathematik.uni-stuttgart.de mailto:Uwe.Semmelmann@mathematik.uni-stuttgart.de 1. Introduction 2. Preliminaries 3. Deformation theory 4. Proof of Theorem 1.1 5. The Aloff-Wallach space References