Metric almost contact 3-structures - Geometric structures and special spinor fields

Since allψ_ileave the vector spaceV := span(ξ₁, ξ₂, ξ₃) invariant and since they are orthonormal, they also leaveV^⊥ invariant. ForX ⊥ξ₁, ξ₂, ξ₃, ∂_rwe have

J1(J2(X)) =ψ1(ψ2(X)) =ψ3(X) =J3(X) =−ψ2(ψ1(X)) =−J2(J1(X)).

Forξ1 we obtain

J1(J2(ξ1)) =−J1(ψ2(ξ1)) =J1(ξ3) =−ψ1(ξ3) =ξ2=J2(ar∂r) =−J2(J1(ξ1)) =ψ3(ξ1) =J3(ξ1) and similarly forξ₂, ξ₃ and ∂_r. For a connection as in Theorem (3.2), we have that all almost hermitian structures are parallel under ¯∇ and forX, Y ∈T M

[X, Y] = ¯∇^¯^g_XY −∇¯^¯^g_YX = ¯∇XY −∇¯YX−T¯(X, Y).

Thus the commutator relations are given by

[ξ1, ξ2] = a²[J1(∂r), J2(∂r)] = a²( ¯∇J1(∂r)J2(∂r)−∇¯J2(∂r)J1(∂r))−T(ξ¯ 1, ξ2)

= a²(J2( ¯∇J1(∂r)∂r)−J1( ¯∇J2(∂r)∂r))−T¯(ξ1, ξ2)

= a²(J2(J1(∂r))−J1(J2(∂r)))−T¯(ξ1, ξ2)

= −2a²J3(∂r)−T¯(ξ1, ξ2) = 2aξ3−T(ξ1, ξ2).

The other relations are to be calculated similarly.

If the almost contact structures are normal, then the almost hermitian structures are normal and with the formula for the torsion given in Remark 3.17 we have

g( ¯∇X, Y, Z) = ¯g( ¯∇^¯^g_XY, Z) +1

2(J_i◦dω_i)(X, Y, Z) for anyi= 1,2,3. This impliesJ₁◦dω₁=J₂◦dω₂=J₃◦dω₃.

Remarks 3.25.

- In [FFUV11, Section 7], the authors obtain a similar result (but without a description of the characteristic connections). Furthermore, they investigate more closely the conditions for the HKT structure to bestrong(J_i◦dω_i is closed).

- If we rescale the metric such thata= 1 and ifT = 0, we have 3 K¨ahlerian structures on ¯M and thus 3 Sasakian structures onM. Then the commutator relations in Theorem 3.24 ensure that the structures onM form a 3-Sasakian structure. This is Lemma 5 of [B¨a93]: A one to one correspondence between hyper-K¨ahler structures on ¯M and 3-Sasaki structures on M.

- We emphasize that it is not necessary that the three characteristic connections ∇^c,i, i = 1,2,3 coincide in order to apply Theorem 3.24, only the connections ∇ⁱ with torsion Tⁱ = T^c,i−2aηi∧Fi have to be equal. If M is a 3-Sasakian manifold, Tⁱ = 0 for i = 1,2,3 and thus∇¹ =∇² =∇³=∇^g. In this case there exists a specialG2 structure onM which will be discussed in Example 4.18.

4 G

₂

structures – Spin(7) structures on the cone

Let (M, g,Ψ, P) be aG2manifold (see Section 4 of Chapter I). For Spin(7) structures there is no one to one correspondence to spinors. Since we want to lift aG2structure to a Spin(7) structure on the cone, in this section it is more convenient to use the description of aG2 structures via the diﬀerential form Ψ rather then via a spinor. Correspondences of geometric structures on non-twisted cones using the diﬀerential forms are considered in many other cases, see [II05] to name but one.

We cite a classical, but for us crucial result by Fernandez and Gray:

Lemma 4.1 ([FG82, Lemma 2.7]).

∗Ψ(V, W, X, Y) =g(P(V, W), P(X, Y))−g(V, X)g(W, Y) +g(V, Y)g(W, X)

= Ψ(V, W, P(X, Y))−g(V, X)g(W, Y) +g(V, Y)g(W, X).

Remark 4.2. In [FG82] this formula is stated diﬀerently,

∗Ψ(V, W, X, Y) = −g(P(V, W), P(X, Y)) +g(V, X)g(W, Y)−g(V, Y)g(W, X).

= −Ψ(V, W, P(X, Y)) +g(V, X)g(W, Y)−g(V, Y)g(W, X).

This is due to the standard 3-form Ψ used by Fern´andez and Gray, which corresponds to the orientation opposite to ours. This changes the sign of the Hodge operator.

Now we are able to prove

Lemma 4.3. For any metric connection∇ with skew torsion onM, the G2 formΨsatisﬁes (∇Z∗Ψ)(V, W, X, Y) = (∇ZΨ)(V, W, P(X, Y)) + (∇ZΨ)(X, Y, P(V, W)).

If ∇ satisﬁes∇Ψ =a∗Ψfor somea >0, we have the simpliﬁed relation

(∇Z∗Ψ)(V, W, X, Y) = a[Ψ(X, Y, V)g(Z, W)−Ψ(X, Y, W)g(Z, V) +Ψ(V, W, X)g(Z, Y)−Ψ(V, W, Y)g(Z, X)].

Proof. For any metric connection with skew torsion we have

(∇Z∗Ψ)(V, W, X, Y) =Z∗Ψ(V, W, X, Y)− ∗Ψ(∇ZV, W, X, Y)− ∗Ψ(V,∇ZW, X, Y)

− ∗Ψ(V, W,∇ZX, Y)− ∗Ψ(V, W, X,∇ZY).

Since∇ is metric,g is parallel and with Lemma 4.1 we get

=ZΨ(V, W, P(X, Y))−Ψ(∇ZV, W, P(X, Y))−Ψ(V,∇ZW, P(X, Y))−Ψ(V, W, P(∇ZX, Y))

−Ψ(V, W, P(X,∇ZY))−Ψ(V, W,∇ZP(X, Y)) + Ψ(V, W,∇ZP(X, Y)).

We have Ψ(V, W,(∇ZP)(X, Y)) =g(P(V, W),(∇ZP)(X, Y)) = (∇ZΨ)(X, Y, P(V, W)) and thus we get

(∇Z∗Ψ)(V, W, X, Y) = (∇ZΨ)(V, W, P(X, Y)) + (∇ZΨ)(X, Y, P(V, W)).

The condition∇Ψ =a∗Ψ implies

(∇Z∗Ψ)(V, W, X, Y) = −a∗Ψ(P(X, Y), Z, V, W)−a∗Ψ(P(V, W), Z, X, Y) and aplying once again Lemma 4.1 yields

(∇Z∗Ψ)(V, W, X, Y) =

= −aΨ(P(X, Y), Z, P(V, W))−aΨ(P(V, W), Z, P(X, Y)) +ag(P(X, Y), V)g(Z, W)

−ag(P(X, Y), W)g(Z, V) +ag(P(V, W), X)g(Z, Y)−ag(P(V, W), Y)g(Z, X)

= a[Ψ(X, Y, V)g(Z, W)−Ψ(X, Y, W)g(Z, V) + Ψ(V, W, X)g(Z, Y)−Ψ(V, W, Y)g(Z, X)], which ﬁnishes the proof.

We deﬁne a 4-form on the cone ¯M via

Φ(∂r, X, Y, Z) := a³r³Ψ(X, Y, Z), Φ(X, Y, Z, W) := a⁴r⁴∗Ψ(X, Y, Z, W)

forX, Y, Z, W ∈T M. Since ∂_ryΦ locally is aG₂-structure on∂_r^⊥, Φ is a Spin(7)-structure on M¯. As in Section 3, given a characteristic connection onM with respect to Ψ, we construct a connection∇with skew symmetric torsionT onM such that its lift ¯∇to ¯M with torsion ¯T is the characteristic connection on ¯M with respect to Φ. Since we have T = ¯T_|_{T M} and∂ryT¯ = 0, we have ¯T =T = 0 in case of a parallel Spin(7) structure with respect to the Levi-Civita connection on ¯M, and thus∇is the Levi-Civita connection onM and thusT measures the diﬀerence of the G2 structure to the nearly parallel case (see Remark 4.8).

Definition 4.4. Let (M, g,Ψ) be aG₂ manifold with characteristic connection∇^c. We deﬁne a metric connection∇with skew symmetric torsionT via

T :=T^c−2a 3 Ψ.

As in the metric almost contact case (see the comments in Deﬁnition 3.1),T cannot be computed abstractly, but it is found through an educated guess and justiﬁed a posteriori from its properties.

Theorem 4.5. The connection∇satisﬁes

∇Ψ =a∗Ψ,

andΦis parallel with respect to∇¯, the appendant connection onM¯. Proof. We have for the Riemannian connection∇^gonM

∇XΨ(Y, Z, W) = XΨ(Y, Z, W)−Ψ(∇^gXY, Z, W)−Ψ(Y,∇^gXZ, W)−Ψ(Y, Z,∇^gXW)

−1

2Ψ(T(X, Y), Z, W)−1

2Ψ(Y, T(X, Z), W)−1

2Ψ(Y, Z, T(X, W))

= (∇^cXΨ)(Y, Z, W) +1

2Ψ((T^c−T)(X, Y), Z, W) +1

2Ψ(Y,(T^c−T)(X, Z), W) +1

2Ψ(Y, Z,(T^c−T)(X, W)) and because∇^cΨ = 0 we have

∇XΨ(Y, Z, W) =

= 1

2[(T^c−T)(X, Y, P(Z, W)) + (T^c−T)(X, Z, P(W, Y)) + (T^c−T)(X, W, P(Y, Z))]

= a

3[Ψ(X, Y, P(Z, W)) + Ψ(X, Z, P(W, Y)) + Ψ(X, W, P(Y, Z))].

With Lemma 4.1 we obtain a∗Ψ(X, Y, Z, W) = a

3[∗Ψ(X, Y, Z, W) +∗Ψ(X, Z, W, Y) +∗Ψ(X, W, Y, Z)]

= a

3[Ψ(X, Y, P(Z, W)) + Ψ(X, Z, P(W, Y)) + Ψ(X, W, P(Y, Z))

−g(X, Z)g(Y, W) +g(X, W)g(Y, Z)−g(X, W)g(Z, Y) +g(X, Y)g(Z, W)

−g(X, Y)g(W, Z) +g(X, Z)g(W, Y)]

= ∇XΨ(Y, Z, W),

which proves the ﬁrst statement. To show ¯∇Φ = 0 on ¯M we look at several cases. Let always beV, W, X, Y, Z∈T M.

Case 1: If∂ris one of the arguments, we compute ( ¯∇WΦ)(∂_r, X, Y, Z) =W a³r³Ψ(X, Y, Z)−1

rΦ(W, X, Y, Z)−r³a³Ψ(∇WX, Y, Z)

−r³a³Ψ(X,∇WY, Z)−r³a³Ψ(X, Y,∇WZ)

= a³r³(∇WΨ)(X, Y, Z)−1

rΦ(W, X, Y, Z) = a⁴r³∗Ψ(W, X, Y, Z)−1

rΦ(W, X, Y, Z) = 0.

Case 2: If the direction of the derivative is equal to∂r, we obtain ( ¯∇∂rΦ)(X, Y, Z, W) =∂_r(a⁴r⁴∗Ψ(X, Y, Z, W))−41

rΦ(X, Y, Z, W)

= 4r³a⁴∗Ψ(X, Y, Z, W)−41

rΦ(X, Y, Z, W) = 0.

Case 3: If the direction of the derivative and one argument are equal to∂rwe compute ( ¯∇∂_rΦ)(∂r, X, Y, Z) = ∂r(a³r³Ψ(X, Y, Z))−3a³r³1

rΨ(X, Y, Z) = 0.

Case 4: OnT M we have:

( ¯∇VΦ)(W, X, Y, Z) =

= a⁴r⁴V ∗Ψ(W, X, Y, Z)−Φ( ¯∇VW, X, Y, Z)−Φ(W,∇¯VX, Y, Z)−Φ(W, X,∇¯VY, Z)

−Φ(W, X, Y,∇¯VZ)

= a⁴r⁴V ∗Ψ(W, X, Y, Z)−Φ(∇VW −1

r¯g(V, W)∂_r, X, Y, Z)−Φ(W,∇VX−1

rg(V, X)∂¯ _r, Y, Z)

−Φ(W, X,∇VY −1

r¯g(V, Y)∂_r, Z)−Φ(W, X, Y,∇VZ−1

r¯g(V, Z)∂_r)

= a⁴r⁴(∇V ∗Ψ)(W, X, Y, Z) +r⁴a⁵[g(V, W)Ψ(X, Y, Z)−g(V, X)Ψ(W, Y, Z) +g(V, Y)Ψ(W, X, Z)−g(V, Z)Ψ(W, X, Y)],

which is equal to zero due to Lemma 4.3.

Conversely, given a Spin(7) structure ( ¯M ,g,¯ Φ,P ,¯ p) on ¯¯ M (see Section 5 of Chapter I for the deﬁnitions), ∂_ryΦ is aG₂ structure with respect to the metrica²g onM =M × {1} ⊂M¯ and thus

Ψ := 1 a³∂ryΦ

deﬁnes a G2 structure onM with respect to the metricg. To prove the following theorem, we need

Lemma 4.6. If ∗ is the Hodge operator onM with respect tog and∗a²g is the Hodge operator onM with respect to the metrica²g, we have for any3-formω

∗a²gω=a∗ω.

Proof. Let ei for i = 1..7 be an orthonormal basis with dual basiseⁱ onM with respect to g.

Then ¹_aeiwith dualaeⁱis a orthonormal basis with respect toa²g. We deﬁnee^{^i,j,k^}:=eⁱ∧e^j∧e^k ande^{^i,j,k,j^}:=eⁱ∧e^j∧e^k∧e^l as well as (se)^{^i,j,k^}:=seⁱ∧se^j∧se^k fors∈Rand (se)^{^i,j,k,j^} respectively. Then we have

∗a²ge^{^i,j,k^}= 1

a³ ∗a²g(ae)^{^i,j,k^}= 1

a³(ae)^{^1,..,7^}\{^i,j,k^}= 1

a³a⁴e^{^1,..,7^}\{^i,j,k^}=a∗e^{^i,j,k^}, which proves the lemma.

Theorem 4.7. Given aSpin(7) structure onM¯ with characteristic connection ∇¯ being the lift of a connection∇ onM, we have for the G2 structureΨinduced by Φ

∇Ψ =a∗Ψ

and the characteristic connection on (M, g,Ψ)is given by T^c=T+^2a₃Ψ.

Proof. We have forW, X, Y, Z ∈T M (∇WΨ)(X, Y, Z) = 1

a³[WΦ(∂_r, X, Y, Z)

−Φ(∂_r,∇WX, Y, Z)−Φ(∂_r, X,∇WY, Z)−Φ(∂_r, X, Y,∇WZ)]

= 1

a³[( ¯∇WΦ)(∂_r, X, Y, Z) + Φ( ¯∇W∂_r, X, Y, Z)] = 1

a³Φ(W, X, Y, Z).

With Lemma 8 of [B¨a93] and the deﬁnition of Ψ we conclude Φ|T M =∗a²g(∂_ryΦ) =∗a²g(a³Ψ) = a⁴∗Ψ, where∗a²g is the Hodge operator on M ⊂M¯ with respect to the metrica²g. The last equality follows from Lemma 4.6. Thus we get

∇Ψ =a∗Ψ.

For the connection∇^c with torsionT^c=T+^2a₃Ψ we calculate as in the proof of Theorem 4.5 (∇^cXΨ)(Y, Z, W) = (∇XΨ)(Y, Z, W) +1

2[(T−T^c)(X, Y, P(Z, W)) + (T−T^c)(X, Z, P(W, Y)) + (T−T^c)(X, W, P(Y, Z))]

=a∗Ψ(X, Y, Z, W)−a

3[Ψ(X, Y, P(Z, W)) + Ψ(X, Z, P(W, Y)) + Ψ(X, W, P(Y, Z))]

which is equal to zero due to Lemma 4.1. Since the characteristic connection of aG2 manifold is unique, this proves the theorem.

Remark 4.8. In the case of an nearly parallelG2 structure we haveT^c= ^2a₃Ψ, i. e. T = 0 and thus∇=∇^g lifts to the Levi-Civita connection on ¯M and the corresponding Spin(7) structure on the cone is integrable. This means, that, as in the metric almost contact case,T =T^c−^2a₃Ψ measures the ’distance’ of theG2 structure from a nearly parallelG2 structure.

4.1 The classification of G

₂

structures and the corresponding classifi-cation of Spin(7) structures on the cone

We will now discuss the classiﬁcation of Fern´andez [Fe86] of Spin(7) structures on ¯M given in Section 5 of Chapter I, and compute the correspondence to the classiﬁcation ofG₂ structures [FG82]. Again we are only interested in structures carrying a characteristic connection (G₂ structures of classW1⊕ W3⊕ W4). We writeX_M for the projection onT M of a vector ﬁeldX inTM¯. We summarize some useful identities:

Lemma 4.9.

1. P can be expressed through Ψon T M:P(Y, Z) = ∑

lΨ(el, Y, Z)el. 2. For any metric connection∇˜ with skew torsion onM, we have:

( ˜∇XΨ)(Y, Z, V) = g(( ˜∇XP)(Y, Z), V), ( ˜∇XP)(Y, Z) = ∑

g(e_l,( ˜∇XP)(Y, Z))e_l=∑

( ˜∇XΨ)(e_l, Y, Z)e_l.

3. For ∇, this can be simpliﬁed to (∇XP)(Y, Z) = a∑

l∗Ψ(X, el, Y, Z)el. 4. P,Ψ, andP¯ are related by(X, Y, Z∈T M)

P¯(∂r, X, Y)⊥∂r, ¯g( ¯P(X, Y, Z), ∂r) = Φ(X, Y, Z, ∂r) =−a³r³Ψ(X, Y, Z), P¯(∂r, X, Y) =arP(X, Y), P(Y, Z, V¯ )M =ar²(∇YP)(Z, V).

5. The derivative ofΦon M¯ can be expressed in terms of ΨonM (X, Y, Z, V, W ∈T M):

( ¯∇^g_X^¯Φ)(∂r, Z, V, W) =a³r³[(∇^g− ∇)XΨ](Z, V, W), ( ¯∇^gX^¯Φ)(Y, Z, V, W) =a⁴r⁴[(∇^g− ∇)X∗Ψ](Y, Z, V, W).

Proof. Statements (1)-(3) are easily checked. To prove statement (4) forX, Y, Z∈T M, we have

g( ¯P(∂_r, X, Y), Z) = Φ(∂_r, X, Y, Z) =a³r³Ψ(X, Y, Z) =ar¯g(P(X, Y), Z), thus ¯P(∂r, X, Y) =arP(X, Y). Furthermore,

g(X,P¯(Y, Z, V)) = Φ(Y, Z, V, X) = a³r⁴(∇YΨ)(Z, V, X) = a³r⁴g(X,(∇YP)(Z, V))

= ar²¯g(X,(∇YP)(Z, V)),

and thus ¯P(Y, Y, V)_M = ar²(∇YP)(Z, V). For (5) and vector ﬁelds X, Y, Z, V, W ∈ T M, we calculate

2( ¯∇^¯^g_XΦ)(∂r, Z, V, W) =

= 2( ¯∇XΦ)(∂_r, Z, V, W) + Φ(∂_r,T(X, Z), V, W¯ ) + Φ(∂_r, Z,T¯(X, V), W) + Φ(∂_r, Z, V,T(X, W¯ ))

= a³r³[Ψ(T(X, Z), V, W) + Ψ(Z, T(X, V), W) + Ψ(Z, V, T(X, W))]

= 2a³r³[Ψ((∇X− ∇^g_X)Z, V, W) + Ψ(Z,(∇X− ∇^g_X)V, W) + Ψ(Z, V,(∇X− ∇^g_X)W)]

= −2a³r³[(∇ − ∇^g)XΨ](Z, V, W), and similarly

( ¯∇^gX^¯Φ)(Y, Z, V, W) =

= 1

2[Φ( ¯T(X, Y), Z, V, W) + Φ(Y,T¯(X, Z), V, W) + Φ(Y, Z,T¯(X, V), W) + Φ(Y, Z, V,T¯(X, W))]

= a⁴r⁴

2 [∗Ψ(T(X, Y), Z, V, W) +∗Ψ(Y, T(X, Z), V, W) +∗Ψ(Y, Z, T(X, V), W) +∗Ψ(Y, Z, V, T(X, W))]

= −a⁴r⁴[(∇ − ∇^g)X∗Ψ](Y, Z, V, W) =a⁴r⁴[(∇^g− ∇)X∗Ψ](Y, Z, V, W), which ﬁnishes the proof.

Remark 4.10. Since the characteristic connection of the Spin(7) structure on ¯M is unique (see Section 5 of Chapter I), we can conclude for any such structure satisfying ¯∇^¯^gΦ = 0 that∇=∇^g and thus∇^gΨ = a∗Ψ and the G2 structure is of class W1. Conversely, given a connection ∇ with skew symmetric torsion and∇Ψ =a∗Ψ we construct∇^cviaT^c:=T−^2a₃Ψ, which satisﬁes

∇^cΨ = 0 and thus is unique. Hence a metric connection with skew symmetric torsion and the property∇Ψ =∗Ψ is unique.

For any tensorR onM we introduce the notationRxX to denoteR(−, X).

We extend the metricgto arbitraryk-tensorsR, S via an orthonormal framee1, . . . , en

g(R, S) :=

∑n i1,..,i_k=1

R(ei₁, .., ei_k)S(ei₁, .., ei_k).

Lemma 4.11. ASpin(7) structure onM¯ is of classU1 if and only if onM

• g(∇^gΨ,∗Ψ) =ag(∗Ψ,∗Ψ), and

• for every X ∈T M we haveg(∗Ψ,[(∇ − ∇^g)∗Ψ]xX) = 3g(Ψ,[(∇ − ∇^g)Ψ]xX).

The structure on M¯ is of class U2 if and only if the following conditions are satisﬁed for X, Y, Z, X₁, .., X₄∈T M and a local orthonormal frame e₁, .., e₇ of T M:

• δΦ|T M = 0 onT M, which is equivalent to0 =

∑7 i=1

[(∇^g− ∇)e_i∗Ψ](ei, X, Y, Z)

• 0 =

∑4 i=1

∑

l<j<8

(−1)ⁱδΨ(el, ej)Ψ(el, ej, Xi)Ψ(X1, ..,Xˆi, .., X4)

• 28[(∇^g− ∇)_W ∗Ψ](X₁, X₂, X₃, X₄)

∑4 i=1

∑

l<j<8

(−1)ⁱ⁺¹δΨ(el, ej)Ψ(el, ej, Xi)∗Ψ(W, X1, ..,Xˆi, .., X4).

Proof. We consider a local ¯g-orthonormal frame ¯e1 = _ar¹e1, ..,e¯7 = _ar¹e7, e8 = ∂r of TM¯ such thate1, .., e7is a local orthonormal frame ofT M. With Lemma 4.2 of [Fe86] a Spin(7) structure is deﬁned to be of classU1 if and only if

0 = −6δΦ(¯p(X)) =

∑8 i,k,j=1

( ¯∇^ge^¯¯_iΦ)(¯ej,¯ek,P¯(¯ei,e¯j,¯ek), X).

ForX ∈T M we have

0 = −6δΦ(¯p(X)) =

∑8 i,k,j=1

( ¯∇^¯^g¯e_iΦ)(¯ej,e¯k,P¯(¯ei,e¯j,e¯k), X)

∑7 i,k,j=1

( ¯∇^ge^¯¯_iΦ)(¯ej,¯ek,P(¯¯ ei,¯ej,¯ek), X) + 2

∑7 i,j=1

( ¯∇^¯^g¯e_iΦ)(¯ej, ∂r,P¯(¯ei,¯ej, ∂r), X)

= 1

a⁶r⁶

∑7 i,k,j=1

( ¯∇^¯^ge_iΦ)(e_j, e_k, ar²(∇eiP)(e_j, e_k) + ¯g( ¯P(e_i, e_j, e_k), ∂_r)∂_r, X)

+2 1 a⁴r⁴

∑7 i,j=1

( ¯∇^¯^geiΦ)(ej, ∂r, arP(ei, ej), X)

= 1

a⁵r⁴

∑7 i,k,j=1

a⁴r⁴[(∇^g− ∇)e_i∗Ψ](ej, ek,(∇e_iP)(ej, ek), X)

− 1 a³r³

∑7 i,k,j=1

Ψ(e_i, e_j, e_k)( ¯∇^¯^ge_iΦ)(e_j, e_k, ∂_r, X)−2a³r³ a³r³

∑7 i,j=1

([∇^g− ∇]_e_iΨ)(e_j, P(e_i, e_j), X)

∑7 i,k,j,l=1

[(∇^g− ∇)e_i∗Ψ](ej, ek,∗Ψ(ei, el, ej, ek)el, X)

−3

∑7 i,k,j=1

Ψ(ei, ej, ek)([∇^g− ∇]e_iΨ)(ej, ek, X)

= g(∗Ψ,(∇^g− ∇)∗ΨxX)−3g(Ψ,(∇^g− ∇)ΨxX).

In caseX =∂_r, we deduce from Lemma 4.9:

0 =

∑7 i,j,k=1

( ¯∇^ge^¯_iΦ)(ej, ek,P¯(ei, ej, ek), ∂r) = ar²

∑7 i,j,k=1

( ¯∇^¯^ge_iΦ)(ej, ek,(∇e_iP)(ej, ek), ∂r)

=a⁴r⁵

∑7 i,j,k,l=1

[(∇^g− ∇)e_iΨ](ej, ek, el)g((∇e_iP)(ej, ek), el)

=−a⁴r⁵[

∑7 i,j,k,l=1

(∇^ge_iΨ)(e_j, e_k, e_l)(∇eiΨ)(e_j, e_k, e_l)−

∑7 i,j,k,l=1

(∇eiΨ)(e_j, e_k, e_l)(∇eiΨ)(e_j, e_k, e_l)]

=−a⁴r⁵[g(∇^gΨ,∇Ψ)−g(∇Ψ,∇Ψ)] =−a⁵r⁵[g(∇^gΨ,∗Ψ)−ag(∗Ψ,∗Ψ)],

and thus we haveg(∇^gΨ,∗Ψ) =ag(∗Ψ,∗Ψ). A Spin(7) structure is of classU2 if it satisﬁes

28( ¯∇^¯^gWΦ)(X1, X2, X3, X4) =−

∑4 i=1

(−1)ⁱ⁺¹[δΦ(¯p(Xi))Φ(W, X1, ..,Xˆi, .., X4) + 7¯g(W, X_i)δΦ(X₁, ..,Xˆ_i, .., X₄)].

(II.9)

SupposeW =X₁=∂_randX₂, X₃, X₄∈T M. For a 3-formξonT M we have

g(¯p(∂_r), ξ) = ¯g(∂_r,P¯(ξ)) =−Φ(∂_r, ξ) =−a³r³Ψ(ξ) = ¯g(−a³r³Ψ, ξ)

and thus ¯p(∂_r) =−a³r³Ψ. Since∂_ryT¯ = 0 we have ¯∇^¯^g∂_rΦ = 0 and the deﬁning relation of the classU2 reduces to

0 =δΦ(p(∂_r))Φ(∂_r, X₂, X₃, X₄)+7δΦ(X₂, X₃, X₄) =δΦ(−a⁶r⁶Ψ(X₂, X₃, X₄)Ψ+7X₂∧X₃∧X₄).

Since a⁶r⁶Ψ(X₂, X₃, X₄)Ψ−7X₂ ∧X₃ ∧X₄ spans Λ³(T M) we have δΦ = 0 on T M. For X, Y, Z∈T M we have

0 =δΦ(X, Y, Z) =−

∑8 i=1

( ¯∇^¯^g¯e_iΦ)(¯ei, X, Y, Z) =− 1 a²r²

∑7 i=1

( ¯∇^ge^¯_iΦ)(ei, X, Y, Z)

=−a²r²

∑7 i=1

[(∇^g− ∇)e_i∗Ψ](ei, X, Y, Z).

ForX ∈T M we have

δΦ(¯p(X)) =δΦ(

∑8 i<j<k=1

g(¯p(X),¯ei∧e¯j∧e¯k)¯ei∧¯ej∧¯ek)

=δΦ( ∑

i<j<8

g(¯p(X),e¯i∧e¯j∧e¯8)¯ei∧e¯j∧e¯8)

= ∑

i<j<8

g(¯p(X),¯ei∧e¯j∧e¯8)δΦ(¯ei,¯ej, ∂r)

=−

∑7 k=1

∑

i<j<8

( ¯∇^¯^g¯e_kΦ)(¯ek,¯ei,e¯j, ∂r)¯g(X,P(¯¯ ei,¯ej, ∂r))

∑7 k=1

∑

i<j<8

a³r³(∇^g¯e_kΨ)(¯ek,¯ei,e¯j)Φ(¯ei,¯ej, ∂r, X)

=a⁶r⁶

∑7 k=1

∑

i<j<8

(∇^ge¯_kΨ)(¯ek,e¯i,e¯j)Ψ(¯ei,e¯j, X)

=− ∑

i<j<8

δΨ(ei, ej)Ψ(ei, ej, X).

SupposeW =∂r andX1, .., X4∈T M. Then equation (II.9) gives us

0 =

∑4 i=1

(−1)ⁱ⁺¹δΦ(¯p(Xi))a³r³Ψ(X1, ..,Xˆi, .., X4)

= a³r³

∑4 i=1

∑

l<j<8

(−1)ⁱδΨ(el, ej)Ψ(el, ej, Xi)Ψ(X1, ..,Xˆi, .., X4).

ForW, Xi∈T M, equation (II.9) reduces to

28( ¯∇^gW^¯ Φ)(X1, X2, X3, X4) = 28a⁴r⁴[(∇^g− ∇)W∗Ψ](X1, X2, X3, X4), which is equal to

−

∑4 i=1

(−1)ⁱ⁺¹[δΦ(¯p(X_i))Φ(W, X₁, ..,Xˆ_i, .., X₄) + 7¯g(W, X_i)δΦ(X₁, ..,Xˆ_i, .., X₄)]

=a⁴r⁴

∑4 i=1

∑

l<j<8

(−1)ⁱ⁺¹δΨ(e_l, e_j)Ψ(e_l, e_j, X_i)∗Ψ(W, X₁, ..,Xˆ_i, .., X₄).

This proves the statement.

Remark 4.12. One can use Lemma 4.3 and Lemma 4.9 to simplify these equations in rather lengthly calculations. The property

0 =

∑7 i=1

[(∇^g− ∇)_e_i∗Ψ](e_i, X, Y, Z) can for example be simpliﬁed to

0 =g((ΨxY)xZ, δΨxX) +g(ΨxX,(∇^gΨxY)xZ)−g(ΨxX,(∗ΨxY)xZ).

Another simpliﬁcation (see Lemma 4.17) will be used in Example 4.18.

Theorem 4.13. If theSpin(7) structure on the coneM¯ is of classU1, then:

• The G2 structureΨonM cannot be of classW3⊕ W4.

• The G2 structure is of classW1 if and only if theSpin(7) structure is integrable.

If the structure onM¯ is of class U2, then the structure onM is never of classW1⊕ W3. Proof. Since the relationg(∇^gΨ,∗Ψ) = 0 deﬁnes the classW2⊕ W3⊕ W4, we conclude the ﬁrst result directly from Lemma 4.11. Now, assume the G2 structure Ψ is of classW1, i. e. nearly parallelG2 (see [FG82]):

∇^gΨ = 1

168g(∇^gΨ,∗Ψ)∗Ψ.

Taking the scalar product with∗Ψ on both sides leads to g(∇^gΨ,∗Ψ) = 1

168g(∇^gΨ,∗Ψ)g(∗Ψ,∗Ψ).

With the Spin(7) structure being of class U1 and the calculation above we get g(∗Ψ,∗Ψ) =

168g(∗Ψ,∗Ψ)g(∗Ψ,∗Ψ) and thus g(∗Ψ,∗Ψ) = 168. Therefore,

∇^gΨ = 1

168g(∇^gΨ,∗Ψ)∗Ψ =a 1

168g(∗Ψ,∗Ψ)∗Ψ =a∗Ψ.

Thus ∇^gΨ =∇Ψ =a∗Ψ and with Remark 4.10 we get∇ =∇^g and ¯∇^g^¯= ¯∇. Since ¯∇Φ = 0 the Spin(7) structure on ¯M is integrable.

Consider a structure on ¯M of classU2. With Lemma 4.11 we get δΦ = 0 onT M. To see that this structure is integrable it is suﬃcient to show∂ryδΦ = 0, see [Fe86]. We have forX, Y ∈T M

(∂ryδΦ)(X, Y) =−

∑8 i=1

( ¯∇^ge^¯¯_iΦ)(¯ei, ∂r, X, Y) =ar

∑7 i=1

((∇^g− ∇)e_iΨ)(ei, X, Y)

=ar

∑7 i=1

[(∇^geiΨ)(ei, X, Y) + Φ(ei, ei, X, Y)] =−arδΨ(X, Y).

This is equal to zero if the structure onMis cocalibrated (of classχ1⊕χ3, deﬁned byδΨ = 0).

4.2 Corresponding spinors on G

manifolds and their cones

Since we haveT−T^c=−^2a₃Ψ, the diﬀerence ¯T−T^c is the lift ofa²r²T−a²r²T^c=−^2a₃a²r²Ψ.

Furthermore, _a₃¹_r₃∂_ryΦ is the lift of Ψ to ¯M, hence we have T¯−T^c=−2

3r∂ryΦ.

Now Lemma 2.4 implies:

Theorem 4.14. For a G2 manifold with characteristic connection∇^c and for α= ¹₂a or α=

−¹₂a, there is

1. a one to one correspondence between Killing spinors with torsion

∇^sXϕ=αXϕ

on M, and parallel spinors of the connection∇¯^s+^4s_3r∂ryΦonM¯ with cone constanta

∇¯^sXϕ+2s

3r(Xy(∂_ryΦ))ϕ= 0.

2. a one to one correspondence between ∇¯^s-parallel spinors on M¯ with cone constant a and spinors onM satisfying

∇^sXϕ=αXϕ+2as

3 (XyΨ)ϕ.

In particular fors= ¹₄ we get the correspondence

spinors on M spinors onM¯

∇^cXϕ=αXϕ ∇¯Xϕ=−_6r¹(Xy(∂_ryΦ))ϕ

∇^cXϕ=αXϕ+^a₆(XyΨ) ϕ ∇¯Xϕ= 0

Remark 4.15. As for metric almost contact structures (see Remark 3.17), one can use the characterisation ¯T =−δΦ−⁷₆∗(θ∧Φ) withθ=¹₇∗(δΦ∧Φ) (see [Iv04]) and the description of T^c given in Theorem 4.8 of [FI02] to rewrite these equations in terms of the geometric data of the Spin(7) structure.

Remark 4.16. In Section 3 of Chapter I we see, that a G2 structure also is given by a spinor ϕ, which is ∇^c-parallel. On the other hand, for any Spin(7) manifold there is a spinor that is parallel with respect to the characteristic connection (Theorem 1.1 of [Iv04]). The G₂ spinor induces the Spin(7) spinor in the following way:

From Lemma 4.2 we know that (XyΨ)·ϕ=−3X·ϕand thus

∇^cXϕ= 0 = 3X·ϕ+ (XyΨ)·ϕ,

which is the identity for a spinor onM inducing a ¯∇-parallel spinor on ¯M in Theorem 4.14. Be cautious that∇^c may have more parallel spinor ﬁelds than just ϕ; for these, we cannot deﬁne a suitable ‘lifted’ spinor on the cone, unless one ﬁnds a similar trick to write the spinor ﬁeld equation in a form covered by Theorem 4.14.

This remark indicates, that one might use the description of a Spin(7) and aG₂ structure via spinors to calculate the correspondences given in Theorem 4.13 as it was done in Section 1 for SU(3) andG2 manifolds. Note that the stabilizer of a spinor in dimension 8 is not always the group Spin(7), so one does not get correspondences as in Lemmas 3.1 and 4.1 of Chapter I.

4.3 Examples

To simplify the calculations in the example we reformulate the second condition for aG2structure onM to imply a Spin(7) structure of classU1on ¯M of Lemma 4.11. So we only have to calculate Ψ,∗Ψ and∇^gΨ to check the conditions.

Lemma 4.17. The second condition of Lemma 4.11

g(∗Ψ,[(∇ − ∇^g)∗Ψ]xX) = 3g(Ψ,[(∇ − ∇^g)Ψ]xX) is equivalent to

0 =

∑7 i,k,j,l,m=1

[

∗Ψ(ei, ej, ek, el)(∇^geiΨ)(ej, ek, em)Ψ(em, el, X)

+∗Ψ(ei, ej, ek, el)(∇^geiΨ)(el, X, em)Ψ(em, ej, ek)− ∗Ψ(ei, ej, ek, el)∗Ψ(ei, ej, ek, em)Ψ(em, el, X)

− ∗Ψ(e_i, e_l, e_j, e_k)∗Ψ(e_i, e_l, X, e_m)Ψ(e_m, e_j, e_k) ]

+ 3

∑7 i,k,j=1

[

−Ψ(e_i, e_j, e_k)(∇^ge_iΨ)(e_j, e_k, X) +aΨ(e_i, e_j, e_k)∗Ψ(e_i, e_j, e_k, X) ]

Proof. We continue the calculation of the proof of Lemma 4.11 and with Lemma 4.3 we get 0 =1

∑7 i,k,j,l=1

(∇^ge_i∗Ψ)(e_j, e_k, a∗Ψ(e_i, e_l, e_j, e_k)e_l, X)

−1 a

∑7 i,k,j,l=1

(∇e_i∗Ψ)(ej, ek, a∗Ψ(ei, el, ej, ek)el, X)

−3

∑7 i,k,j=1

Ψ(e_i, e_j, e_k)(∇^ge_iΨ)(e_j, e_k, X) + 3a

∑7 i,k,j=1

Ψ(e_i, e_j, e_k)∗Ψ(e_i, e_j, e_k, X)

∑7 i,k,j,l=1

∗Ψ(ei, el, ej, ek)(∇^ge_i∗Ψ)(ej, ek, el, X)

−

∑7 i,k,j,l=1

∗Ψ(ei, el, ej, ek)(∇e_i∗Ψ)(ej, ek, el, X)

−3

∑7 i,k,j=1

Ψ(e_i, e_j, e_k)(∇^ge_iΨ)(e_j, e_k, X) + 3a

∑7 i,k,j=1

Ψ(e_i, e_j, e_k)∗Ψ(e_i, e_j, e_k, X)

=−

∑7 i,k,j,l=1

∗Ψ(ei, el, ej, ek)(∇^geiΨ)(ej, ek, P(el, X))

−

∑7 i,k,j,l=1

∗Ψ(e_i, e_l, e_j, e_k)(∇^ge_iΨ)(e_l, X, P(e_j, e_k))

∑7 i,k,j,l=1

∗Ψ(ei, el, ej, ek)(∇e_iΨ)(ej, ek, P(el, X))

∑7 i,k,j,l=1

∗Ψ(e_i, e_l, e_j, e_k)(∇eiΨ)(e_l, X, P(e_j, e_k))

−3

∑7 i,k,j=1

Ψ(ei, ej, ek)(∇^ge_iΨ)(ej, ek, X) + 3a

∑7 i,k,j=1

Ψ(ei, ej, ek)∗Ψ(ei, ej, ek, X) which is equal to the condition stated in the lemma.

Example 4.18. Let (M, ξ1, ξ2, ξ3, η1, η2, η3) be a 7 dimensional 3-Sasaki manifold with corre-sponding 2-formsFi, i= 1,2,3. Letηi fori= 1, ..,7 be the dual of a local basis{e1=ξ1, e2 = ξ2, e3=ξ3, e4, .., e7}, such that

F₁=−η₂₃−η₄₅−η₆₇, F₂=η₁₃−η₄₆+η₅₇, F₃=−η₁₃−η₄₇−η₅₆.

Here for η_i ∧..∧η_j we write η_i,..,j. In [AF10] it is explained that there is no characteristic connection as such, but one can construct a cocalibratedG₂ structure

Ψ =η1∧F1+η2∧F2+η3∧F3+ 4η1∧η2∧η3=η123−η145−η167−η246+η257−η347−η356

with characteristic connection∇^c and torsion T^c =η1∧dη1+η2∧dη2+η3∧dη3 that is very well adapted to the 3-Sasakian structure. It is therefore called thecanonical G2 structure of the

underlying 3-Sasakian structure. Remark 4.16 ensures then the existence of a ¯∇-parallel spinor ﬁeld on ¯M.

We calculate the class of the Spin(7) structure on ¯M of the canonicalG2structure using Lemma 4.11.

Theorem 4.19. TheSpin(7)structure on the cone constructed from the canonicalG2 structure of a3-Sasakian manifold is of classU1 if and only if the cone constant is a=¹⁵₁₄.

Proof. Due to the formulation of the second condition of Lemma 4.11 given in Lemma 4.17, we just need to calculate∗Ψ and∇^gΨ. Obviously∗Ψ is given by

∗Ψ =η4567−η2367−η2345−η1357+η1346−η1256−η1247. To get∇^gΨ we observe

∇^ge_jΨ = (∇^ge_jη₁)∧F₁+ (∇^ge_jη₂)∧F₂+ (∇^ge_jη₃)∧F₃ +η₁∧(∇^ge_jF₁) +η₂∧(∇^ge_jF₂) +η₃∧(∇^ge_jF₃)

+ 4(∇^ge_jη1)∧η2∧η3+ 4η1∧(∇^ge_jη2)∧η3+ 4η1∧η2∧(∇^ge_jη3)

and since (ηi, Fi) are Sasakian structures we have (∇^ge_jFi)(Y, Z) =g(ej, Z)ηi(Y)−g(ej, Y)ηi(Z).

Thus∇^ge_jF_i =η_j∧η_i fori= 1,2,3 andj= 1, ..,7 implyingη_i∧(∇^ge_jF_i) = 0. Since (∇^g_Xη_i)Y =g(Y,∇^g_Xξ_i) =g(Y,−Ψ_iX) =F_i(X, Y)

we have∇^g_Xηi=XyFi and get

∇^gejΨ =(ejyF1)∧F1+ (ejyF2)∧F2+ (ejyF3)∧F3

+ 4(e_jyF₁)∧η₂∧η₃+ 4η₁∧(e_jyF₂)∧η₃+ 4η₁∧η₂∧(e_jyF₃).

This gives us

∇^ge1Ψ = −η346+η357+η247+η256, ∇^ge2Ψ = η345+η367−η147−η156,

∇^ge₃Ψ = −η₂₄₅−η₂₆₇+η₁₄₆−η₁₅₇, ∇^ge₄Ψ = 3 (−η₂₃₅+η₅₆₇+η₁₃₆−η₁₂₇),

∇^ge5Ψ = 3 (η234−η467−η137−η126), ∇^ge6Ψ = 3 (−η237+η457−η134+η125),

∇^ge₇Ψ = 3 (η₂₃₆−η₄₅₆+η₁₃₅+η₁₂₄).

Using an appropriate computer algebra system we easily calculate g(∇^gΨ,∗Ψ) = 180, g(∗Ψ,∗Ψ) = 168,

thus the ﬁrst condition of Lemma 4.11 is satisﬁed ifa = ¹⁵₁₄. Using the formulation given in Lemma 4.17 of the second condition one easily checks that the this condition is satisﬁed for any a.

We expect that for all other values of the cone constanta, the structure is of generic classU1⊕U2, but the system of equations that one obtains is extremely involved.

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