Adapted connections

Let (M⁷, g, ϕ) be a 7-dimensional spin manifold with Levi-Civita connection ∇. As usual we identify (3,0)- and (2,1)-tensors usingg. As in section 3.4 we consider the spaceA^g of all metric connections and define the map

Ξ :End(T M)→ A^g, S7→2

3Ψϕ(S., ., .).

The prescription

∇ⁿ :=∇+2 3SyΨϕ

defines a canonical G2-connection (it preserves ϕ), since (XyΨϕ)ϕ = −3Xϕ, Ψϕϕ = 7ϕ and Ψϕϕ^∗=−ϕ^∗,∀ϕ^∗⊥ϕ(see Lemma 4.2):

∇ⁿXϕ=∇Xϕ+1

3Ψ_ϕ(S(X), ., .)·ϕ=S(X)·ϕ−S(X)·ϕ= 0.

The endomorphismS encodes the intrinsic torsion. Abiding by Cartan’s formalism, the set of all metric connections is isomorphic to (compare description ofR⁷, Λ³(R⁷) andT in Section 3.4)

R⁷

|{z}

Λ¹R⁷

⊕(R⊕R⁷⊕S₀²R⁷)

| {z }

Λ³R⁷

⊕(g₂⊕S₀²R⁷⊕R⁶⁴)

| {z }

T

underG2, and this allows us to see

Proposition 4.13. The four pure kinds of aG₂-manifold correspond to∇ⁿ living in:

W1 ⇐⇒ ∇ⁿ∈Λ³ W2 ⇐⇒ ∇ⁿ∈ T W3 ⇐⇒ ∇ⁿ∈Λ³⊕ T W4 ⇐⇒ ∇ⁿ∈Λ¹⊕Λ³. Proof. Since forA∈G₂ we have

A⁻¹SA7→Ψϕ(A⁻¹SA., ., .) = Ψϕ(AA⁻¹SA., A., A.) = Ψϕ(SA., A., A.)

and Ξ is G₂ equivariant. Comparing the dimensions of the corresponding modules, in the cases W1 andW2 the connection ∇ⁿ must have skew symmetric torsion and cyclic traceless torsion.

Algebraic computations show that forS ∈ W3 the Λ³(R⁷) part does not vanish, 0̸=1

3

X,Y,Z

S Ψ_ϕ(SX, Y, Z)∈Λ³(R⁷) and neither does theT part

0̸= Ψϕ(X, Y, Z)−1 3

X,Y,Z

S Ψϕ(SX, Y, Z)∈ T.

ForS∈ W4 we haveS(X) =V ×X for some vectorV and thus with Lemma 4.4 we get Ψϕ(SX, Y, Z) =g(V ×X, Y ×Z) =g(V, Y)g(X, Z)−g(V, Z)g(X, Y)− ∗Ψϕ(V, X, Y, Z), which is contained inR⁷⊕Λ³(R⁷).

The connection of pure type T of the W2 case is discussed in greater detail in [CI07]. Now among allG2-connections, ensuing from the proposition above, there exists a unique connection

∇^c with skew-symmetric torsion. Therefore we may write

S=λId +S₃+S₄∈ W1+W3+W4

withS3∈S₀²(T M⁷) andS4=VyΨϕ for some vectorV.

Theorem 4.14. Let (M⁷, g, ϕ)be a RiemannianG2 manifold of type W134. The characteristic torsion reads

T^c(X, Y, Z) =−1 3

XY Z

S Ψϕ((2λId + 9S3+ 3S4)X, Y, Z).

Proof. Consider the projections

T^∗M⁷⊗g^⊥₂ →^κ Λ³(T^∗M⁷)→^Θ T^∗M⁷⊗g^⊥₂ Ψϕ(SX, Y, Z)→ 1

3

XY Z

S Ψϕ(SX, Y, Z), T → ∑

iei⊗(eiyT)_g⊥ 2

A little computation shows that the composite Θ ◦κ is the identity map with eigenvalues 1,0,2/9,2/3 on the four respective summandsWi. But from [FI02] we know that if−2Γ = Θ(T) for some 3-formT, thenT is the characteristic torsion.

5 Spin(7) structures

In a similar spirit as in the case of the other structures, an 8-dimensional oriented Riemannian manifold (M, g) is called a Spin(7) manifold if it has a reduction to Spin(7)⊂SO(8), and this is equivalent to the choice of a 4-form Φ which, in a local frame e₁, . . . , e₈, can be written as

Φ =ϕ+∗ϕ, andϕ=e1278+e3478+e5678+e2468−e2358−e1458−e1368.

We define a skew symmetric endomorphismP(X, Y, .) onT M via g(P(X, Y, Z), V) = Φ(X, Y, Z, V).

We extend the metricgto 3-forms onT M in the usual way, i. e. g(W1∧W2∧W3, V1∧V2∧V3) = det(g(Wi, Vi)) for Vi, Wj ∈T M. For 3-forms ξ = ∑

i<j<k

ξijkeijk andη = ∑

i<j<k

ηijkeijk let η(ξ) be defined as

η(ξ) := ∑

i<j<k

ξijkη(ei, ej, ek) = ∑

i<j<k

ξijkηijk =g(η, ξ).

We definep(X) via

g(X, P(ξ)) = g(p(X), ξ)

for X ∈ T M and a 3-form ξ on M (P(ξ) is well defined, since P is totally skew symmetric).

Spin(7) manifolds were classified by Fern´andez in [Fe86]: they split in the two classesU1andU2. S. Ivanov proves in [Iv04] that such a manifold always carries a characteristic connection∇^c. He also states, that there always is a∇^c-invariant spinorϕ, thus defining the structure. As done in Sections 3 and 4 for SU(3) and G₂ structures one could translate the defining relations for the classes of classification to spinorial equations. But note, that a spinor in dimension 8 does not always have Spin(7) as its stabilizer. So there is no correspondence of spinors to Spin(7) structures as we used them in dimensions 6 and 7.

Hypersurfaces, cones and generalized Killing spinors

In this chapter we want to focus on hypersurfaces M ⊂M¯. We always consider a Gstructure on M, which corresponds to a ¯Gstructure on ¯M. We are able to calculate the corresponding classes. In the first section we look at 6 dimensional M where G = SU(3) and ¯G = G2 on M¯. Since in this case we have spinorial characterisations of both structures, we are able to do the calculations for a general hypersurface to then restrict ourself to the case of a twisted cone over an SU(3) manifold. For the two cases of almost contact structures and G2 structures on M (corresponding to almost hermitian and Spin(7) structures on ¯M) we have to choose another technique, which we can only use on (twisted) cones. So in Section 2 we introduce this technique to use it in Sections 3 (the almost contact case) and 4 (theG2 case).

In the back of any section we will discuss (generalized) Killing spinors with torsion. Therefore we consider aGstructure with Levi-Civita connection∇, such that there exists a characteristic connection with torsionT. We define the one-parameter family of metric connections

∇^s:=∇+ 2sT

and recall thatϕ^∗ is called ageneralized Killing spinor with torsion (gKS) if

∇^sXϕ^∗=A(X)·ϕ^∗

for some symmetric endomorphismA. In the case whereA=λId is a multiple of the identity, this is the definition of a Killing spinor with torsion. The cases= ¹₄ corresponds to the characteristic connection; however, there are many geometric situations in which the Killing equation holds for valuess̸= 1/4. Especially the equation fors= _4(nⁿ⁻₋¹₃₎ giving the limiting case of the eigenvalue inequality for the Dirac operator with torsion (see [ABBK13]) is interesting.

If additionally we haves= 0 we know that forλreal this is the definition of a real Killing spinor andλis constant. Such spinors realize the equality case of the inequality for the eigenvalue of the Dirac operator (see [Fr80]) and in dimensions 6 and 7 this spinors correspond to nearly K¨ahler structures and nearly parallelG2 structures (see Chapter I).

Ifs= 0 andAis arbitrary symmetric, this is the equation of a so called generalized Killing spinor (see [BGM05] and [FK01]). We saw in Chapter I that such spinors give half flat and cocalibrated structures in dimensions 6 and 7. As the Weingarten map of a hypersurfaceM ⊂M¯ is symmetric, the hypersurface theory can be used to construct generalized Killing spinors with (and without, being the case wheres= 0) torsion.

31

1 SU(3) hypersurfaces in G

₂

manifolds

Let ( ¯M⁷,¯g, ϕ) be a 7-dimensional G₂ manifold and M⁶ a hypersurface with transverse unit directionV

TM¯⁷=T M⁶⊕ ⟨V⟩. (II.1)

By restriction the spin bundle ¯Σ of ¯M⁷ gives a Spin(6)-bundle Σ over M⁶, and similarly the Clifford multiplication· of M⁶ is X·ϕ =V Xϕ in terms of the one on ¯M⁷ (whose symbol we suppress, as usual). In particular, this implies that any σ ∈ Λ^2kM⁶ ⊂Λ^2kM¯⁷ of even degree will satisfy σ·ϕ = σϕ. This notation was also used in [BGM05] to describe almost Killing spinors (compare Section 1.2 in this chapter). The second fundamental formg(W(X), Y) of the immersion (W is the Weingarten map) accounts for the difference between the two Riemannian structures, and in ¯Σ we can compare

∇¯Xϕ=∇Xϕ−1

2V W(X)ϕ,

where ∇ and ¯∇ are the Levi-Civita connections of M⁶ and ¯M⁷ A global spinor ϕ on ¯M⁷ (a G₂-structure) restricts to a spinorϕonM⁶(an SU(3)-structure). The next lemma explains how both the almost complex structure and the spin structure are – in practice – induced byϕand the normalV.

Lemma 1.1. For any sectionsϕ^∗∈Σand vectors X∈T M⁶ i) V ϕ^∗=j(ϕ^∗)

ii) V Xϕ= (JϕX)ϕ

Proof. The volume form σ7 satisfiesσ7ϕ^∗ =−ϕ^∗ for any spinor ϕ^∗ ∈ Σ. Therefore V j(Xϕ) = σ7(Xϕ) =−Xϕ.

Another way of interpreting the structure onM⁶ is to say that theG2-form Ψϕ definesJ by VyΨϕ=−ω.

Lemma 1.2. With respect to decomposition (II.1) theG2-endomorphism ofM¯⁷ has the form

S¯=



JϕS−¹₂JϕW ∗

η ∗∗



 (II.2)

where(S, η)are the intrinsic tensors ofM⁶,Jϕ the almost complex structure,W the Weingarten map.

This result was first proved in [CS06] using Cartan-K¨ahler theory. Our alternative argument is much simpler:

Proof. From the definition ∇Xϕ=V S(X)ϕ+η(X)V ϕand Lemma 1.1 we get ¯∇Xϕ=∇Xϕ−

1

2V W(X)ϕ=JϕS(X)ϕ−¹₂JϕW(X)ϕ+η(X)V ϕ.

The starred terms in matrix (II.2) should point to the half-obvious fact that the derivative∇Vϕ cannot be reconstructed fromS and η. As a matter of fact, later we will need to know that the bottom row of ¯S is controlled by the product (∇ϕ, V ϕ), so that the entry∗∗vanishes when

∇Vϕ= 0.

Now we are ready for the main theorems, which explain how to go fromM⁶ to ¯M⁷ (Theorem 1.4) and backwards (Theorem 1.5). The run-up to those requires the following preparatory definition. Recall that the mapSis symmetric if the SU(3) structure is of typeχ¯1¯23(see Lemma 3.6, Chapter I).

Definition 1.3. The symmetry of the Weingarten endomorphismW expresses half-flatness, i.e.

classχ₁₂₃, by [CS06]. Motivated by that we shall say that a hypersurfaceM⁶⊂M¯⁷ has (0) type zero ifW is the trivial map (meaning ¯∇=∇),

(I) type one ifW is of classχ¯1, (II) type two ifW is of classχ¯2, (III) type three ifW is of classχ3.

Due to the freedom in choosing entries in (II.2), we will take the easiest option (probably also the most meaningful one, geometrically speaking) and consider embeddings where∇Vϕ= 0.

Theorem 1.4. Embed(M⁶, g, ϕ)in some ( ¯M⁷,g, ϕ)¯ as in (II.1), and suppose the G₂-structure to be parallel in the normal direction: ∇¯Vϕ= 0.

Then the classesWα of( ¯M⁷,g, ϕ)¯ depend on the column position (the class ofM⁶) and the row position (the Weingarten type ofM⁶) as in the table

χ⁺₁ χ⁻₁ χ⁺₂ χ⁻₂ χ₃ χ₄ χ₅

0 W13 W4 W3 W2 W3 W2 W234

I W134 W4 W34 W24 W34 W24 W234

II W123 W24 W23 W2 W23 W2 W234

III W13 W34 W3 W23 W3 W23 W234

Proof. LetAbe an endomorphism ofR⁶andθa covector. Then the endomorphism ¯A=(_JϕA0

θ 0

) ofR⁷ is of typeW4 if and only ifθ= 0 andA is a multiple of the identity, sinceJϕ is given by g(X, J_ϕY) = ¹₂Ψ_ϕ(V, X, Y).

With other similar and easy implications we show that the type of ¯A=(_JϕA0

θ 0

) is determined by the class of the intrinsic tensors (A, θ) onM⁶ in the following way:

(A, θ)∈ χ1 χ⁻₁ χ⁺₂ χ⁻₂ χ3 χ4 χ5

(JϕA, θ)∈ χ¯1 χ⁺₁ χ⁻₂ χ⁺₂ χ3 χ4 χ5

A¯∈ W13 W4 W3 W2 W3 W2 W234

Now the theorem can be proved:: Consider for example an SU(3) structure (S, η) of typeχ3on a hypersurface of type one. Then

(J_ϕS0 η 0

)

is of typeW3and sinceW is a multiple of the identity, (_J

ϕW 0 0 0

)is of typeW4. This immediately states that ¯S=

(J_ϕS−¹₂J_ϕW 0

η 0

)

and thus the type of theG2structure isW34.

We will now do the opposite: start from the ambient space ( ¯M⁷,¯g, ϕ) and infer the structure of its codimension-one submanifoldsM⁶. By inverting formula (II.2) we immediately see from

S¯_{|T M}6=J S+1

2J W implyingJϕS¯_{|T M}6 =−S−1 2W

that

JϕS¯_{|T M}6+1

2W =−S and thus

S=−J_ϕS¯

T M⁶+1

2W, η(X) =g( ¯SX, V) for anyX∈T M⁶.

To conclude, we can state the final result on hypersurfaces (which can be found, in a different form, in [C06], Section 4).

Theorem 1.5. Let ( ¯M⁷,¯g, ϕ)be a Riemannian spin manifold of classWα. Then a hypersurface M⁶ orthogonal toV for some V ∈TM¯⁷ has an induced spin structure ϕ⁺: its class is an entry in the matrix below that is determined by its column (the Weingarten type) and row position (Wα)

W1 W2 W3 W4

0 χ₁ χ₁₂₄₅ χ₁₂₃₅ χ₁₄₅ I χ₁₁ χ₁₂₄₅ χ₁₁₂₃₅ χ₁₄₅ II χ₁₂ χ₁₂₄₅ χ₁₂₂₃₅ χ₁₂₄₅ III χ₁₃ χ₁₂₃₄₅ χ₁₂₃₅ χ₁₃₄₅

Proof. To proceed as in Theorem 1.4, we prove that the class of an endomorphism ¯A=(_JϕA∗

θ ∗

) onR⁷ determines the class of (A, θ) on aR⁶ in the following way:

A¯∈ W1 W2 W3 W4

(A, θ)∈ χ1 χ₁₂₄₅ χ₁₂₃₅ χ₁₄₅ If ¯A∈ W1we have ¯A=λId and thusη= 0 and A=λJϕ.

If ¯A∈ W2thenJ_ϕAis skew-symmetric, andAhas typeχ₁₂₄. If ¯A is of type W3 we have ¯S =

(JϕA η η −tr(JϕA)

)

for some symmetric JϕA. Therefore J A is of typeχ₁₂₃ implying the typeχ₁₂₃forA.

Suppose ¯A∈ W4, so there is a vectorZ such thatg(X,AY¯ ) = Ψϕ(Z, X, Y), whence (XY Zϕ, ϕ) = ( ¯AY ϕ, Xϕ)

for everyX, Y ∈R⁷. Restrict this equation toX, Y ∈R⁶ and putZ =λV +Z1, Z1∈R⁶. Then J_ϕA=λJ_ϕ+A₁ with (XY Z₁ϕ, ϕ) = (A₁Y ϕ, Xϕ). SinceA₁ is skew we have

g(X, A₁J_ϕY) = (Z₁XJ_ϕY ϕ, ϕ) = (Z₁XV Y ϕ, ϕ) =−(Z₁Y V Xϕ, ϕ)

= −(Z1Y JϕXϕ, ϕ) =−g(Y, A1JϕX) =−g(X, JϕA1Y), soA1Jϕ=−JϕA1 andA1 has typeχ4. ThusJϕA∈χ14.

From this table it becomes clear that we cannot have aW1-manifold if the derivative ofϕalong V vanishes.

Moreover, in case ∇Vϕ = 0 the χ₅-component disappears from everywhere, simplifying the matter a little.

Im Dokument Geometric structures and special spinor fields (Seite 40-47)