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The Quarterly Journal of Mathematics
, Volume 69 (1) – Mar 1, 2018

30 pages

/lp/ou_press/second-order-deformations-of-associative-submanifolds-in-nearly-MmO9CPa7rP

- Publisher
- Oxford University Press
- Copyright
- © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hax038
- Publisher site
- See Article on Publisher Site

Abstract Associative submanifolds A in nearly parallel G2-manifolds Y are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over Y has the holonomy group contained in Spin(7) and the Riemannian cone over A is a Cayley submanifold. Infinitesimal deformations of associative submanifolds were considered by the author. This paper is a continuation of the work. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly. As an application, we show that the infinitesimal deformations of a homogeneous associative submanifold in the 7-sphere given by Lotay, which he called A3, are unobstructed to second order. 1. Introduction Associative submanifolds A in nearly parallel G2-manifolds are minimal 3-submanifolds in spin 7-manifolds Y with a real Killing spinor. The Riemannian cone over Y has the holonomy group contained in Spin(7) and the Riemannian cone over A is a Cayley submanifold. There are many examples of associative submanifolds. For example, special Legendrian submanifolds and invariant submanifolds in the sense of [3, Section 8.1] in Sasaki–Einstein manifolds are associative. Lagrangian submanifolds in the sine cones of nearly Kähler 6-manifolds are also associative [16, Propositions 3.8, 3.9 and 3.10]. We are interested in deformations of associative submanifolds in nearly parallel G2-manifolds. Since associative deformations are equivalent to Cayley cone deformations, it may help to develop the deformation theory of a Cayley submanifold with conical singularities. This study can also be regarded as an analogous study of associative submanifolds in torsion-free G2-manifolds. The standard 7-sphere S7 has a natural nearly parallel G2-structure. Lotay [16] studied associative submanifolds in S7 intensively. In particular, he classified homogeneous associative submanifolds [16, Theorem 1.1], in which he gave the first explicit homogeneous example which does not arise from other geometries. He called it A3. This is the only known example of this property up to the Spin(7)-action. Hence, A3 is a very mysterious example. It would be very interesting to see whether it is possible to obtain other new associative submanifolds not arising from other geometries by deforming it. It is known that the expected dimension of the moduli space of associative submanifolds is 0. However, there are many examples which have non-trivial deformations as pointed out in [16, Theorem 1.3]. In [11], the author studied infinitesimal associative deformations of homogeneous associative submanifolds in S7. Infinitesimal associative deformations of other homogeneous examples than A3 are unobstructed (namely, they extend to actual deformations) or reduced to the Lagrangian deformation problems in a totally geodesic S6 [11, Theorems 1.1 and 1.2]. However, we did not know whether infinitesimal associative deformations of A3 are unobstructed or not [11, Theorem 1.1]. The associative submanifold A3 does not arise from other known geometries so its deformations are more complicated. In this paper, we study second-order deformations of associative submanifolds. Second-order deformations of other geometric objects are considered by many people. For example, see [5, 13, 20]. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly (Lemma 3.6 and Proposition 3.8). As an application, we obtain the following. Theorem 1.1 All of the infinitesimal deformations of the associative submanifold A3defined by (4.2) in S7are unobstructed to second order. As stated above, the expected dimension of the moduli space of associative submanifolds is 0. Thus, we will expect that an associative submanifold does not admit associative deformations generically. Theorem 1.1 is an unexpected result because it implies that infinitesimal associative deformations of A3 might extend to actual deformations (for example, by the action of some group). Unfortunately, we have no idea currently. If all infinitesimal associative deformations of A3 are unobstructed, we will be able to know the type of singularities of Cayley submanifolds in some cases. Namely, as in [15, Theorem 1.1], we can expect that if a Cayley integral current has a multiplicity one tangent cone of the form R>0×A3 with isolated singularity at an interior point p, then it has a conical singularity at p. Moreover, as in [15, Theorem 1.3], it might be useful to construct Cayley submanifolds with conical singularities in compact manifolds with Spin(7) holonomy. Remark 1.2 In [12], the author classified homogeneous associative submanifolds and studied their associative deformations in the squashed 7-sphere, which is a 7-sphere with another nearly parallel G2-structure. In this case, all of homogeneous associative submanifolds arise from pseudoholomorphic curves of the nearly Kähler CP3. Thus, the deformation problems are easier and all infinitesimal associative deformations of homogeneous associative submanifolds in the squashed S7 are unobstructed [12, Theorem 1.6]. This paper is organized as follows. In Section 2, we review the fundamental facts of G2 and Spin(7) geometry. In Section 3, we recall the infinitesimal deformations of associative submanifolds and consider their second-order deformations. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order (Lemma 3.6) and describe it explicitly (Proposition 3.8). In Section 4, we prove Theorem 1.1 by using Proposition 3.8 and the Clebsch–Gordan decomposition. We also describe the trivial deformations (deformations given by the Spin(7)-action) of A3 explicitly. Notation Let (M,g) be a Riemannian manifold. We denote by i(·) the interior product. For a tangent vector v∈TM, define a cotangent vector v♭∈T*M by v♭=g(v,·). For a cotangent vector α∈T*M, define a tangent vector α♯∈TM by α=g(α♯,·). For a vector bundle E over M, we denote by C∞(M,E) the space of all smooth sections of E→M. 2. G2 and Spin(7) geometry First, we review the fundamental facts of G2 and Spin(7) geometry. Definition 2.1 Define a 3-form φ0 on R7 by φ0=dx123+dx1(dx45+dx67)+dx2(dx46−dx57)−dx3(dx47+dx56),where (x1,…,x7) is the standard coordinate system on R7 and wedge signs are omitted. The Hodge dual of φ0 is given by ∗φ0=dx4567+dx23(dx67+dx45)+dx13(dx57−dx46)−dx12(dx56+dx47). Decompose R8=R⊕R7 and denote by x0 the coordinate on R. Define a self-dual 4-form Φ0 on R8 by Φ0=dx0∧φ0+∗φ0. Identifying R8≅C4 via R8∋(x0,…,x7)↦(x0+ix1,x2+ix3,x4+ix5,x6+ix7)≕(z1,z2,z3,z4)∈C4, (2.1) Φ0 is described as Φ0=12ω0∧ω0+ReΩ0,where ω0=i2∑j=14dzjj¯ and Ω0=dz1234 are the standard Kähler form and the holomorphic volume form on C4, respectively. The stabilizers of φ0 and Φ0 are the Lie groups G2 and Spin(7), respectively: G2={g∈GL(7,R);g*φ0=φ0},Spin(7)={g∈GL(8,R);g*Φ0=Φ0}. The Lie group G2 fixes the standard metric g0=∑i=17(dxi)2 and the orientation on R7. They are uniquely determined by φ0 via 6g0(v1,v2)volg0=i(v1)φ0∧i(v2)φ0∧φ0, (2.2)where volg0 is a volume form of g0 and vi∈T(R7). Similarly, Spin(7) fixes the standard metric h0=∑i=07(dxi)2 and the orientation on R8. We have the following identities: Φ02=14volh0,(i(w2)i(w1)Φ0)2∧Φ0=6∥w1∧w2∥h02volh0, (2.3)where volh0 is a volume form of h0 and wi∈T(R8). Definition 2.2 Let M7 be an oriented 7-manifold and φ be a 3-form on M7. A 3-form φ is called a G2-structure on M7 if for each p∈M7, there exists an oriented isomorphism between TpM7 and R7 identifying φp with φ0. From (2.2), φ induces the metric g and the volume form on M7. Similarly, for an oriented 8-manifold with a 4-form Φ, we can define a Spin(7)-structure by Φ0. Lemma 2.3 A G2-structure φis calledtorsion-freeif dφ=d∗φ=0. A Spin(7)-structure Φis calledtorsion-freeif dΦ=0. It is well-known that a G2- or Spin(7)-structure is torsion-free if and only if the holonomy group is contained in G2or Spin(7). This is also equivalent to saying that φor Φis parallel with respect to the Levi–Civita connection of the induced metric. Definition 2.4 ([1, Proposition 2.3]) Let (M7,φ,g) be a manifold with a G2-structure. Let ∇ be the Levi–Civita connection of g. A G2-structure φ is called a nearly parallel G2-structure if one of the following equivalent conditions is satisfied: dφ=4∗φ, ∇φ=14dφ, ∇φ=∗φ, ∇v(∗φ)=−v♭∧φ for any v∈TM, i(v)∇vφ=0 for any v∈TM, The Riemannian cone C(M)=R>0×M admits a torsion-free Spin(7)-structure Φ=r3dr∧φ+r4∗φ with the induced cone metric g¯=dr2+r2g.We call a manifold with a nearly parallel G2-structure a nearly parallel G2-manifold for short. Definition 2.5 Let (M7,φ,g) be a manifold with a G2-structure. Define the cross product ·×·:TM×TM→TM and a tangent bundle valued 3-form χ∈Ω3(M,TM) by g(x×y,z)=φ(x,y,z),g(χ(x,y,z),w)=∗φ(x,y,z,w)for x,y,z,w∈TM. They are related via χ(x,y,z)=−x×(y×z)−g(x,y)z+g(x,z)y. (2.4) Next, we summarize the facts about submanifolds in G2 and Spin(7) settings. Let M7 be a manifold with a G2-structure φ and the induced metric g. Lemma 2.6 ([8]) For every oriented k-dimensional subspace Vk⊂TpM7, where p∈M7and k=3,4, we have φ∣V3≤volV3,∗φ∣V4≤volV4.An oriented 3-submanifold L3⊂M7is calledassociativeif φ∣TL3=volL3, which is equivalent to χ∣TL3=0and φ∣TL3>0. An oriented 4-submanifold L4⊂M7is calledcoassociativeif ∗φ∣TL4=volL4, which is equivalent to φ∣TL4=0and ∗φ∣TL4>0. Associative submanifolds have the following good properties with respect to the cross product. Lemma 2.7 Let L3⊂M7be an associative submanifold and ν→Lbe the normal bundle of L3in M7. Then, we have TL×TL⊂TL,TL×ν⊂ν,ν×ν⊂TL.Here, the left-hand sides are the spaces given by the cross product of elements of TL or ν. Definition 2.8 Let X be a manifold with a Spin(7)-structure Φ. Then, for every oriented 4-dimensional subspace W⊂TxX, where x∈X, we have Φ∣W≤volW. An oriented 4-submanifold N⊂X is called Cayley if Φ∣TN=volN. Lemma 2.9 ([8]) If a G2-structure is torsion-free, φand ∗φdefine calibrations. Hence, compact (co)associative submanifolds are volume minimizing in their homology classes, and hence, minimal. We also know that any (not necessarily compact) (co)associative submanifolds are minimal. Similar statement holds for Cayley submanifolds in a manifold with a torsion-free Spin(7)-structure. Lemma 2.10 Let (M7,φ,g)be a nearly parallel G2-manifold. Then, there are no coassociative submanifolds in M [16, Lemma 3.2]. An oriented 3-dimensional submanifold L⊂Mis associative if and only if C(L)=R>0×L⊂R>0×M=C(Y)is Cayley. In particular, L is minimal. 3. Deformations of associative submanifolds 3.1. Infinitesimal deformations of associative submanifolds First, we describe the infinitesimal deformation space explicitly again. The arguments here are based on [7, Section 2], [9, Section 6.1], [18, Section 3.1]. Let (M7,φ,g) be a manifold with a G2-structure and let L3⊂M7 be a compact associative submanifold. Let ν→L be the normal bundle of L3 in M7. By the tubular neighborhood theorem, there exists a neighborhood of L in M which is identified with an open neighborhood T⊂ν of the zero section by the exponential map. Set C∞(L,T)={v∈C∞(L,ν);vx∈Tforanyx∈L}. The exponential map induces the embedding expV:L↪M by expV(x)=expx(Vx) for V∈C∞(L,T) and x∈L. Let PV:TM∣L→TM∣expV(L)forV∈C∞(L,T)be the isomorphism given by the parallel transport along the geodesic [0,1]∋t↦expx(tVx)∈M, where x∈L, with respect to the Levi–Civita connection of g. Let ⊥:TM∣L=TL⊕ν→ν be the orthogonal projection and νV⊂TM∣expV(L) be the normal bundle of expV(L). Consider the orthogonal projection ⊥∣PV−1(νV):PV−1(νV)→ν. The condition for this map to be an isomorphism is open and it is an isomorphism for V=0. Thus, shrinking T if necessary, we may assume that ϕV:C∞(L,νV)→C∞(L,ν),ϕV(W)=(PV−1(W))⊥is an isomorphism for V∈C∞(L,T). Then, define the first-order differential operator F:C∞(L,T)→C∞(L,ν) by F(V)=ϕV((expV*χ)(e1,e2,e3)), (3.1)where {e1,e2,e3} is a local oriented orthonormal frame of TL. Then, expV(L)⊂M is associative if and only if F(V)=0. Thus, a neighborhood of L in the moduli space of associative submanifolds is identified with that of 0 in F−1(0) (in the C1 sense). Set D=(dF)0:C∞(L,ν)→C∞(L,ν),which is the linearization of F at 0. The operator D is computed as follows. Proposition 3.1 ([19, Section 5], [7, Theorem 2.1]) Let (M7,φ,g)be a manifold with a G2-structure and let L3⊂M7be a compact associative submanifold. The operator D above is given by DV=∑i=13ei×∇ei⊥V+((∇V∗φ)(e1,e2,e3,·))♯,where {e1,e2,e3}is a local oriented orthonormal frame of TL satisfying ei=ei+1×ei+2for i∈Z/3, ∇⊥is the connection on the normal bundle νinduced by the Levi–Civita connection ∇of (M,g). Proof For simplicity, we write exptV=ιt. Then, DV=ddt(PtV−1(ιt*χ)(e1,e2,e3)∣t=0)⊥=(∇ddt(ιt*χ)(e1,e2,e3)∣t=0)⊥,where ∇ddt is the covariant derivative along the geodesic [0,1]∋t↦expx(tVx)∈M, where x∈L, induced from the Levi–Civita connection of g. Let {ηj}j=17 be a local orthonormal frame of TM. Then, we have χ=−∑j=17i(ηj)∗φ⊗ηj.We further compute DV=−∑j(∇ddt((∗φ◦ιt)(ηj◦ιt,(ιt)*e1,(ιt)*e2,(ιt)*e3)ηj◦ιt)∣t=0)⊥=−∑j((∇V∗φ)(ηj,e1,e2,e3)ηj+∑i∈Z/3∗φ(ηj,∇ddt(ιt)*ei∣t=0,ei+1,ei+2)ηj)⊥,where we use ∗φ(e1,e2,e3,·)=0, since L is associative. Note that ∇ddt(ιt)*ei is the restriction of the covariant derivative ∇ddt(ι¯)*ei along the map ι¯:L×[0,1]∋(x,t)↦ιt(x)∈M. Then, the standard equations of the covariant derivative along the map imply that ∇ddt(ιt)*ei∣t=0=∇ei(ιt)*(ddt)∣t=0=∇eiV,∗φ(∇eiV,ei+1,ei+2,ηj)=∗φ(∇ei⊥V,ei+1,ei+2,ηj)=g(χ(∇ei⊥V,ei+1,ei+2),ηj)=g(−∇ei⊥V×(ei+1×ei+2),ηj)=g(ei×∇ei⊥V,ηj),where we use the fact that L is associative, (2.4) and ei=ei+1×ei+2. Then, we obtain the statement.□ We can also describe the last term of DV as follows. Lemma 3.2 By [4, Section 4], we have an endomorphism T∈C∞(M,End(TM))given by ∇vφ=i(T(v))∗φ (3.2)for any v∈TM.Then, we have ((∇V∗φ)(e1,e2,e3,·))♯=(T(V))⊥. Proof We easily see that ∇v∗φ=∗(∇vφ)=−(T(v))♭∧φ. Then, ((∇V∗φ)(e1,e2,e3,·))♯=−((T(v))♭∧φ)(e1,e2,e3,·)♯=φ(e1,e2,e3)T(v)−∑i∈Z/3g(T(v),ei)φ(ei+1,ei+2,·)♯=(T(v))⊥,where we use φ(e1,e2,e3)=1 and φ(ei+1,ei+2,·)♯=ei+1×ei+2=ei.□ Using this lemma, we see the following. Lemma 3.3 If d∗φ=0, D is self-adjoint. Proof For any normal vector fields V,W∈C∞(L,ν), we compute g(DV,W)=g(∑i=13ei×∇eiV+T(V),W)=Lemma2.7∑i=13g(ei×∇eiV,W)+g(T(V),W),∑i=13g(ei×∇eiV,W)=−∑i=13φ(∇eiV,ei,W)=∑i=13(−ei(φ(V,ei,W))+(∇eiφ)(V,ei,W)+φ(V,∇eiei,W)+φ(V,ei,∇eiW)).Define a 1-form α on L by α=φ(V,·,W). Then, =d*α+∑i=13(∇eiφ)(V,ei,W)+g(V,ei×∇eiW)=(3.2)d*α+∑i=13∗φ(T(ei),V,ei,W)+g(V,ei×∇eiW).By (2.4), it follows that ∗φ(T(ei),V,ei,W)=g(χ(ei,V,W),T(ei))=−g(ei×(V×W),T(ei)).By Lemma 2.7, ei×(V×W) is a (local) tangent vector field to L. Then, ∑i=13∗φ(T(ei),V,ei,W)=∑i,j=13g(T(ei),ej)∗φ(ej,V,ei,W).Hence, we obtain g(DV,W)=g(V,DW)+g(T(V),W)−g(V,T(W))+∑i,j=13g(T(ei),ej)∗φ(ej,V,ei,W)+d*α.Then, we see that D is self-adjoint if T is symmetric. In terms of [10, Section 2.5], this is the case ∇φ∈W1⊕W27, which is equivalent to d∗φ=0 by [10, Table 2.1].□ Remark 3.4 If a G2-structure is torsion-free, we have T=0, and hence ((∇V∗φ)(e1,e2,e3,·))♯=0. If a G2-structure is nearly parallel G2, we have T=idTM and ((∇V∗φ)(e1,e2,e3,·))♯=V. We can also deduce this by Definition 2.4 [11, Lemma 3.5]. In these cases, D is self-adjoint as stated in Lemma 3.3. We easily see that the operator D is elliptic, and, hence, Fredholm. Since L is 3-dimensional, the index of D is 0. Thus, if D is surjective, the moduli space of associative submanifolds is 0-dimensional. See [7, Proposition 2.2]. To understand the moduli space of associative submanifolds more, we consider their second-order deformations in the next subsection. 3.2. Second-order deformations of associative submanifolds Use the notation in Section 3.1. The principal task in deformation theory is to integrate given infinitesimal (first order) deformations V∈kerD. Namely, to find a one-parameter family {V(t)}⊂C∞(L,ν) such that F(V(t))=0andddtV(t)∣t=0=V. In general, this is not possible. In this subsection, we define the second-order deformations of associative submanifolds and give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order. Definition 3.5 Let M7 be a manifold with a G2-structure and L3⊂M7 be a compact associative submanifold. An infinitesimal associative deformation V1∈kerD⊂C∞(L,ν) is said to be unobstructed to second order if there exists V2∈C∞(L,ν) such that d2dt2F(tV1+12t2V2)∣t=0=0.In other words, tV1+12t2V2 gives an associative submanifold up to terms of the order o(t2). We easily compute d2dt2F(tV1+12t2V2)∣t=0=d2dt2F(tV1)∣t=0+D(V2).Since D is elliptic and L is compact, we have an orthogonal decomposition C∞(L,ν)=ImD⊕CokerD with respect to the L2 inner product. Then, we obtain the following. Lemma 3.6 Let π:C∞(L,ν)→CokerDbe an orthogonal projection with respect to the L2inner product. Then, an infinitesimal deformation V1∈kerDis unobstructed to second order if and only if π(d2dt2F(tV1))∣t=0=0. (3.3)In other words, we have ⟨d2dt2F(tV1)∣t=0,W⟩L2=0for any W∈CokerD. Remark 3.7 Since D is elliptic and hence Fredholm, we can construct a Kuranishi model for associative deformations of a compact associative submanifold L [18, Section A.4] as in the case of Lagrangian deformations in nearly Kähler manifolds [14, Theorem 4.10]. Namely, there is a real analytic map τ:U→V, where U⊂kerD and V⊂CokerD are open neighborhoods of 0, satisfying τ(0)=0 and (dτ)0=0 such that the moduli space of associative deformations of L is locally homeomorphic to the kernel of τ (hence, the moduli space is locally a finite dimensional analytic variety). Then, we obtain (3.3) by taking the second derivative of τ at 0 as in [14, Proposition 4.17]. From Lemma 3.6, we have to understand d2dt2F(tV1)∣t=0 for the second-order deformations. It is explicitly computed as follows. Proposition 3.8 Use the notation in Proposition3.1. For V∈kerD, we have d2dt2F(tV)∣t=0=((∇V∇∗φ)(V))(e1,e2,e3,·)♯+2∑i∈Z/3((∇V∗φ)(∇ei⊥V,ei+1,ei+2,·)♯)⊥+∑i=13ei×(R(V,ei)V)⊥+2∑i,j=13g(V,Π(ei,ej))ei×∇ej⊥V,where R is the curvature tensor of (M,g)and Πis the second fundamental form of L in M. If a G2-structure φis torsion-free or nearly parallel G2, we have d2dt2F(tV)∣t=0=∑i=13ei×(R(V,ei)V)⊥+2∑i,j=13g(V,Π(ei,ej))ei×∇ej⊥V. Proof Use the notation in the proof of Proposition 3.1. Setting χj=−i(ηj)∗φ, we have χ=∑j=17χj⊗ηj. Then, ddtF(tV)=(PtV−1∇ddt((ιt*χ)(e1,e2,e3)))⊥,d2dt2F(tV)∣t=0=(∇ddt∇ddt((ιt*χ)(e1,e2,e3)))⊥∣t=0=∑j(∇ddt∇ddt(ιt*χj(e1,e2,e3)ηj◦ιt))⊥∣t=0=∑j(d2dt2ιt*χj(e1,e2,e3)∣t=0ηj+2ddtιt*χj(e1,e2,e3)∣t=0∇Vηj+χj(e1,e2,e3)∇ddt∇ddtηj◦ιt∣t=0)⊥. (3.4)Since ddtιt*χj(e1,e2,e3)∣t=0=g(DV,ηj) by the proof of Proposition 3.1 and χj(e1,e2,e3)=0, we only have to compute d2dt2ιt*χj(e1,e2,e3)∣t=0. Then, d2dt2ιt*χj(e1,e2,e3)∣t=0=−d2dt2((∗φ◦ιt)(ηj◦ιt,(ιt)*e1,(ιt)*e2,(ιt)*e3))∣t=0=−ddt((∇ddt∗φ◦ιt)(ηj◦ιt,(ιt)*e1,(ιt)*e2,(ιt)*e3)+(∗φ◦ιt)(∇ddtηj◦ιt,(ιt)*e1,(ιt)*e2,(ιt)*e3)+∑i∈Z/3(∗φ◦ιt)(ηj◦ιt,∇ddt(ιt)*ei,(ιt)*ei+1,(ιt)*ei+2))∣t=0=−(∇ddt∇ddt∗φ◦ιt)∣t=0(ηj,e1,e2,e3)−2∑i∈Z/3(∇V∗φ)(ηj,∇ddt(ιt)*ei∣t=0,ei+1,ei+2)−∗φ(∇ddt∇ddtηj◦ιt∣t=0,e1,e2,e3)−2(∇V∗φ)(∇Vηj,e1,e2,e3)−2∑i∈Z/3∗φ(∇Vηj,∇ddt(ιt)*ei∣t=0,ei+1,ei+2)−∑i∈Z/3∗φ(ηj,∇ddt∇ddt(ιt)*ei∣t=0,ei+1,ei+2)−2∑i∈Z/3∗φ(ηj,∇ddt(ιt)*ei∣t=0,∇ddt(ιt)*ei+1∣t=0,ei+2).By the same argument as in the proof of Proposition 3.1, we have ∗φ(e1,e2,e3,·)=0,∇ddt(ιt)*ei∣t=0=∇eiV,∇ddt∇ddt(ιt)*ei∣t=0=R(V,ei)V+∇ei∇ddt(ιt)*(ddt)∣t=0=R(V,ei)V,where we use ∇ddt(ιt)*(ddt)=0 because ιt=exp(tV) is a geodesic. By the definition of the induced connection, we have ∇ddt∇ddt∗φ◦ιt∣t=0=∇ddt((∇dιtdt∗φ)◦ιt)∣t=0=∇ddt(((∇∗φ)◦ιt)(dιtdt))∣t=0=(∇V∇∗φ)(V),where we use ∇ddtdιtdt=0. Moreover, by the proof of Proposition 3.1, we have (∇V∗φ)(∇Vηj,e1,e2,e3)+∑i∈Z/3∗φ(∇Vηj,∇ddt(ιt)*ei∣t=0,ei+1,ei+2)=−g(DV,∇Vηj).Thus, it follows that d2dt2ιt*χj(e1,e2,e3)∣t=0=−((∇V∇∗φ)(V))(ηj,e1,e2,e3)−2∑i∈Z/3(∇V∗φ)(ηj,∇eiV,ei+1,ei+2)−∑i∈Z/3∗φ(ηj,R(V,ei)V,ei+1,ei+2)−2∑i∈Z/3∗φ(ηj,∇eiV,∇ei+1V,ei+2)+2g(DV,∇Vηj). (3.5)Hence, from (2.4), (3.4) and (3.5), we obtain d2dt2F(tV)∣t=0=((∇V∇∗φ)(V))(e1,e2,e3,·)♯+2∑i∈Z/3((∇V∗φ)(∇eiV,ei+1,ei+2,·)♯)⊥+∑i=13ei×(R(V,ei)V)⊥+2∑i∈Z/3χ(∇eiV,∇ei+1V,ei+2)⊥+2∑jg(DV,∇Vηj)ηj⊥+2∑jg(DV,ηj)(∇Vηj)⊥. (3.6) Next, we compute ∑i∈Z/3χ(∇eiV,∇ei+1V,ei+2)⊥. Let ⊤:TM∣L→TL be the projection. Since L is associative, we have χ(∇eiV,∇ei+1V,ei+2)⊥=χ(∇ei⊤V,∇ei+1⊥V,ei+2)⊥+χ(∇ei⊥V,∇ei+1⊤V,ei+2)⊥+χ(∇ei⊥V,∇ei+1⊥V,ei+2)⊥. The first term is computed as χ(∇ei⊤V,∇ei+1⊥V,ei+2)=−∑j=02g(V,Π(ei,ei+j))χ(ei+j,∇ei+1⊥V,ei+2)=(2.4)−∑j=02g(V,Π(ei,ei+j))∇ei+1⊥V×(ei+j×ei+2)=−g(V,Π(ei,ei))ei+1×∇ei+1⊥V+g(V,Π(ei,ei+1))ei×∇ei+1⊥V. The second term is computed as χ(∇ei⊥V,∇ei+1⊤V,ei+2)=−∑j=02g(V,Π(ei+1,ei+j))χ(∇ei⊥V,ei+j,ei+2)=(2.4)∑j=02g(V,Π(ei+1,ei+j))∇ei⊥V×(ei+j×ei+2)=g(V,Π(ei,ei+1))ei+1×∇ei⊥V−g(V,Π(ei+1,ei+1))ei×∇ei⊥V. The third term is computed as χ(∇ei⊥V,∇ei+1⊥V,ei+2)=χ(ei+2,∇ei⊥V,∇ei+1⊥V)=(2.4)−ei+2×(∇ei⊥V×∇ei+1⊥V),which is a section of TL by Lemma 2.7. Then χ(∇ei⊥V,∇ei+1⊥V,ei+2)⊥=0. Hence, we obtain ∑i∈Z/3χ(∇eiV,∇ei+1V,ei+2)⊥=−∑i∈Z/3{g(V,Π(ei+1,ei+1))+g(V,Π(ei+2,ei+2))}ei×∇ei⊥V+∑i∈Z/3g(V,Π(ei,ei+1))(ei×∇ei+1⊥V+ei+1×∇ei⊥V)=∑i,j=13g(V,Π(ei,ej))ei×∇ej⊥V−(∑i=13g(V,Π(ei,ei)))(∑j=13ej×∇ej⊥V). (3.7) Thus, using the equation ∑i∈Z/3(∇V∗φ)(∇eiV,ei+1,ei+2,·)=∑i∈Z/3(∇V∗φ)(∇ei⊥V,ei+1,ei+2,·)−(∑i=13g(V,Π(ei,ei)))(∇V∗φ)(e1,e2,e3,·),we obtain from Propositions 3.1, (3.6) and (3.7) d2dt2F(tV)∣t=0=((∇V∇∗φ)(V))(e1,e2,e3,·)♯+2∑i∈Z/3((∇V∗φ)(∇ei⊥V,ei+1,ei+2,·)♯)⊥+∑i=13ei×(R(V,ei)V)⊥+2∑i,j=13g(V,Π(ei,ej))ei×∇ej⊥V+2∑jg(DV,∇Vηj)ηj⊥+2∑jg(DV,ηj)(∇Vηj)⊥−2(∑i=13g(V,Π(ei,ei)))DV,which implies the first equation of Proposition 3.8. If a G2-structure φ is torsion-free, the second equation of Proposition 3.8 is obvious. If φ is nearly parallel G2, we have by Definition 2.4 (∇V∇∗φ)(V)=∇V∇V∗φ−∇∇VV∗φ=∇V(−V♭∧φ)+(∇VV)♭∧φ=−V♭∧i(V)∗φ,which implies that ((∇V∇∗φ)(V))(e1,e2,e3,·)=0. Similarly, we have by Definition 2.4 (∇V∗φ)(∇ei⊥V,ei+1,ei+2,·)♯=−(V♭∧φ)(∇ei⊥V,ei+1,ei+2,·)♯=−g(V,∇ei⊥V)ei,where we use φ(ei+1,ei+2,·)♯=ei+1×ei+2=ei. Hence, ∑i∈Z/3((∇V∗φ)(∇eiV,ei+1,ei+2,·)♯)⊥=0.Thus, we obtain the second equation of Proposition 3.8.□ Remark 3.9 Using the endomorphism T given by (3.2), we have ((∇V∇∗φ)(V))(e1,e2,e3,·)♯+2∑i∈Z/3((∇V∗φ)(∇ei⊥V,ei+1,ei+2,·)♯)⊥=((∇VT)(V))⊥+T(V)⊤×T(V)⊥−2∇T(V)⊤⊥Vby a direct computation. If φ is torsion-free ( T=0), these terms obviously vanish. If φ is nearly parallel G2 ( T=idTM), these terms vanish again because ∇idTM=0 and (idTM(V))⊤=0 for a normal vector field V. 4. Associative submanifolds in S7 In this section, we give a proof of Theorem 1.1. The standard 7-sphere S7 has a natural nearly parallel G2-structure [16, Section 2]. Homogeneous associative submanifolds in S7 are classified by Lotay [16, Theorem 1.1]. As noted in the introduction, there is a mysterious homogeneous example called A3 which does not arise from other geometries. First, we summarize the facts for A3 from [11, Example 6.3, Section 6.3.3]. Define ρ3:SU(2)↪SU(4) by ρ3((a−b¯ba¯))=(a3−3a2b¯3ab¯2−b¯33a2ba(∣a∣2−2∣b∣2)−b¯(2∣a∣2−∣b∣2)3a¯b¯23ab2b(2∣a∣2−∣b∣2)a¯(∣a∣2−2∣b∣2)−3a¯2b¯b33a¯b23a¯2ba¯3), (4.1)where a,b∈C such that ∣a∣2+∣b∣2=1. This is an irreducible SU(2)-action on C4. By using the notation of Appendix A, ρ3 is the matrix representation of ρ3:SU(2)→GL(V3)≅GL(4,C) with respect to the basis {v0(3),…,v3(3)}. Then, A3=ρ3(SU(2))·12t(0,1,i,0)≅SU(2) (4.2)is an associative submanifold in S7. Define the basis of the Lie algebra su(2) of SU(2) by E1=(01−10),E2=(0ii0),E3=(i00−i), (4.3)which satisfies the relation [Ei,Ei+1]=2Ei+2 for i∈Z/3. Denote by e1,e2,e3 the left invariant vector fields on SU(2)≅A3 induced by 17E1,17E2,E3, respectively. Then, they define a global orthonormal frame of T A3. Explicitly, we have at p0=12t(0,1,i,0) e1=114t(3,2i,−2,−3i),e2=114t(3i,−2,2i,−3),e3=12t(0,i,1,0),and (ei)ρ3(g)·p0=ρ3(g)·(ei)p0 for g∈SU(2). Set (η1)p0=12t(i,0,0,1),(η3)p0=142t(−23i,−3,3i,23),(η2)p0=12t(−1,0,0,−i),(η4)p0=142t(−23,3i,−3,23i), (4.4)which is an orthonormal basis of the normal bundle at p0. Setting (ηj)ρ3(g)·p0=ρ3(g)·(ηj)p0 for g∈SU(2), we obtain an orthonormal frame {ηj}j=14 of the normal bundle ν. 4.1. Second-order deformations of A3 Now, we consider the second-order deformations of A3. First, we describe the second derivative of the deformation map in a normal direction V∈C∞(A3,ν) explicitly using Proposition 3.8. Since S7 with the round metric ⟨·,·⟩ has constant sectional curvature 1, we have R(x,y)z=⟨y,z⟩x−⟨x,z⟩yforx,y,z∈TS7,which implies that (R(V,ei)V)⊥=0. Then, by Proposition 3.8, it follows that d2dt2F(tV)∣t=0=2∑i,j=13⟨V,Π(ei,ej)⟩ei×∇ej⊥V. We will compute this. By [11, Lemma 6.20] and its proof, we have the following. Lemma 4.1 (∇ei⊥ηj)1≤i≤3,1≤j≤4=37(−η4−η3η2η1η3−η4−η1η27η2−7η1−5η45η3),(ei×ηj)1≤i≤3,1≤j≤4=(η4η3−η2−η1−η3η4η1−η2η2−η1η4−η3),(Π(ei,ej))1≤i,j≤3=237(η1η2−2η3η2−η12η4−2η32η40). Then, d2dt2F(tV)∣t=0 is described explicitly as follows. Lemma 4.2 Set V=∑j=14Vjηj∈kerD,d2dt2F(tV)∣t=0=∑j=14Fjηj,where Vj,Fj∈C∞(A3)are smooth functions on A3. Denoting V1=V1+iV2, V2=V3−iV4, we have F1+iF2=437{−(ie1+e2)(V1V2)+V¯2(−ie1+e2)V1+(ie3−247)(V22)},F3−iF4=437{V¯1(−ie1+e2)V1+12(ie1+e2)(V22)+V¯2((2ie3−487)V1−(−ie1+e2)V2)}. Proof By the third equation of Lemma 4.1, we have ∑i,j=13⟨V,Π(ei,ej)⟩ei×∇ej⊥V=237V1(e1×∇e1⊥V−e2×∇e2⊥V)+237V2(e1×∇e2⊥V+e2×∇e1⊥V)−437V3(e1×∇e3⊥V+e3×∇e1⊥V)+437V4(e2×∇e3⊥V+e3×∇e2⊥V). By the first and the second equations of Lemma 4.1, we have ∇e1⊥V=∑j=47e1(Vj)ηj+37(−V1η4−V2η3+V3η2+V4η1),∇e2⊥V=∑j=47e2(Vj)ηj+37(V1η3−V2η4−V3η1+V4η2),∇e3⊥V=∑j=47e3(Vj)ηj+37(7V1η2−7V2η1−5V3η4+5V4η3). Then, by the second equation of Lemma 4.1 and a straightforward computation, we obtain d2dt2 F(tV)∣t=0=437V1{(−e1(V4)−e2(V3))η1+(−e1(V3)+e2(V4))η2+(e1(V2)+e2(V1))η3+(e1(V1)−e2(V2))η4}+437V2{(−e2(V4)+e1(V3))η1+(−e2(V3)−e1(V4))η2+(e2(V2)−e1(V1))η3+(e2(V1)+e1(V2))η4}−837V3{(−e3(V4)−e1(V2)+127V3)η1+(−e3(V3)+e1(V1)−127V4)η2+(e3(V2)−e1(V4)+247V1)η3+(e3(V1)+e1(V3)−247V2)η4}+837V4{(e3(V3)−e2(V2)+127V4)η1+(−e3(V4)+e2(V1)+127V3)η2+(−e3(V1)−e2(V4)+247V2)η3+(e3(V2)+e2(V3)+247V1)η4}.Hence, F1+iF2=437V1(−e1(V4+iV3)−e2(V3−iV4))+437V2(−e2(V4+iV3)+e1(V3−iV4))−837V3(−e3(V4+iV3)+e1(iV1−V2)+127(V3−iV4))+837V4(e3(V3−iV4)+e2(iV1−V2)+127(V4+iV3))=437V1(−ie1−e2)(V3−iV4)+437V2(e1−ie2)(V3−iV4)−837V3(ie1(V1+iV2)+(−ie3+127)(V3−iV4))+837V4(ie2(V1+iV2)+(e3+127i)(V3−iV4)), F3−iF4=437V1(e1(V2−iV1)+e2(V1+iV2))+437V2(e2(V2−iV1)−e1(V1+iV2))−837V3(e3(V2−iV1)−e1(V4+iV3)+247(V1+iV2))+837V4(−e3(V1+iV2)−e2(V4+iV3)+247(V2−iV1))=437V1(−ie1+e2)(V1+iV2)+437V2(−e1−ie2)(V1+iV2)−837V3((−ie3+247)(V1+iV2)−ie1(V3−iV4))+837V4((−e3−247i)(V1+iV2)−ie2(V3−iV4)).Using 2i(−V3e1+V4e2)=−V2(ie1+e2)+V¯2(−ie1+e2), we obtain the statement.□ By [11, (6.24) and (6.25)] and the proof of [11, Proposition 6.22], we know the following about kerD, where D is given in Proposition 3.1. Note that D in this paper corresponds to D+idν in [11]. Lemma 4.3 For V=∑j=14Vjηj∈C∞(L,ν), set V1=V1+iV2and V2=V3−iV4. Then, DV=0is equivalent to (ie3−87)V1+(−ie1+e2)V2=0,−(ie1+e2)V1+(−ie3+4)V2=0. (4.5)By using the notation in AppendixA, elements of kerDare explicitly described as V1=−i710⟨ρ6(·)v5(6),u1⟩−2i76⟨ρ4(·)v3(4),u2⟩,V2=⟨ρ6(·)v4(6),u1⟩+⟨ρ4(·)v2(4),u2⟩+⟨ρ4(·)v4(4),u3⟩ (4.6)for u1∈V6,u2,u3∈V4. Lemma 4.4 For V,W∈kerD, the L2inner product of d2dt2 F(tV)∣t=0and W is given by ⟨d2dt2 F(tV)∣t=0,W⟩L2=437Re(I(V,W)+I(V+W,V)−I(V,V)−I(W,V)).Here, I(V,W)=∫SU(2)(V1V2·(−ie1+e2)W1¯+12V22·(3ie3−8)W1¯)dg,where V=∑j=14Vjηj,W=∑j=14Wjηj, V1=V1+iV2,V2=V3−iV4, W1=W1+iW2and W2=W3−iW4. Proof Use the notation in Lemma 4.2. First note that ⟨d2dt2 F(tV1)∣t=0,W⟩L2=Re∫SU(2)((F1+iF2)·W¯1+(F3−iF4)·W¯2)dg.By using Lemma A.2, we can integrate by parts to obtain −∫SU(2)(ie1+e2)(V1V2)·W¯1dg=∫SU(2)V1V2·(−ie1+e2)W1¯dg, ∫SU(2)((ie3−247)(V22)·W¯1+12(ie1+e2)(V22)·W¯2)dg =∫SU(2)V22·{(ie3−247)W1−12(−ie1+e2)W2}¯dg=(4.5)12∫SU(2)V22·(3ie3−8)W1¯dg.We also have V¯2((2ie3−487)V1−(−ie1+e2)V2)=(4.5)V¯2·(3ie3−8)V1.Thus, it follows that ⟨d2dt2 F(tV)∣t=0,W⟩L2=437Re∫SU(2)(V1V2·(−ie1+e2)W1¯+12V22·(3ie3−8)W1¯+(V¯2W¯1+V¯1W¯2)·(−ie1+e2)V1+V¯2W¯2·(3ie3−8)V1)dg.From the equations V2W1+V1W2=(V1+W1)(V2+W2)−(V1V2+W1W2) and 2V2W2=(V2+W2)2−V22−W22, the proof is done.□ Thus, we only have to calculate I(V,W) for any V,W∈kerD to compute ⟨d2dt2 F(tV)∣t=0,W⟩L2. In fact, we have the following. Lemma 4.5 For V,W∈kerD, we have I(V,W)=0. Proof For V=∑j=14Vjηj and W=∑j=14Wjηj, set V1=V1+iV2,V2=V3−iV4, W1=W1+iW2 and W2=W3−iW4. By Lemma 4.3, we may assume that V1,V2 are given by (4.6) for u1∈V6,u2,u3∈V4 and W1,W2 are given by the right-hand side of (4.6), where we replace uj with wj for j=1,2,3 and w1∈V6,w2,w3∈V4. By (A.5) and {e1,e2,e3}={E1/7,E2/7,E3}, note that (−ie1+e2)W1=235⟨ρ6(·)v6(6),w1⟩+86⟨ρ4(·)v4(4),w2⟩,(3ie3−8)W1=−4i710⟨ρ6(·)v5(6),w1⟩+4i76⟨ρ4(·)v3(4),w2⟩.Then, by Lemmas B.3 and B.4, we compute I(V,W)=∫SU(2)(V1V2·(−ie1+e2)W1¯+12V22·(3ie3−8)W1¯)dg=∫SU(2)(−2i76⟨ρ4(g)v3(4),u2⟩⟨ρ4(g)v4(4),u3⟩·235⟨ρ6(g)v6(6),w1⟩¯−i710⟨ρ6(g)v5(6),u1⟩⟨ρ4(g)v2(4),u2⟩·86⟨ρ4(g)v4(4),w2⟩¯−2i76⟨ρ4(g)v3(4),u2⟩⟨ρ6(g)v4(6),u1⟩·86⟨ρ4(g)v4(4),w2⟩¯+4i710⟨ρ4(g)v2(4),u2⟩⟨ρ4(g)v4(4),u3⟩·⟨ρ6(g)v5(6),w1⟩¯−4i76⟨ρ6(g)v4(6),u1⟩⟨ρ4(g)v2(4),u2⟩·⟨ρ4(g)v3(4),w2⟩¯)dg=(A.4)∫SU(2)(−4i710⟨ρ4(g)v3(4),u2⟩⟨ρ4(g)v4(4),u3⟩·⟨ρ6(g)v6(6),w1⟩¯+i710·86⟨ρ4(g)v2(4),u2⟩⟨ρ4(g)v0(4),w2*⟩·⟨ρ6(g)v1(6),u1*⟩¯−2i76·86⟨ρ4(g)v3(4),u2⟩⟨ρ4(g)v0(4),w2*⟩·⟨ρ6(g)v2(6),u1*⟩¯+4i710⟨ρ4(g)v2(4),u2⟩⟨ρ4(g)v4(4),u3⟩·⟨ρ6(g)v5(6),w1⟩¯+4i76⟨ρ4(g)v2(4),u2⟩⟨ρ4(g)v1(4),w2*⟩·⟨ρ6(g)v2(6),u1*⟩¯)dg.By Lemmas B.2 and B.5, we further compute =17(−4i710⟨v3(4)⊗v4(4),α4,4,1(v6(6))⟩·⟨u2⊗u3,α4,4,1(w1)⟩¯+i710·86⟨v2(4)⊗v0(4),α4,4,1(v1(6))⟩·⟨u2⊗w2*,α4,4,1(u1*)⟩¯−2i76·86⟨v3(4)⊗v0(4),α4,4,1(v2(6))⟩·⟨u2⊗w2*,α4,4,1(u1*)⟩¯+4i710⟨v2(4)⊗v4(4),α4,4,1(v5(6))⟩·⟨u2⊗u3,α4,4,1(w1)⟩¯+4i76⟨v2(4)⊗v1(4),α4,4,1(v2(6))⟩·⟨u2⊗w2*,α4,4,1(u1*)⟩¯) =17(4i710·c4,4,1(−245+245)·⟨u2⊗u3,α4,4,1(w1)⟩¯+4i76·c4,4,1(210·(−245)+46·243−242)·⟨u2⊗w2*,α4,4,1(u1*)⟩¯)=0.□ Theorem 1.1 follows from these lemmas. Proof of Theorem 1.1 Recall that D given in Proposition 3.1 is self-adjoint by Lemma 3.3. Then, by Lemma 3.6, we only have to show that ⟨d2dt2 F(tV)∣t=0,W⟩L2=0 for any V,W∈kerD. This equation is satisfied by Lemmas 4.4 and 4.5.□ 4.2. Deformations of A3 arising from Spin(7) To see whether infinitesimal associative deformations of A3 extend to actual deformations, it would be important to understand the trivial deformations (deformations given by the Spin(7)-action) of A3. Since A3≅SU(2), the dimension of the subgroup of Spin(7) preserving A3 is at least 3. We show that it is 4-dimensional. More precisely, we have the following. Lemma 4.6 Use the notation in (4.1), (4.4), LemmasC.1andC.2. Set p0=12t(0,1,i,0). Then, we have {X∈spin(7);⟨X·ρ3(g)·p0,(ηi)ρ3(g)·p0⟩=0foranyg∈SU(2)andi=1,…,4}=W1spin(7)⊕W3su(4). Proof Since the left-hand side is SU(2)-invariant, it is a direct sum of Wkspin(7)’s or Wlsu(4)’s. Thus, we only have to see whether an element in Wkspin(7) or Wlsu(4) is contained in the left-hand side. By definition, W3su(4) is contained in the left-hand side. Via the identification of C4≅R8 given by (2.1), we see that (ρ3(g−1)H0ρ3(g))·p0=H0·p0=12t(0,0,0,1,1,0,0,0)=(ρ3)*(E3)·p0for any g∈SU(2). Hence, W1spin(7) is contained in the left-hand side. For X=(0i00i000000−i00−i0)∈W5su(4),Y=(0010000−1−10000100)∈W7su(4),and Z=(00−10000110000−100)⊕(0010000−1−10000100)∈W5spin(7),we have ⟨X·p0,(η1)p0⟩=1,⟨Y·p0,(η1)p0⟩=1 and ⟨Z·p0,(η4)p0⟩=2/7. Note that via the identification of C4≅R8 given by (2.1) p0=12t(0,0,1,0,0,1,0,0)and(η4)p0=142t(−23,0,0,3,−3,0,0,23).Hence, W5su(4), W7su(4) and W5spin(7) are not contained in the left-hand side.□ Hence, by Lemma 4.6, we see that the space of trivial deformations of A3 is isomorphic to spin(7)/(W1spin(7)⊕W3su(4))≅W5spin(7)⊕W5su(4)⊕W7su(4),which is a 17-dimensional subspace of the 34-dimensional space kerD. Remark 4.7 Use the notation in (A.1), Lemmas A.1, C.1 and C.2. By tedious calculations, we can describe elements of kerD given by spin(7)/(W1spin(7)⊕W3su(4))≅W5spin(7)⊕W5su(4)⊕W7su(4). Elements in kerD are of the form (4.6). In the following table, each space in the left-hand side corresponds to the elements in kerD given by the right-hand side. kerD W7su(4) u1∈(1−j)V6,u2=u3=0 W5su(4) u1=0,u2∈(1−j)V4,u3=(26/3)·u2* W5spin(7) u1=0,u2∈(1+j)V4,u3=(26/3)·u2* kerD W7su(4) u1∈(1−j)V6,u2=u3=0 W5su(4) u1=0,u2∈(1−j)V4,u3=(26/3)·u2* W5spin(7) u1=0,u2∈(1+j)V4,u3=(26/3)·u2* Funding This work was supported by JSPS KAKENHI Grant nos. JP14J07067 and JP17K14181. Acknowledgements The author would like to thank Hông Vân Lê for suggesting the problems in this paper. He thanks the referee for the careful reading of an earlier version of this paper and useful comments on it. References 1 B. Alexandrov and U. Semmelmann, Deformations of nearly parallel G2-structures, Asian J. Math. 16 ( 2012), 713– 744. Google Scholar CrossRef Search ADS 2 M. Al Nuwairan, SU(2)-irreducibly covariant and EPOSIC channels, math.PH/1306.5321. 3 D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics 203, Birkhäuser Boston, Inc., Boston, 2002. Google Scholar CrossRef Search ADS 4 M. Fernández and A. Gray, Riemannian manifolds with structure group G2, Ann Mat. Pura Appl. 32 ( 1982), 19– 45. Google Scholar CrossRef Search ADS 5 L. Foscolo, Deformation theory of nearly Kähler manifolds, J. Lond. Math. Soc. 95 ( 2017), 586– 612. Google Scholar CrossRef Search ADS 6 W. Fulton and J. Harris, Representation Theory: A first Course , Springer, Berlin, 1991. 7 D. Gayet, Smooth moduli spaces of associative submanifolds, Q. J. Math. 65 ( 2014), 1213– 1240. Google Scholar CrossRef Search ADS 8 R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 ( 1982), 47– 157. Google Scholar CrossRef Search ADS 9 D. Huybrechts, Complex Geometry , Springer, Berlin, 2005. 10 S. Karigiannis, Deformations of G2 and Spin(7)-structures, Can. J. Math. 57 ( 2005), 1012– 1055. Google Scholar CrossRef Search ADS 11 K. Kawai, Deformations of homogeneous associative submanifolds in nearly parallel G2-manifolds, Asian J. Math. 21 ( 2017), 429– 462. Google Scholar CrossRef Search ADS 12 K. Kawai, Some associative submanifolds of the squashed 7-sphere, Q. J. Math. 66 ( 2015), 861– 893. Google Scholar CrossRef Search ADS 13 N. Koiso, Rigidity and infinitesimal deformability of Einstein metrics, Osaka J. Math. 19 ( 1982), 643– 668. 14 H. V. Lê and L. Schwachöfer, Lagrangian submanifolds in strict nearly Kähler 6-manifolds, math.DG/1408.6433. 15 J. D. Lotay, Stability of coassociative conical singularities, Commun. Anal. Geom. 20 ( 2012), 803– 867. Google Scholar CrossRef Search ADS 16 J. D. Lotay, Associative submanifolds of the 7-sphere, Proc. Lond. Math. Soc. 105 ( 2012), 1183– 1214. Google Scholar CrossRef Search ADS 17 K. Mashimo, Minimal immersions of 3-dimensional spheres into spheres, Osaka J. Math. 21 ( 1984), 721– 732. 18 D. Mcduff and D. Salamon, J-Holomorphic Curves and Symplectic Topology , American Mathematical Society, Providence, RI, 2004. Google Scholar CrossRef Search ADS 19 R. C. McLean, Deformations of calibrated submanifolds, Commun. Anal. Geom. 6 ( 1998), 705– 747. Google Scholar CrossRef Search ADS 20 M. Mukai, The deformation of harmonic maps given by the Clifford tori, Kodai Math. J. 20 ( 1997), 252– 268. Google Scholar CrossRef Search ADS 21 A. L. Onishchik, Lectures on Real Semisimple Lie algebras and Their Representations, EMS, 2004. 22 C. Procesi, Lie groups: An Approach through Invariants and Representations, Springer, New York, 2007. Appendix A. Representations of SU(2) In this section, we summarize the results about representations of SU(2). First, we recall the C-irreducible representations of SU(2). Let Vn be a C-vector space of all complex homogeneous polynomials with two variables z1,z2 of degree n, where n≥0, and define the representation ρn:SU(2)→GL(Vn) as (ρn(a−b¯ba¯)f)(z1,z2)=f((z1,z2)(a−b¯ba¯)).Define the Hermitian inner product ⟨,⟩ of Vn such that {vk(n)=z1n−kz2k/k!(n−k)!}0≤k≤nis a unitary basis of Vn. Denoting by SU(2)^ the set of all equivalence classes of finite dimensional irreducible representations of SU(2), we know that SU(2)^={(Vn,ρn);n≥0}. Then, every C-valued continuous function on SU(2) is uniformly approximated by the C-linear combination of {⟨ρn(·)vi(n),vj(n)⟩;n≥0,0≤i,j≤n},which are mutually orthogonal with respect to the L2 inner product. Next, we review the R-irreducible representations of SU(2) by [17, Section 2]. A more general reference of this topic is [21]. Define the map j:Vn→Vn by (jf)(z1,z2)=f(−z¯2,z¯1)¯, (A.1)which is a C-antilinear SU(2)-equivariant map satisfying j2=(−1)n. This map j is called a structure map [17, Section 2]). When n is even, we have j2=1 and Vn decomposes into two mutually equivalent real irreducible representations: Vn=(1+j)Vn⊕(1−j)Vn. When n is odd, Vn is also irreducible as a real representation. All of the real irreducible representations are given in this way, and hence their dimensions are given by 4m or 2n+1 for m,n≥0. Denote by Wk, where k∈4Z∪(2Z+1), the k-dimensional R-irreducible representation of SU(2). It follows that V2m+1=W4m+4,V2m=W2m+1⊕W2m+1form≥0. (A.2) The characters χVn of Vn are determined by the values on the maximal torus {ha=(a00a−1);a∈C,∣a∣=1}of SU(2). It is well known that χVn(ha)=∑k=0na2k−n=an+1−a−(n+1)a−a−1.By (A.2), the characters χWk of Wk on the maximal torus are given by χW4m+4(ha)=2χV2m+1(ha)=2∑k=02m+1a2k−(2m+1)=2(a2m+2−a−(2m+2))a−a−1,χW2m+1(ha)=χV2m(ha)=∑k=02ma2k−2m=a2m+1−a−(2m+1)a−a−1. (A.3) Finally, we summarize technical lemmas. Lemma A.1 ([11, Lemma 6.9]) For u=∑l=0nClvl(n)∈Vn, set u*=ju=∑l=0n(−1)n−lC¯n−lvl(n)∈Vn. Then, for any n≥0,0≤k≤n,u∈Vn, we have ⟨ρn(·)vk(n),u⟩¯=(−1)k⟨ρn(·)vn−k(n),u*⟩. (A.4) Let {E1,E2,E3}be the basis of the Lie algebra su(2)of SU(2)given by (4.3). Identify Ei∈su(2)with the left invariant differential operator on SU(2). Then (−iE1+E2)⟨ρn(·)vk(n),u⟩={2i(k+1)(n−k)⟨ρn(·)vk+1(n),u⟩,(k<n)0,(k=n)(iE1+E2)⟨ρn(·)vk(n),u⟩={2ik(n−k+1)⟨ρn(·)vk−1(n),u⟩,(k>0)0,(k=0)iE3⟨ρn(·)vk(n),u⟩=(−n+2k)⟨ρn(·)vk(n),u⟩. (A.5) Since the Haar measure is SU(2)-invariant, we have the following. Lemma A.2 For any X∈su(2)and a smooth function f on SU(2), we have ∫SU(2)X(f)(g)dg=0. Appendix B. Clebsch–Gordan decomposition Use the notation in Section A. In the computation in Section 4, we need the irreducible decomposition of Vm⊗Vn for m,n≥0. This is well known as the Clebsch–Gordan decomposition: Vm⊗Vn=⊕h=0min{m,n}Vm+n−2h.Identify Vm⊗Vn with the vector subspace of polynomials in (z1,z2,w1,w2) consisting of homogeneous polynomials of degree m in (z1,z2) and of degree n in (w1,w2). Then, the inclusion Vm+n−2h→Vm⊗Vn is explicitly given as follows. Lemma B.1 ([22, p. 46], [2, Section 2.1.2]) For 0≤h≤min{m,n}, define the map αm,n,h:Vm+n−2h→Vm⊗Vnby αm,n,h(f(z1,z2))=cm,n,h(z1w2−z2w1)h(w1∂∂z1+w2∂∂z2)n−h(f(z1,z2)),where cm,n,h>0is given in [2, Section 2.2.2]. Then, the map αm,n,his SU(2)equivariant and isometric. Denote by ρm,n the induced representation of SU(2) on Vm⊗Vn. Since we know that ⟨ρm(g)um,um′⟩⟨ρn(g)un,un′⟩=⟨ρm,n(g)(um⊗un),um′⊗un′⟩for um,um′∈Vm,un,un′∈Vn and g∈SU(2), we have the following by Lemma B.1 and the Schur orthogonality relations. Lemma B.2 Set r=m+n−2h. Then, we have ∫SU(2)⟨ρm(g)um,um′⟩⟨ρn(g)un,un′⟩⟨ρr(g)ur,ur′⟩¯dg=1r+1⟨um⊗un,αm,n,h(ur)⟩⟨um′⊗un′,αm,n,h(ur′)⟩¯for uj,uj′∈Vj. The next lemma is very useful for the computation in Section 4. Lemma B.3 ∫SU(2)⟨ρm(g)va(m),um′⟩⟨ρn(g)vb(n),un′⟩⟨ρr(g)vc(r),ur′⟩¯dg=0for any um′∈Vm,un′∈Vn,ur′∈Vrif a+b≠c+h(=c+m+n−r2). Proof We compute (r!(r−c)!/cm,n,h)αm,n,h(vc(r))=(z1w2−z2w1)h(w1∂∂z1+w2∂∂z2)n−h(z1r−cz2c)=∑i=0h∑j=0n−h(hi)(n−hj)(z1w2)i(−z2w1)h−iw1jw2n−h−j(∂∂z1)j(∂∂z2)n−h−j(z1r−cz2c)∈span{v(h−i)+c−(n−h−j)(m)⊗vi+(n−h−j)(n);0≤i≤h,0≤j≤n−h}⊂span{vd(m)⊗ve(n);d+e=c+h},which gives the proof.□ In this paper, the case of (m,n,h)=(4,4,1) or (6,6,3) is important. Recall that the character of the induced representation on the second symmetric power S2(Vn) is given by (χVn(g)2+χVn(g2))/2 (for example, see [6, Exercise 2.2]). By computing the character of S2(V4) and S2(V6), we see that S2(V4)=V8⊕V4⊕V0,S2(V6)=V12⊕V8⊕V4⊕V0.Thus, we have α4,4,1(V6)⊂Λ2V4, α6,6,3(V6)⊂Λ2V6 and we obtain the following. Lemma B.4 Suppose that m=4or6and um,uˆm,um′,uˆm′∈Vm. If um=uˆmor um′=uˆm′, we have ∫SU(2)⟨ρm(g)um,um′⟩⟨ρm(g)uˆm,uˆm′⟩⟨ρ6(g)v6,v6′⟩¯dg=0,for any v6,v6′∈V6. The next lemma is straightforward and we omit the proof. Lemma B.5 α4,4,1(v0(6))=c4,4,1·245v0(4)∧v1(4),α4,4,1(v1(6))=c4,4,1·245v0(4)∧v2(4),α4,4,1(v2(6))=c4,4,1·24(3v0(4)∧v3(4)+2v1(4)∧v2(4)),α4,4,1(v3(6))=c4,4,1·24(v0(4)∧v4(4)+2v1(4)∧v3(4)),α4,4,1(v4(6))=c4,4,1·24(3v1(4)∧v3(4)+2v2(4)∧v3(4)),α4,4,1(v5(6))=c4,4,1·245v2(4)∧v4(4),α4,4,1(v6(6))=c4,4,1·245v3(4)∧v4(4). Appendix C. Irreducible decomposition of spin(7) In this section, we give an irreducible decomposition of the Lie algebra spin(7) of Spin(7) under an SU(2)-action. First, we study the su(4)⊂spin(7) case. Lemma C.1 Use the notation in AppendixA. Let SU(2)act on su(4)by the composition of ρ3:SU(2)↪SU(4)given by (4.1) and the adjoint action of SU(4)on su(4). Then, we have su(4)≅W3⊕W5⊕W7.More explicitly, Wkcorresponds to the k-dimensional SU(2)-invariant subspace Wksu(4)of su(4), where k=3,5,7, given by W3su(4)=(ρ3)*su(2)={(3ia3z00−3z¯ia2z00−2z¯−ia3z00−3z¯−3ia);z∈C,a∈R},W5su(4)={(iazw0−z¯−ia0w−w¯0−ia−z0−w¯z¯ia);z,w∈C,a∈R},W7su(4)={(iaz1z2z3−z¯1−3ia−3z1−z2−z¯23z¯13iaz1−z¯3z¯2−z¯1−ia);z1,z2,z3∈C,a∈R}. Proof First, we compute the character of this representation on the maximal torus by using (4.1) and (A.3). Then, we see that it is given by χW3+χW5+χW7. This is a straightforward computation, so we omit it. Hence, we obtain the first statement. We easily see that the three spaces above are invariant by the adjoint action of (ρ3)*su(2). Hence, the three spaces above are SU(2)-invariant and the proof is done.□ The explicit description of spin(7) is given in [16, Proposition 4.2]. It is straightforward to deduce the following, so we omit the proof. Lemma C.2 We have spin(7)=su(4)⊕W1spin(7)⊕W5spin(7),where W1spin(7)=RH0,H0=(000000100000000−10000−10000000010000100000000−10000−1000000001000000),W5spin(7)={(00−a1−a2−a3−a40−a500−a2a1−a4a3−a50a1a2000−a5−a3a4a2−a100−a50a4a3a3a40a500a1−a2a4−a3a5000−a2−a10a5a3−a4−a1a200a50−a4−a3a2a100);a1,⋯.a5∈R}.By using the notation in AppendixA, H0is the structure map j:V3→V3given in (A.1) with respect to the basis {v0(3),iv0(3),…,v3(3),iv3(3)}. Lemma C.3 Use the notation in AppendixA. Let SU(2)act on spin(7)by the composition of ρ3:SU(2)↪SU(4)⊂Spin(7)given by (4.1) and the adjoint action of Spin(7)on spin(7). Then, we have spin(7)≅(W1⊕W5)⊕(W3⊕W5⊕W7).The subspaces W1and the first W5correspond to W1spin(7)and W5spin(7)in LemmaC.2, respectively. The subspace W3⊕W5⊕W7corresponds to su(4), whose irreducible decomposition is given in LemmaC.1. Proof By Lemma C.2, we only have to prove that H0 is invariant under the SU(2)-action and W5spin(7) is an irreducible 5-dimensional representation of SU(2). Since H0 is the structure map, it is invariant under the SU(2)-action. We easily see that W5spin(7) is invariant by the adjoint action of (ρ3)*su(2). Hence, it is SU(2)-invariant. As in the proof of Lemma C.1, we can compute the character of the SU(2)-representation on W5spin(7) and it is equal to χW5. Hence, the proof is done.□ © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com

The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Mar 1, 2018

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