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On the $G



On the G2 bundle of a Riemannian 4-manifold
arXiv:0808.1714v1 [math.DG] 12 Aug 2008
R. Albuquerque? rpa@dmat.uevora.pt August 12, 2008

Abstract We expose the theory of th

e construction of a natural G2 structure on the unit sphere tangent bundle SM of any given orientable Riemannian 4-manifold M , as laid in [4]. This time we work in the context of metric connections, or geometry with torsion.

Key Words: metric connections, sphere bundle, G2 structure, Einstein manifold, holonomy. MSC 2000: Primary: 53C10, 53C20, 53C25; Secondary: 53C05, 53C28

The author acknowledges the support of Funda? c? ao Ci? encia e Tecnologia, either through POCI/MAT/60671/2004 and through CIMA-UE, Centro de Investiga? c? ao em Matem? atica ? e Aplica? c? oes da Universidade de Evora.

1

Recalling the theory

By a G2 manifold it is understood a 7 dimensional Riemannian manifold with holonomy group contained in G2 = Aut(O). The structure is characterized by a smooth 3-form which at each point is given by φ(u, v, w ) = uv, w , where uv is the octonionic product, and it is known that there exists a non-degenerate φ if the ?rst two Stiefel-Whitney classes vanish. Besides this topological determination, by general principles of Riemannian geometry, the holonomy is in G2 if and only if φ is parallel. It was proved the latter di?erential condition corresponds with φ being harmonic and it is a particularly di?cult problem to
? Departamento de Matem? atica da Universidade de Evora and Centro de Investiga? ca ?o em Matem? atica ? e Aplica? co ?es (CIMA), Rua Rom? ao Ramalho, 59, 7000 Evora, Portugal.
?

R. Albuquerque

2

?nd examples of G2 holonomy, one of the possibilities shown in the celebrated list of M. Berger, as one may deduce from [8, 10]. Leaving aside such distinguished conditions, it is possible to ?nd G2 structures on Riemannian 7-manifolds which solve rather less stringent equations. These follow from the decomposition of the ∧? under the action of the exceptional group, which yields particular features in degrees 4 and 5, those of dφ and d ? φ, with equivalents in degrees 2 and 3 by the Hodge operator duality. Such representation theory was ?rst discovered by [13]. In this work we recall a particular construction, found in [4], on the unit sphere tangent bundle SM of a Riemannian 4-manifold M . It assigns smoothly the structure of the imaginary part of an octonionic structure to each Tu SM, u ∈ SM , having uR as the real line. So far, it has been deduced how the respective 3-form φ is written, its exterior di?erential and co-di?erential. By topological reasons, such structure cannot be harmonic. In the present article we see what happens to the equations when the torsion is of vectorial type, a simple type of torsion. Thus the ?rst purpose is to analyze the variations of the G2 structure on the sphere bundle depending of the torsion on M . There is manifest interest on co-calibrated structures, de?ned by the weaker condition δφ = 0, as we see from recent work of Th. Friedrich and S. Ivanov in [15]. The stricter case of nearly parallel structures, dφ = c ? φ, with c constant, has also been thoroughly studied, as the reader may see in [14]. A manifold admits a G2 structure if, and only if, its two ?rst Stiefel-Whitney classes w1 , w2 vanish. We remark also the interesting feature of SM which follows by changing the sign of the metric only on the ?bre direction. This corresponds with the split octonions and the non-compact dual of G2 . Hence we may construct a G2 structure using the same method.

2

The sphere bundle, its metric and Levi-Civita connection

Let (M, , ) be a smooth orientable Riemannian manifold of dimension 4. Let D be a metric connection on M , ie. a linear connection for which the metric is parallel. We denote SM = u ∈ T M | u = 1 , (1) the total space of the unit sphere bundle, and let f denote the projection onto the base M . Notice SM is always orientable. The connection D induces a direct sum decomposition T SM = V ⊕ HD into vertical and horizontal distribuitions: V = ker df and HD ? f ? T M . Furthermore, the bundle V may be identi?ed with the subvector bundle of f ?1 T M (occasionaly we use the notation f ?1 to refer to the vertical side) such that Vu = u⊥ ?
df

R. Albuquerque

3

Tf (u) M . In the following we denote by U the ubiquous section U : SM → f ?1 T M de?ned by Uu = u. We recall that HD = {X ∈ T SM | (f ? D )X U = 0} and f ? D Y U = Y for any Y ∈ V , a result whose complete proof the reader may see in [4]. In this regard, it is worthwile to notice that U only varies in vertical directions, as the identity map, and that, for a given section X ∈ ?0 (T M ), ie. a vector ?eld, the vertical part of dX (Z ) is precisely DZ X , for any Z ∈ Tx M, x ∈ M . Indeed, in this case X ? Ux = UX (x) = Xx , therefore (dX (Z ))v = f ? DdX (Z ) U = (X ? f ? D )Z X ? U = DZ X . Now, we endow SM with the unique Riemaniann structure for which the two distribuitions sit orthogonally inside T SM and the identi?cations with f ? T M are isometric. We use · v and · h to denote the obvious projections. The tangent bundle T SM also inherits a metric connection via the pull-back connection f ? D preserving the decomposition above, which we shall denote by D ? . Notice the star, due to the fact that on vertical tangent directions we must add a correction term D ? X v = f ? DX v ? f ? DX v , U U (2)

in order to have a metric linear connection. Notice ? f ? DY X, U = X, f ? DY U = X v , Y v for X, Y ∈ T SM . We shall use explicitly the following mirror isomorphism θ : HD → f ?1 T M ? V de?ned by f ? D parallel isometric identi?cation of the pull-back bundle of T M . We may extend θ by 0 to the vertical tangent bundle, so in fact we have θ : T T M ?→ T T M,
?

θ t θ (X h ) = X h ,

θθt (X v ) = X v .

(3)

Let R? = f ? RD = Rf D denote the curvature of f ? D . The latter identity follows by tensoriality. We use RD (X, Y ) = DX DY ? DY DX ? D[X,Y ] and view R? U as the following V -valued 2-form R over SM :
3

Ru (X, Y ) = R? (X, Y )Uu =
i=1

RD (f? X, f? Y )u, ei ei+3 ,

?u ∈ SM,

(4)

where e0 , . . . , e3 is a local orthonormal frame of HD such that e0 = θt U and ei+3 = θei , i ≥ 1. Notice R? U ⊥ U because D is an so(4)-connection. Let T denote the HD -valued form f ? T D , the pull-back of the torsion tensor of D : T D (X, Y ) = DX Y ? DY X ? [X, Y ]. Like R? , T varies only on two horizontal directions. ? We remark T D = T + R. Only the vertical part here is unexpected. It follows from applying f ? D· U as the projection map. Theorem 2.1. The Levi-Civita connection ? of SM is given by 1 ? ? X Y = DX Y ? R(X, Y ) + A(X, Y ) + τ (X, Y ) 2 where A, τ are HD -valued tensors de?ned by A(X, Y ), Z h = 1 1 R(X h , Z h ), Y v + R(Y h , Z h ), X v 2 2 (6) (5)

R. Albuquerque and τ (X, Y, Z ) = τ (X, Y ), Z h = 1 T (Y, X, Z ) ? T (Z, X, Y ) + T (Y, Z, X ) , 2

4

(7)

with T (X, Y, Z ) = f ? T D (X, Y ), Z , for any vector ?elds X, Y, Z over T M . Proof. This is a straightforward veri?cation of identities d X, Y = ?X, Y + X, ?Y and T ? = 0. This checking was done in [4] for the torsion free case; the present one being equally trivial. In sum, D ? preserves types, R takes vertical values and A and τ take horizontal values. Moreover, A(X, Y ) = 0 if X, Y are both horizontal or both vertical, whereas R and τ vanish if one direction is vertical. Notice the well known result following trivially from the formula above: HD corresponds with an integrable distribuition if, and only if, D is ?at. Also, by the skew-symmetries in X, Y , τ = 0 if, and only if, T D = 0.

3

The canonical G2 structure or the g-twoistor space

Any quaternionic Hermitian structure on an oriented Euclidian 4-vector space Q arises from a unit direction u ∈ S 3 ? Q, which plays the role of the generator of the real line in the quaternions; indeed, the volume form volQ coupled with u determines an exterior product in u⊥ , and that is su?cient to have a quaternionic Hermitian line. An octonionic structure follows on Q ⊕ Q, by the well known Cayley-Dickson process, cf. [17], and hence a Lie group G2 = Aut Q associated to u is also determined. Continuing with the study of our space SM , we ?rst de?ne an octonionic structure in each Tu T M ? Tx M ⊕ Tx M , for u ∈ Sx M . The imaginary part of such non-associative division algebra is precisely T SM and this is what we call the canonical G2 structure of an oriented Riemannian 4-manifold (cf. [4]). Because of the somehow tautological construction and its resemblance with twistor theory we write the following de?nition. De?nition 1. We call g-twoistor bundle of M associated to D to the unit sphere tangent bundle SM of an oriented Riemannian 4-manifold M together with its canonical G2 structure induced from a linear metric connection D .

Remark. Given any p-tensor η and any endomorphisms Bi of the tangent bundle we let η ? (B1 ∧ . . . ∧ Bp ) denote the new p-tensor de?ned by η ? (B1 ∧ . . . ∧ Bp )(Y1 , . . . , Yp ) =
σ∈Sp

sg(σ )η (B1 Yσ1 , . . . , Bp Yσp ).

(8)

It is easy to check such contraction is parallel and that it obeys a simple Leibniz rule under covariant di?erentiation, with no ? signs attached. For instance, if η is a p-form,

R. Albuquerque

5

then η ? ∧p 1 = p! η . Furthermore, one veri?es that for a wedge of 1-forms, η1 ∧ . . . ∧ ηp ? (B1 ∧ . . . ∧ Bp ) = σ∈Sp η1 ? Bσ1 ∧ . . . ∧ ηp ? Bσp . Now, any G2 structure on a 7 dimensional Riemannian manifold is entirely determined by a non-degenerate 3-form, say φ. As in [4], we de?ne a 3-form α = U f ?1 volM ∈ ?0 (SM ; Λ3 V ? ) ? ?3 SM ; a 1-form ? by ?(X ) = U, θ (X ) and a 2-form β such that β (X, Y ) = θX, Y ? θY, X = θX h , Y v ? θY h , X v . Using the natural contraction, we de?ne αi ∈ ?3 , for i = 1, 2, by 1 α1 = α ? (θ ∧ 1 ∧ 1), 2 1 α2 = α ? (θ ∧ θ ∧ 1) 2 (9)

where 1 is the identity map. To understand this notice α is a 3-form and hence α1 = + α(θX, Y, Z ), the cyclic sum. Finally the associated 3-form φ of the g-twoistor bundle
X,Y,Z

of M and its image under Hodge-? are given respectively by φ = α + ? ∧ β ? α2 and 1 2 ? φ = vol? M ? β ? ? ∧ α1 , 2 (10)

? where we denote vol? M = f volM . In [4] one ?nds the proof that φ corresponds to the 3-form of the g-twoistor structure. The following result is very useful.

Proposition 3.1 (?rst structure equations, [4]). We have the following basic relations: ?α = vol? ?α1 = ?? ∧ α2 , ?α2 = ? ∧ α1 , M, 1 ?β = ? ? ∧ β 2 , ?β 2 = ?2? ∧ β, β 3 ∧ ? = ?6VolSM , 2 1 β ∧ αi = β ∧ ?αi = α0 ∧ αi = 0, ?i = 0, 1, 2, α1 ∧ α2 = 3 ? ? = ? β 3 , 2

(11)

where α0 = α. Henceforth α ∧ φ = α2 ∧ φ = ?α1 ∧ φ = 0 and ?α ∧ φ = α ∧ ?φ = VolSM . To study these forms we use a frame. Let e0 = θt U, e1 , . . . , e6 denote the direct orthonormal basis of T SM induced from e0 , . . . , e3 , a direct orthonormal basis of HD , as in (4). It is easy to prove the existence of such frames: we ?x a direct o.n. frame f1 , . . . , f4 on an open set in M and then take Cartesian coordinates (u1 , . . . , u4 ) ∈ S 3 to write e0 = u = uifi . Then e1 , e2 , e3 follow by a well known transformation in Sp(1) of the ui , cf. section 3.1. Now, by de?nition, α = e456 and ? = e0 . Thus ?α = e0123 = vol? M. 14 25 36 It is also trivial to see β = e + e + e . From direct inspection on α composed with θ we ?nd α1 = e156 + e264 + e345 and α2 = e126 + e234 + e315 . (12) Hence ?α1 = ?e0234 + e0135 ? e0126 = ?? ∧ α2 and ?α2 = e0345 + e0156 + e0264 = ? ∧ α1 . Now β 3 = (2e1425 +2e1436 +2e2536 ) ∧ β = 6e142536 = ?6 ? ?. Finally, ?β = e02356 + e01346 + e01245 = 1 ?2 ? ∧ β 2 and ?β 2 = ?2 ? (e1245 + e1346 + e2356 ) = ?2(e036 + e025 + e014 ) = ?2? ∧ β . The other 6 +2 )VolSM = 7. relations in (11) follow as easily. And thereby |φ|2 = ?(φ ∧ ?φ) = ?(1 + 6 2

R. Albuquerque

6

Proposition 3.2. For any vector ?eld X over SM : 1 1. ?X α = 4 α ? (R(X, ·) ∧ 1 ∧ 1) = AX α. 1 (AX α) ? θ ∧ θ ∧ 1 ? α ? θ(AX + τX ) ∧ θ ∧ 1. 2. ?X α2 = 2 ?X (α ? θ ∧ θ ∧ 1) = 1 2 1 1 3. ?X α1 = 2 ?X (α ? θ ∧ 1 ∧ 1) = 2 (AX α) ? θ ∧ 1 ∧ 1 ? 1 α ? θ(AX + τX ) ∧ 1 ∧ 1. 2 ? 4. ?X ? = X ? θ + ? ? (AX + τX ). 1 3 5. d vol? M = 0, d? = ?β + ?(T ) and δ? = 6 ? dβ . 6. dβ (X, Y, Z ) = d(?T )(X, Y, Z ) = + X, θT (Y, Z ) + ?(R(X, Y )Z ) , where R is the
X,Y,Z

curvature in the horizontal subbundle. 7. If D is torsion free, then dβ = 0, d? = ?β, δ? = 0.
1 ? R(X, Yi ) + AX Yi + τX Yi for any three vector ?elds Proof. 1. We have ?X Yi = DX Yi ? 2 Y1 , Y2 , Y3 on SM and thus ? ?1 ? ?X α(Y1 , Y2, Y3 ) = (DX f volM )(U, Y1 , Y2 , Y3) + f ?1 volM (DX U, Y1 , Y2 , Y3) 1 + α(R(X, Y1 ), Y2 , Y3) + α(Y1, R(X, Y2 ), Y3 ) + α(Y1 , Y2, R(X, Y3 )) . 2 ? The ?rst two terms on the sum vanish because D volM = 0 and because DX U = Xv and the Yiv ∈ V are linearly dependent. For the second identity, if we see α = e456 as previously, then 1 α ? (R(X, ·) ∧ 1 ∧ 1) = (AX e4 )? ∧ e56 ? (AX e5 )? ∧ e46 + (AX e6 )? ∧ e45 , 4 ie. AX acts as a derivation of α. ? = ? + f ? D, U U , ie. the Levi-Civita connection of T M . Since α is 0 when we 2. Let ? take one direction proportional to U , the derivative ? of α2 we have to compute can be ? . Since ? ? X θ = [AX + τX , θ], the usual Leibniz rules which one may check made with ? directly yields

? Xθ ∧ θ ∧ 1 ?X (α ? θ ∧ θ ∧ 1) = ?X α ? θ ∧ θ ∧ 1 + 2α ? ? = (AX α) ? θ ∧ θ ∧ 1 ? 2α ? θ(AX + τX ) ∧ θ ∧ 1. 3. This result is analogous to the previous one. 4. Here we just have to compute: (?X ?)Y = X (?Y ) ? ?(?X Y ) = f ? DX U, θY + U, f ?DX (θY ) ? U, θ(f ?DX Y ) ? U, θ(AX Y + τX Y ) = X, θY ? ?(AX Y + τX Y ). 5. Since A is symmetric and τijk ? τjik = ?Tijk , we get d?(X, Y ) = (?X ?)Y ? (?Y ?)X = X, θY ? ?(τX Y ) ? Y, θX + ?(τY X ) = ?β (X, Y ) + ?(T (X, Y )).
1 Furthermore, δ? = ? ? d ? ? = 6 ? dβ 3 . Of course, d vol? M = 0. 6. It is well known that for any linear connection D , dD 1 = T D and dD T D =Bianchi ? identity, say BRD . Now in our case it is easy to see that df D T = BR? . Hence

d(?(T ))(X, Y, Z ) =

+ X (?(T (Y, Z ))) ? ?(T ([X, Y ], Z ))
?D

= ?(df =

T )(X, Y, Z ) + + X (?(T (Y, Z ))) ? ?(f ?DX (T (Y, Z )))

= ?(BR(X, Y, Z )) + + (f ? DX ?)(T (Y, Z )) + ?(R(X, Y )Z ) + X, θT (Y, Z )

R. Albuquerque and the result follows. Notice this R is in HD .

7

Recall the Ricci tensor of M is de?ned by r (X, Y ) = Tr R·D ,X Y . It is also given by an endomorphism Ric ∈ ?(End T M ) satisfying r (X, Y ) = X, Ric Y . Recall r is symmetric if T D = 0, ie. in the case of D being the Levi-Civita connection. We shall denote by r the function r (U, U ) on SM and set τ ?1 = (Ric U )? ∈ ?(V ? ) as a 1-form vanishing on HD and restricted to vertical tangent directions. Proposition 3.3. 1. dα = Rα. T α1 and ? ∧ Rα1 = ?vol? ?1 . 2. dα1 = Rα1 + 1 M ∧τ 2 ? 3. dα2 = ?r volM + T α2 . Proof. 1. We continue to use the frame e0 , . . . , e6 . Let Rijkl = Rei ,ej ek , el . Since Ai e4 , ej + Aj e4 , ei = R? (ei , ej )U, e4 = Rij 01 , then
6

dα =
i=0 6

1 e ∧ ?ei α = 4
i

6

ei ∧ α ? (R(ei , ·) ∧ 1 ∧ 1) =
i=0

=
i=0

ei ∧ (Aei e4 )? ∧ e56 + (Aei e5 )? ∧ e64 + (Aei e6 )? ∧ e45 = =
0≤i<j ≤3

Rij 01 eij 56 + Rij 02 eij 64 + Rij 03 eij 45 = Rα.

2. Since ej +3 θ = ej and A has only horizontal values, we have
6

dα1 = = =
i=0

ei ∧ ?ei α1 =
i=0 6

1 2

ei ∧ (Ai α) ? θ ∧ 1 ∧ 1 ? α ? θAi ∧ 1 ∧ 1 ? α ? θτi ∧ 1 ∧ 1
i=0

3

ei ∧ (Aei e4 )? ∧ (e26 + e53 ) ? (Aei e5 )? ∧ (e16 + e43 ) + (Aei e6 )? ∧ (e15 + e42 ) 1 1 ei ∧ α ? (θ(Aei ej )ej ∧ 1 ∧ 1) ? τij 1 eij 56 + τij 2 eij 64 + τij 3 eij 45 . ? 2 i,j =0 2 i,j =0
6 3

The second sum above has a contraction of a symmetric derivation A within a skew tensor, so its contribuition is null. Also, using the symmetries of τ , τijk ? τjik = ?Tijk , we ?nd dα1 =
0≤i<j ≤3

Rij 01 (eij 26 + eij 53 ) ? Rij 02 (eij 16 + eij 43 ) + Rij 03 (eij 15 + eij 42 ) + 1 1 Tij 1 eij 56 + Tij 2 eij 64 + Tij 3 eij 45 = Rα1 + T α1 . 2 0≤i<j ≤3 2

Now if we couple ? with the ?rst sum above, then we get ? ∧ Rα1 = ?(R1301 + R2302 )e01236 ? (R1201 + R3203 )e01235 ? (R2102 + R3103 )e01234

R. Albuquerque

8

just by using the two skew-symmetries in R which do not depend of T D . Since τ ?1 = 3 i+3 , it is immediate to conclude the result. i=1 r (ei , e0 )e 3. In this case, after careful inspection, we get
6

dα2 =
i=0

ei ∧

1 (Ai α) ? θ ∧ θ ∧ 1 ? α ? θAi ∧ θ ∧ 1 ? α ? θτi ∧ θ ∧ 1 2

=
0≤i<j ≤3

Rij 01 eij 23 + Rij 02 eij 31 + Rij 03 eij 12 +
0≤i<j ≤3

Tij 1 (eij 26 + eij 53 ) ? Tij 2 (eij 16 + eij 43 ) + Tij 3 (eij 15 + eij 42 )

which is easy to see to be equal to ?r (U, U )vol? M + T α2 . Since β ∧ α1 = 0 and ??T ∧ α1 + ? ∧ T α1 = T (? ∧ α1 ), we ?nd dφ = Rα ? β 2 + (?T ) ∧ β ? ? ∧ dβ + r vol? M ? T α2 , 1 ?1 + ? ∧ T α1 . d ? φ = ?dβ ∧ β ? ?T ∧ α1 ? vol? M ∧τ 2 (13)

The proof is immediate and hence the following result on the torsion free case; recall dβ = d(?T ). Theorem 3.1 (cf. [4]). Let D be the Levi-Civita connection of M . Then dφ = Rα ? β 2 + r vol? M . In particular, the g-twoistor space is never a G2 manifold, ie. its holonomy does not reduce to the exceptional group. We have δφ = ? ? d ? φ = Ric U α. In particular, SM is co-calibrated if, and only if, M is an Einstein manifold. Indeed, if δφ = 0, then Ric U must be a multiple of U , and reciprocally. Thus Ric = c1 and the last conclusion follows. There is also an interesting case with the so called metric connections with vectorial ? Cartan, such connections are characterized by their di?erence torsion. Introduced by E. with the Levi-Civita connection being the tensor B ∈ ?1 (End T M ) BX Y = X, Y ν ? ? ν (X )Y (14)

with ν ? a ?xed vector ?eld. Vectorial torsion is one of the three types appearing under orthogonal group-representation of the space of torsion tensors. Then it is easy to see that ?(T ) = ? ∧ ν, T α1 = α1 ∧ ν, T α2 = 2α2 ∧ ν and the reader may complete for dφ and d ? φ, cf. section 4.1. However, one must notice that di?erent connections induce di?erent horizontal distributions and thus further perturbation in φ.

R. Albuquerque

9

3.1

The trivial g-twoistor space

Before going further it is interesting to observe the case of R4 with canonical metric and trivial connection d. Here SM = R4 × S 3 ? R8 , in which we use coordinates (x, u) = ? are orthonormal and horizontal, hence the previously (x1 , . . . , x4 , u1 , . . . , u4). Now ?x i 0 1 6 announced co-frame e , e , . . . , e may be given by the identities e0 = ? = ui dxi and ? ? ? dx1 du1 ? ? ? e1 e4 ?u2 u1 ?u4 u3 ? ? ? ? dx2 du2 ? ? 2 5 ? ? (15) u1 ?u2 ? ? ?. ? e e ? = ? ?u3 u4 ? dx3 du3 ? 3 6 ?u4 ?u3 u2 u1 e e dx4 du4 Now a long but simple computation yields: α = u1 du234 ? u2 du134 + u3 du124 ? u4 du123 ? ∧ β = u1 (ξ122 + ξ133 + ξ144 ) + u2 (ξ211 + ξ233 + ξ244 ) + u3 (ξ311 + ξ322 + ξ344 ) + u4 (ξ411 + ξ422 + ξ433 ) α2 = 3(u1 ξ234 ? u2 ξ134 + u3 ξ124 ? u4 ξ123 ), where duijk = dui ∧ duj ∧ duk and ξijk = dxi ∧ dxj ∧ duk . Notice these results are valid 2 subject to the condition u2 1 + · · · + u4 = 1. Their purpose was to explicitly describe the 3-form φ = α + ? ∧ β ? α2 , which has dφ = ?β 2 .

4

Representation of G2

We are now going to ?nd the four torsion forms τ0 , τ1 , τ2 , τ3 which classify G2 structures. The irreducible decomposition of Λ? R7 as a G2 representation module may be seen in well known references such as [9, 13]. Since the star operator commutes with the group product, the problem resumes to say degrees 2 and 3. We have
2 ?2 = ?2 7 ⊕ ?14 , 3 3 ?3 = ?3 1 ⊕ ?7 ⊕ ?27 ,

(16)

2 2 2 3 where ?2 7 = {γ ∈ ? | γ ∧ φ = ?2 ? γ }, ?14 = {γ ∈ ? | γ ∧ φ = ?γ } ? g2 , ?1 = Rφ, 1 3 3 ?3 7 = {?(γ ∧ φ)| γ ∈ ? }, ?27 = {γ ∈ ? | γ ∧ φ = γ ∧ ?φ = 0}, with the indices below standing for the dimensions. The unique components of dφ and d ? φ on the space of representations are called the torsion forms. Fortunately one of them occurs in two places:

dφ = τ0 ? φ + 3τ1 ∧ φ + ?τ3 ,

d ? φ = 4τ1 ∧ ?φ + ?τ2 ,

(17)

3 with τi ∈ ?i , τ2 ∈ ?2 14 , τ3 ∈ ?27 . Hence there are in principle sixteen classes of G2 structures. There is a very helpful formula in [13]: if k5 ∈ ?5 , then the respective k1 ∈ ?1 for the second decomposition in (17) is given by 1 ? (? k 5 ∧ ? φ ). (18) 12 Notice the wedge on 1-forms with φ or ?φ is a G2 -equivariant monomorphism.

R. Albuquerque

10

4.1

Vectorial torsion

For brevity of notation, from this point on, we omit the wedge product symbol. Suppose D has vectorial torsion, being thus characterized by ?T = ? ∧ ν = ?ν for some ?xed vector ?eld ν ? . Indeed, following the expression in (14), we deduce ek (T ) = ij k 0≤i<j ≤3 Tijk e = e ν , where k = 0, . . . , 3. Henceforth T α1 = α1 ν, T α2 = 2α2 ν . Now, formally, dφ and d ? φ di?er from those expressions given for the torsion free case, say φ0 , as follows: dφ = dφ0 + (?T )β ? ?dβ ? T α2 = Rα ? β 2 + r vol? M + 2?νβ + 2να2 and, respectively, d ? φ = 1 3 = d ? φ0 ? (dβ )β ? (?T )α1 + ?(T α1 ) = ?vol? ?1 + β 2 ν + ?(dν )β + ?α1 ν, Mτ 2 2 see (13). This is due to the fact that d? = ?β + ?T and hence dβ = d(?ν ) = (?β + ?ν )ν ? ?dν = ?βν ? ?dν. Now we have the following proposition, whose proof is a matter of patient checking. Proposition 4.1. 2?νβ + 2να2 has no component in ??3 1 and decomposes as 1 2 3 3 1 3 ? νφ + 2?νβ + νφ + 2να2 = ? νφ ? ? ν ? (3vol? M + β + ?α1 ) 2 2 2 2 2 the sum within brackets being the one in ??3 27 . We may also call attention to the following two decompositions with no ??3 7 part: 6 1 1 6 β 2 = ? ? φ + ? ?β 2 + φ , vol? ? φ + ? vol? (23) M = M ? φ . 7 7 7 7 Notice dα = Rα describes completely any given metric curvature tensor. In order to see it in g-twoistor space we de?ne unique forms ξj such that Rα = ξ0 ? φ + 3ξ1 φ + ?ξ3 following (17). Theorem 4.1. Suppose D has vectorial torsion. Then the torsion di?erential forms of the g-twoistor space are: 1 6 2 τ1 = ?τ ?1 + (dν )? α2 ? ?, ν ? ? 5ν , τ0 = r + , 7 7 12 1 1 ? τ2 = Ric U (φ ? 3α) ? ν (?β + α2 ) ? (dν )? β 2 3 2 ??(?? (dν + ?M dν )) + dν + ?, ν β , τ3 = ξ3 + 1 1 2 1 (r ? 1)6α + (8 ? r)?β + (r + 6)α2 ? ν ? 3vol? M + β + ?α1 7 2 2
r 7

(19)

(20)

(21)

(22)

(24)

(25)

where τ ?1 = (Ric U )? . Moreover ξ0 =

and ξ1 = τ1 + ν . 2

R. Albuquerque

11

Proof. By direct inspection on formulae given in (11,12) and in propositions 3.1,3.3, we get (dφ)α = r VolSM , (dφ)α2 =
0≤i<j ≤3

Rij 01 eij 56234 + Rij 02 eij 64315 + Rij 03 eij 45126

= (R0101 + R0202 + R0303 )VolSM = ?r VolSM , (dφ)?β =
0≤i<j ≤3

Rij 01 eij 56014 + Rij 02 eij 64025 + Rij 03 eij 45036 ? ? ∧ β 3

= (?R2310 ? R3120 ? R1230 + 6)VolSM = 6VolSM , since the previous cyclic sum in 1,2,3 in R is ?d(?T )123 = 0. Indeed, formula 6 in proposition 3.2 gives dβ = d(?T ) =
0≤i<j ≤3

BRijk0 eijk + Tij 1 eij 4 + Tij 2 eij 5 + Tij 3 eij 6 .

And therefore BRe1 ,e2 e3 , e0 = d(?T )123 = 0 by (21). Notice also the less obvious equation which follows easily: the parcel with the curvature tensor is precisely ??dν . The computation above also follows, afterall, from (dφ)?β = φd(?β ) = ??β 3 = 6VolSM . Since ?(dφ ∧ φ) = τ0 |φ|2, we then ?nd 2r + 6 = 7τ0 . From formula (18) and since ei+3,i+4 ? φ = ei+3,i+4 vol? M , ?i ≥ 1, we deduce ?1 + κ5 ) ? φ) = ?τ ?1 + (dν )? α2 ? ?, ν ? ? 5ν 12τ1 = ?(?(?vol? Mτ
3 where κ5 = β 2 ν + ?(dν )β + 2 ?α1 ν . Indeed one proves ?rstly that ?(?(β 2 ν ) ? φ) = ?4 ?, ν ? 2ν , ?(?(?β dν ) ? φ) = (dν )? α2 , ?(?(?α1ν ) ? φ) = 2 ?, ν ? ? 2ν . Henceforth, using the general formula ?(ωX ?) = X ? ω , ? 3τ2 = 3 ? d ? φ ? 12 ? (τ1 ? φ) = ?3 ? (vol? ?1 ) + 3 ? κ5 ? 12τ1 φ Mτ 3 ? ? = ?3? τ1 α + ?3(β 2ν + ?(dν )β + ?α1 ν ) + τ ?1 φ ? ((dν )? α2 )? φ + ?, ν β + 5ν ? φ 2 3 1 9 = Ric U (φ ? 3α) + ν ? (?6?β + α2 + 5φ) ? (dν )? β 2 + (dν )? β 2 2 2 2 ? ? ??(? (?M dν )) + ? (?dν ) + ?, ν β 1 = Ric U (φ ? 3α) ? ν ? (?β + α2 ) ? (dν )? β 2 2 ? ??(? (dν + ?M dν )) + dν + ?, ν β.

For the above one computes aside, e.g., the identities (dν ? α2 )? α2 = e0 (?dν ) = dν ? ?(e0 dν ) and ((dν )? α2 )? (α + ?β ) = (v12 e6 + v23 e4 + v31 e5 ) (α + ?β ) = v12 e45 + v23 e56 + v31 e64 + v12 e03 + v23 e01 + v31 e02 1 = (dν )? ? β 2 + ?(e0 (?M dν ) 2

R. Albuquerque where dν = 0≤i<j ≤3 vij eij and ?M is the star operator on M . Finally we combine (19,22,23,24) to ?nd dφ = (ξ 0 +

12

ν 1 1 2 6 6 r r 2 + ) ? φ + 3(ξ1 ? )φ + ? ξ3 ? ν ? (3vol? M + β + ?α1 ) ? ?β ? φ + rα ? φ 7 7 2 2 2 7 7

and thence the result. We recall here a result for the case ν = 0. Proposition 4.2 ([4]). If M has constant sectional curvature C , then τ1 = τ2 = 0 and 6 τ0 = (C + 1), 7 τ3 = (3C ? τ0 )α + (2 ? τ0 )? ∧ β ? (C ? τ0 )α2 . (26)

4.2

Relation to SU (3)-structures

In [12] we have a set of ideas relating SU (3) with G2 -structures.

5

Conformal change of the metric

∞ Let us assume the notation M ′ = M , , ′ = e2? , , ? ∈ CM , to denote a conformal change of the metric on the given 4-manifold. Thence the same manifold has two associated g-twoistor spaces, when we consider the respective Levi-Civita connections D, D ′ of the given metric and its conformal change. In [5] one ?nds the recipes for the conformal change of D, RD , r D . However, SM and SM ′ are . Let

h : SM ?→ SM ′ ,

h(u) = u′ = e?? u

(27)

with obvious notation (e.g. ? is composed with f ). Proposition 5.1. Let X be any vector ?eld on SM and consider the di?erential h? : ′ T SM → T SM ′ . It satis?es the identities (h? X )h = X h by identi?cation of horizontal subspaces and (h? X )v = e?? (X v + d?(U )θX ? ?(X )grad ?) ∈ V . Proof. We known that D ′ = D + C where CX Y = d?(X )Y + d?(Y )X ? X, Y grad ?. ′ Then h? X h = (df ′ )?1 (df (X )) and this is what we again call X h . The vertical part is
′ ′ f ′? Dh U ′ = h? f ′? DX h? U ′ = f ? DX (e?? U ) + f ? CX e?? U ? (X )

= d(e?? )(X )U + e?? f ? DX U + e?? f ? CX U = e?? X v + e?? d?(U )θX ? e?? θX, U grad ?,
′ = e?? u. The result follows. since h? Uu

R. Albuquerque

13

References
[1] I. Agricola, S. Chiossi and A. Fino, Solvmanifolds with integrable and non-integrable G2 structures, Di?. Geom. Appl. 25 (2006), 125-135. [2] I. Agricola and T. Friedrich, Geometric Structures of Vectorial Type, J. Geom. Phys. 56 (2006), 2403-2414. [3] I. Agricola and C. Thier, The geodesics of metric connections with vectorial torsion, Ann. Global Anal. Geom. 26 (2004), no. 4, 321–332. [4] R. Albuquerque and I. Salavessa, The G2 sphere bundle of a 4-manifold Agosto 2006, arxiv: DG/0608282. [5] A. L. Besse, Einstein Manifolds, Springer-Verlag Berlin Heidelberg 1987. [6] M. Bobienski and P. Nurowski, Irreducible SO (3) geometry in dimension ?ve, J. Reine Ang. Math. (to appear). [7] E. Bonan, Sur les vari? et? es riemanniennes ` a groupe d’holonomie G2 ou Spin(7), C. R. Acad. Sci. Paris 262 (1966), 127–129. [8] R. L. Bryant, Metrics with exceptional holonomy, Annals of Mathematics (2), vol. 126 no. 3 (1987), 525–576. [9] R. L. Bryant, Some remarks on G2 structures, Proceedings of the 2004 Gokova Conference on Geometry and Topology (May, 2003). [10] R. L. Bryant and S. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J., vol. 58 no. 3 (1989), 829–850. [11] S. Chiossi and A. Fino, Special metrics in G2 geometry, Proceedings of the “II Workshop in Di?erential Geometry” (June 6-11, 2005, La Falda, Cordoba, Argentina), Revista de la Union Matematica Argentina. [12] S. Chiossi and S. Salamon, The intrinsic torsion of SU (3) and G2 structures, Di?erential Geometry, Valencia 2001, World Sci. Publishing, 115-133, (2002). [13] M. Fern? andez and A. Gray, Riemannian manifolds with structure group G2 , Ann. Mat. Pura Appl. 4 132 (1982), 19-45. [14] Th. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel G2 structures, Journ. Geom. Phys. 23 (1997), 259-286. [15] Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian Jour of Math., 6 (2002), n.2, 303–335

R. Albuquerque

14

[16] Th. Friedrich and S. Ivanov, Killing spinor equations in dimension 7 and geometry of integrable G2 -manifolds, Journ. Geom. Phys. 48 (2003), 1-11. [17] R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47–157. [18] D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press (2000).



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