3.2. INTRINSIC TORSION OF PARTICULARG-STRUCTURES 66
3.2. INTRINSIC TORSION OF PARTICULARG-STRUCTURES 67 of this section. We do not give an explicit description of the summands here. Instead, we present in the relevant cases a description of the intrinsic torsion solely in terms of the exterior derivatives of the dening dierential forms and other related dierential forms and indicate how the dierent components of the intrinsic torsion appear as certain components of these derivatives. Some of the results have, to the best of the author's knowledge, only been written down explicitly in the literature for the Riemannian case. We transfer them to the pseudo-Riemannian case without going into much detail concerning this transfer since in all cases one may literally write down the same proof as in the Riemannian case.
Notation 3.24. For aG-structureP on ann-dimensional manifoldM withG⊆O(p, n− p) such thatg⊆so(p, n−p) is non-degenerate with respect to the Killing form of so(p, n− p), we denote by capital Latin letters (e.g. W) components of the decomposition of the G-module (Rn)∗ ⊗g⊥. The corresponding subbundles of T∗M ⊗g⊥(P) are denoted by calligraphic letters (e.g. W) as well as the class of G-structures with intrinsic torsion everywhere in the subbundleW. Finally, we denote by small Latin letters (e.g. w) the part of the intrinsic torsion lying in W.
Remark 3.25. • All real nite-dimensional representations of real semisimple Lie groups are completely reducible, cf. [K]. This applies to SU(p, m−p), G2 and Spin(7).
• The condition that g is a non-degenerate subspace of so(p, n−p) with respect to the Killing form ofso(p, n−p) is fullled in the casesg=u(p, m−p),g=su(p, m−p), g=g2 and g=spin(7). Therefore, note that the assertion is obviously true for the Euclidean cases g = u(m),su(m),g2,spin(7). Moreover, for an arbitrary real Lie subalgebra g of so(p, n−p), g is non-degenerate in so(p, n−p) if and only if the complexication gC is non-degenerate in so(p, n−p)C=so(n,C)
3.2.1 Intrinsic torsion of SU(p, m−p)-structures
Before we discuss the intrinsic torsion of SU(p, m−p)-structures, we summarise what is known about the intrinsic torsion ofU(p, m−p)-structures. Recall that anU(p, m−p) -structure may be described by a pair of a two-form ω and a pseudo-Riemannian metricg satisfying a certain compatibility relation. Hence, its intrinsic torsion can be described by
∇gω according to Proposition 3.17 since ∇gg = 0. By Equation (3.3), the U(p, m−p) -module we want to decompose is given by
W :=n
α∈ R2m∗
⊗Λ2 R2m∗
α(u, Jv, Jw) =α(u, v, w)o
= R2m∗
⊗hh
Λ2,0 R2m∗ii
with J being the standard -complex structure on R2m. The decomposition of W into irreducible summands in the case U(m) is due to Gray and Hervella [GH]. If m ≥ 3,
3.2. INTRINSIC TORSION OF PARTICULARG-STRUCTURES 68 then W = W1 ⊕W2 ⊕W3⊕W4 with irreducible non-zero U(m)-submodules Wi. The classesW1, . . . , W4 can literally be dened, with the obvious sign changes, also in the cases U(p, m−p)for arbitrarypandU1(p, m−p)∼= GL(m,R), and againW =W1⊕. . .⊕W4as U(p, m−p)-submodules. For =−1, the decomposition stays irreducible, cf. [SHPhD], and for = 1, the spaces Wi decompose further into two irreducible U1(p, m−p) ∼= GL(m,R)-summands, cf. [GM].
We like to mention some of the classes. First of all, the holonomy principle shows that the class {0}consists exactly of the pseudo-Kähler or para-Kähler manifolds, respectively.
Moreover, the class W1 is the class of nearly pseudo-Kähler or nearly para-Kähler man-ifolds, respectively, the class W2 are exactly those with dω = 0 and these run under the name almost pseudo-Kähler manifolds or almost para-Kähler manifolds, respectively, and the classW3⊕ W4 consists of those with integrableJ, i.e. they are the pseudo-Hermitian or para-Hermitian manifolds, respectively.
Next, we consider the intrinsic torsion of SU(p, m−p)-structures. Since (su(p, n−p))⊥= (u(p, n−p))⊥⊕RJ asSU(p, m−p)-modules, we get
V := R2m∗
⊗(su(p, n−p))⊥∼=W1⊕W2⊕W3⊕W4⊕ R2m∗
=
5
X
i=1
Wi (3.4) asSU(p, m−p)-modules withW5 := R2m∗
.
Consider rst the case m= 3. This case has been treated in [ChiSa] and it has been shown thatW1=W1+⊕W1− and W2 =W2+⊕W2− as irreducibleSU(3)-modules and that W3, W4 and W5 are irreducible. The classes Wi+ and Wi−, i = 1,2, can again literally be dened as in the SU(3)-case also for the SU(p,3−p)-case and for SU1(p,3−p) ∼= SL(3,R). Moreover, the decomposition V =W1+⊕W1−⊕W2+⊕W2−⊕W3⊕W4⊕W5 is irreducible forSU(p,3−p)with arbitrary p, cf. [SHPhD]. W1 is the real two-dimensional SU(p,3−p)-module
Λ3,0 R6∗
and it decomposes into the two real one-dimensional trivial SU(p,3−p)-modules R·ρ and R·(J)∗ρ. Hence, we may identify w+1 and w−1 with functions on M, which we do in the following. The space W2 is a 16-dimensional SU(p,3−p)-module isomorphic to
Λ3,0 R6∗
⊗h
Λ1,10 R6∗i
, and so isomorphic to 2
h
Λ1,10 R6∗i
. Here, h
Λ1,10 R6∗i
are the real forms of type (1,1)whose wedge product withω02 is 0. Thus,w+2 and w−2 are real two forms on M whose wedge product withω2 is 0. The12-dimensionalSU(p,3−p)-moduleW3 is equivalent to the SU(p,3−p)-module hh
Λ2,10 R6∗ii
, which are the real forms of type(2,1)and(1,2)whose wedge product with ω0 vanishes. Hence,w3is a three-form onM such that the wedge product withωvanishes.
W4 and W5 are both equivalent to the SU(p,3−p)-module R6∗
. Thus,w4 and w5 are one-forms on M. By [ChiSa] and [SHPhD], we have the following decomposition of the SU(p,3−p)-modules of all three-forms and four-forms onR6intoSU(p,3−p)-submodules,
3.2. INTRINSIC TORSION OF PARTICULARG-STRUCTURES 69
which is irreducible for=−1:
Λ3 R6∗
=R·ρ⊕R·Jρ∗ρ⊕hh
Λ2,10 R6∗ii
⊕ R6∗
∧ω0 Λ4 R6∗
=R·ω02⊕h
Λ1,10 R6∗i
∧ω0⊕ R6∗
∧ρ.
Using the above mentioned identications of the dierent components of the intrinsic tor-sion with certain dierential forms on M and the just mentioned decompositions of the three- and four-forms, one can show, cf. [ChiSa] for SU(3) and [SHPhD] for arbitrary SU(p,3−p), that the components of the intrinsic torsion can be recovered from the the exterior derivatives of the dening forms (ω, ρ)∈Ω2M×Ω3M and of the pullback Jρ∗ρas follows:
dω=3
2w−1ρ−3
2w+1Jρ∗ρ+w3+w4∧ω, dρ=w+1ω2+w+2 ∧ω+w5∧ρ,
d(Jρ∗ρ) =w−1ω2+w−2 ∧ω−(Jρ∗w5)∧ρ.
(3.5)
Equation (3.5) gives us the following characterisation of the torsion-free SU(p,3−p) -structures:
Corollary 3.26. Let(ω, ρ)∈Ω2M×Ω3M be anSU(p,3−p)-structure on a six-dimension-al manifold M. Then (ω, ρ) is torsion-free if and only if dω= 0, dρ= 0 andd(Jρ∗ρ) = 0.
Many interesting classes of SU(p,3−p)-structures naturally appear by distinguishing them via their intrinsic torsion. In this thesis, we are only interested in the following class.
Denition 3.27. Let (ω, ρ) ∈ Ω2M ×Ω3M be an SU(p,3 − p)-structure on a six-dimensional manifold M. (ω, ρ) is called half-at if the intrinsic torsion lies entirely in W1−⊕ W2−⊕ W3, i.e. if w1+=w2+ =w4 =w5 = 0. By Equation (3.5), this is equivalent to dω2 = 2dω∧ω = 0 and dρ= 0. Therefore, note that a direct computation in a basis as in Lemma 2.1 shows that the wedge-product of a one-form with ω2 vanishes if and only if the one-form itself is 0.
Remark 3.28. W1−⊕W2−⊕W3 is a 21-dimensional SU(p,3−p)-submodule of the 42 -dimensionalSU(p,3−p)-module R6∗
⊕su(p,3−p). In this sense half-atSU(p,3−p) -structures are half torsion-free.
For m ≥4, we restrict to the SU(m)-case. This case has been considered by Martín Cabrera in [MC4] and he showed that the decompositionV =P5
i=1Wi is a decomposition into irreducibleSU(m)-modules. Moreover, he proves a nice characterisation of torsion-free SU(m)-structures, which will play an important role in Section 7.2 to prove a reduction result for the holonomy of the Riemannian manifold obtained via the Hitchin ow on almost Abelian Lie algebras.
3.2. INTRINSIC TORSION OF PARTICULARG-STRUCTURES 70 Proposition 3.29. Let M be a 2m-dimensional manifold, m ≥ 4 and (ω,Ψ) ∈ Ω2M × ΩmM⊗Cbe an SU(m)-structure on M. Then (ω,Ψ) is torsion-free if and only if dω= 0 and dRe(Ψ) = 0.
3.2.2 Intrinsic torsion of G2-structures
By Proposition 3.17, the intrinsic torsion of aG2-structureϕ∈Ω3M on a seven-dimension-al manifoldM is given by∇gϕϕ,gϕ being the induced pseudo-Riemannian metric. Denote for A ∈ {O,Os} by F : Im(A) → R7 the linear isomorphism dened in Denition 1.19 and set ×−1 := F∗×O and ×1 := F∗×Os, where ×A is the real two-fold cross product on Im(A), gA|Im(A)
. Then ×−1 is a two-fold cross product on R7,h·,·i7
and ×1 is a two-fold cross product on R7,h·,·i3,4
. The G2-module X := η0
R7∗
⊗(g2)⊥
dened in Equation (3.3) is given by
X :=n
α∈ R7∗
⊗Λ3 R7∗
α(u, v, w, v×w) = 0∀u, v, w∈R7 o
.
The decomposition of the G2-module X into irreducible submodules has been done by Fernández and Gray in [FG]. We have X = X1 ⊕X2 ⊕X3 ⊕X4 with irreducible G2 -modules Xi, i= 1, . . . ,4. We can dene G∗2-submodules Xi of X literally as the ones in theG2-case and getX =X1⊕X2⊕X3⊕X4asG∗2-modules. The dimensions of the modules are given by dim(X1) = 1,dim(X2) = 14, dim(X3) = 27 and dim(X4) = 7 and they are also irreducible in the G∗2-case by [Kath1]. Note that there is, up to equivalence, exactly one irreducible G2-module in each of the dimensions 1, 7, 14 and 27. Hence, X1 is the trivial representation,X2is the adjoint representation,X3 is the representationS02 R7
of trace-free symmetric two-tensors and X4 the standard representation on R7. Thus, x1 is a function on M andx4 is a one-form on M. To interpretx2 and x3 as dierential forms, we recall that by [FG] and [Kath1] we have
Λ2 R7∗
= Λ27⊕Λ214, Λ3 R7∗
=R·ϕ⊕Λ37⊕Λ327 with
Λ27 :=n
ω ∈Λ2 R7∗
ω∧ϕ = 2?ϕωo
, Λ214:=n
ω∈Λ2 R7∗
ω∧ϕ=−?ϕωo , Λ37 :=
n
?ϕ(α∧ϕ)
α∈ R7∗o
, Λ327:=
n
ψ∈Λ3 R7∗
ψ∧ϕ= 0, ψ∧?ϕϕ = 0 o
as decompositions of the two- and the three-forms on R7 into irreducible G2-modules.
Hence, x2 may be considered as a two-form on M with x2 ∧ϕ = −?ϕx2 and x3 as a three-form on M with x3∧ϕ = 0 and x3∧?ϕϕ = 0. Moreover, by applying the Hodge star operator, we get corresponding decompositions of the ve- and four-forms. Using these decompositions, one can show, cf. [Br5] and [MC2], that the intrinsic torsion of a G2-structure is encoded in the exterior derivatives ofϕand ?ϕϕas follows:
dϕ= x1?ϕϕ+ 3x4∧ϕ+?ϕx3, d ?ϕϕ= 4x4∧?ϕϕ+x2∧ϕ. (3.6)
3.3. GEOMETRIC STRUCTURES ON LIE ALGEBRAS 71