A higher-dimensional generalisation of the Goldberg …A higher-dimensional generalisation of the...

A higher-dimensional generalisation of the Goldberg-Sachstheorem

based on arXiv:1011.6168 and arXiv:1107.2283

Arman Taghavi-ChabertMasaryk University, Brno

23 September 2011 / Canberra

What is the Goldberg-Sachs theorem?

I A number of versions: Goldberg, Sachs (1962); Newman, Penrose (1962); Kundt,Thompson (1963); Robinson, Schild (1963); Plebanski, Hacyan (1975);Przanowski, Broda (1983); Apostolov, Gauduchon (1997); Ivanov, Zamkovoy(2005); Gover, Hill, Nurowski (2009);

Goldberg-Sachs TheoremA real or complex four-dimensional (pseudo-)Riemannian Einstein manifold admits a(locally) integrable distribution of self-dual null 2-planes if and only if its self-dual Weyltensor is algebraically special.

I Applications: mass production of Einstein metrics (e.g. Kerr metric)I Aim:

Higher-dimensional generalisation?A real or complex (2m + ε)-dimensional (pseudo-)Riemannian Einstein manifold(ε ∈ {0, 1}) admits a (locally) integrable distribution of null m-planes if and only if itsWeyl tensor is algebraically special.

I But what is ‘algebraically special’?I Other possible partial generalisations (Lorentzian only): Durkee, Reall (2009)

Four-dimensional Kerr-NUT-AdS metric

I 4-parameter family of Lorentzian Einstein metrics (subclass ofPlebanski-Demianski metric (1976)) with coordinates {r , y , φ, ψ}:

g = −R(dφ+ y2dψ)2

r2 + y2+

Y (dφ− r2dψ)2

r2 + y2+

r2 + y2

Ydy2 +

r2 + y2

where R = a− 2mr + br2 − cr4 and Y = a− 2ny + by2 − cy4 for arbitraryparameters a, b, c, m and n.

I Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

where R = R(r), Y = Y (y).

Four-dimensional Kerr-NUT-AdS metric

I 4-parameter family of Lorentzian Einstein metrics (subclass ofPlebanski-Demianski metric (1976)) with coordinates {r , y , φ, ψ}:

g = −R(dφ+ y2dψ)2

r2 + y2+

Y (dφ− r2dψ)2

r2 + y2+

r2 + y2

Ydy2 +

r2 + y2

where R = a− 2mr + br2 − cr4 and Y = a− 2ny + by2 − cy4 for arbitraryparameters a, b, c, m and n.

I Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

where R = R(r), Y = Y (y).

Four-dimensional Kerr-NUT-AdS metricI Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

where R = R(r), Y = Y (y).I Conjugate pair of distributions of complex 2-planesN1(p) = {X p ∈ C⊗ TpM : ιXp θ

1 = ιXpθ2 = 0}, p ∈M, and N12 = N1;

N1 and N1 are null, i.e. g|N1 = 0 = g|N1.

N1 ∩N1 =: C⊗K where K real null line bundleI N1 and N1 integrable, i.e. [Γ(N1), Γ(N1)] ⊂ Γ(N1).K defines a congruence of null geodesics,M/K is CR manifold.Robinson (or optical) structure. (Nurowski-Trautman (2002))

I Another Robinson structure defined by distributionN0(p) = {X ∈ C⊗ TpM : ιXpθ

1 = ιXpθ2 = 0} and N2 = N0;

I Robinson structures: Lorentzian analogues of Hermitian structures...

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

1 = ιXpθ2 = 0}, p ∈M, and N12 = N1;

1 = ιXpθ2 = 0} and N2 = N0;

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

1 = ιXpθ2 = 0}, p ∈M, and N12 = N1;

1 = ιXpθ2 = 0} and N2 = N0;

√y2 + r2

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dr −

Ry2 + r2

(dφ+ y2dψ)

√y2 + r2

(dy + i

Yy2 + r2

(dφ− r2dψ)

), θ2 = θ2 ,

1 = ιXpθ2 = 0}, p ∈M, and N12 = N1;

1 = ιXpθ2 = 0} and N2 = N0;

Four-dimensional Euclidean Kerr-NUT-AdS metric

I Wick transform to Euclidean 4-dim metric:{r , y , φ, ψ} → {x1 = ir , x2 = y , ψ0 = φ, ψ1 = ψ}:

g = 2θ1 � θ1 + 2θ2 � θ2 ,

θµ =

√Uµ2Xµ

(dxµ + i

XµUµ

(σµ(0)

dψ0 + σµ(1)

), θµ = θµ , µ = 1 , 2 ,

with Uµ, Xµ and σµ(k)

are functions of xν .

I 2 conjugate pairs of distributions of null complex 2-planes:N0(p) = {X ∈ C⊗ TpM : ιXpθ

1 = ιXpθ2 = 0} and N12 = N0;

N1(p) = {X ∈ C⊗ TpM : ιXp θ1 = ιXpθ

2 = 0} and N2 = N1;

I N0 ∩N0 = {0} and N1 ∩N1 = {0}N0, N1, N2, N12 integrable⇒ 2 conjugate pairs of Hermitian structures.

I Known as ambihermitian metric (Apostolov, Calderbank, Gauduchon (2010))

g = 2θ1 � θ1 + 2θ2 � θ2 ,

θµ =

√Uµ2Xµ

(dxµ + i

XµUµ

(σµ(0)

dψ0 + σµ(1)

), θµ = θµ , µ = 1 , 2 ,

1 = ιXpθ2 = 0} and N12 = N0;

2 = 0} and N2 = N1;

g = 2θ1 � θ1 + 2θ2 � θ2 ,

θµ =

√Uµ2Xµ

(dxµ + i

XµUµ

(σµ(0)

dψ0 + σµ(1)

), θµ = θµ , µ = 1 , 2 ,

1 = ιXpθ2 = 0} and N12 = N0;

2 = 0} and N2 = N1;

g = 2θ1 � θ1 + 2θ2 � θ2 ,

θµ =

√Uµ2Xµ

(dxµ + i

XµUµ

(σµ(0)

dψ0 + σµ(1)

), θµ = θµ , µ = 1 , 2 ,

1 = ιXpθ2 = 0} and N12 = N0;

2 = 0} and N2 = N1;

(2m + ε)-dim Euclidean Kerr-NUT-AdS metric (ε ∈ {0,1})

I Chen, Lü, Pope (2005): Generalisation to higher dimensions: (2m + ε)-parameterfamily of Euclidean Einstein metrics (coordinates {xµ, ψk}):

g = 2m∑µ=1

θµ � θµ + εe0 ⊗ e0 ,

θµ =

√Uµ2Xµ

dxµ + iXµUµ

m−1∑k=0

σµ(k)

, θµ = θµ , e0 =

√cσ(m)

m∑k=0

σ(k)dψk .

I For each of the 2m subsets I ⊂ S := {1 , . . . ,m}, define

NI(p) :={

X p ∈ C⊗ TpM : ιXpθµ = ιXp θ

ν = 0 = ειXp e0 , ∀ν ∈ I , µ ∈ S \ I}

Each NI is a distribution of totally null m-planes.I All 2m distributions NI are integrable.I All 2m distributions N⊥I are integrable.I Known as multihermitian structures (Mason, TC 2008) or canonical null structures

g = 2m∑µ=1

θµ � θµ + εe0 ⊗ e0 ,

θµ =

√Uµ2Xµ

dxµ + iXµUµ

m−1∑k=0

σµ(k)

, θµ = θµ , e0 =

√cσ(m)

m∑k=0

σ(k)dψk .

NI(p) :={

ν = 0 = ειXp e0 , ∀ν ∈ I , µ ∈ S \ I}

g = 2m∑µ=1

θµ � θµ + εe0 ⊗ e0 ,

θµ =

√Uµ2Xµ

dxµ + iXµUµ

m−1∑k=0

σµ(k)

, θµ = θµ , e0 =

√cσ(m)

m∑k=0

σ(k)dψk .

NI(p) :={

ν = 0 = ειXp e0 , ∀ν ∈ I , µ ∈ S \ I}

g = 2m∑µ=1

θµ � θµ + εe0 ⊗ e0 ,

θµ =

√Uµ2Xµ

dxµ + iXµUµ

m−1∑k=0

σµ(k)

, θµ = θµ , e0 =

√cσ(m)

m∑k=0

σ(k)dψk .

NI(p) :={

ν = 0 = ειXp e0 , ∀ν ∈ I , µ ∈ S \ I}

g = 2m∑µ=1

θµ � θµ + εe0 ⊗ e0 ,

θµ =

√Uµ2Xµ

dxµ + iXµUµ

m−1∑k=0

σµ(k)

, θµ = θµ , e0 =

√cσ(m)

m∑k=0

σ(k)dψk .

NI(p) :={

ν = 0 = ειXp e0 , ∀ν ∈ I , µ ∈ S \ I}

Null structure

Definition(2m + ε)-dim real pseudo-Riemannian manifold (M,g) (ε ∈ {0, 1}).

I An almost null structure N onM is a maximal totally null subbundle of thecomplexified tangent bundle C⊗ TM, i.e.

M⊂ N ⊂ N⊥ ⊂ C⊗ TM

with g|N = 0 and rank N = m.I N is called a null structure if both N and N⊥ are integrable, i.e.

[Γ(N ), Γ(N )] ⊂ Γ(N ) , [Γ(N⊥), Γ(N⊥)] ⊂ Γ(N⊥)

Null structure

Definition(2m + ε)-dim complex Riemannian manifold (M,g) (ε ∈ {0, 1}).

I An almost null structure N onM is a maximal totally null holomorphic subbundleof the holomorphic tangent bundle TM, i.e.

M⊂ N ⊂ N⊥ ⊂ TM

with g|N = 0 and rank N = m.I N is a null structure if both N and N⊥ are integrable, i.e.

[Γ(N ), Γ(N )] ⊂ Γ(N ) , [Γ(N⊥), Γ(N⊥)] ⊂ Γ(N⊥)

PropertiesI Even dim: N⊥ = N and N self-dual or anti-self-dual;I Odd dim: N⊥/N has rank 1;I N and N⊥ integrable⇒ totally geodetic foliation in the sense that

g(∇X Y ,Z ) = 0 , g(∇Z Y ,X) = 0 ,

for all X ,Y ∈ Γ(N ), Z ∈ Γ(N⊥); (∇ Levi-Civita connection)I Conformally invariant.

Null structure

Definition(2m + ε)-dim complex Riemannian manifold (M,g) (ε ∈ {0, 1}).

I An almost null structure N onM is a maximal totally null holomorphic subbundleof the holomorphic tangent bundle TM, i.e.

M⊂ N ⊂ N⊥ ⊂ TM

with g|N = 0 and rank N = m.I N is a null structure if both N and N⊥ are integrable, i.e.

[Γ(N ), Γ(N )] ⊂ Γ(N ) , [Γ(N⊥), Γ(N⊥)] ⊂ Γ(N⊥)

PropertiesI Even dim: N⊥ = N and N self-dual or anti-self-dual;I Odd dim: N⊥/N has rank 1;I N and N⊥ integrable⇒ totally geodetic foliation in the sense that

g(∇X Y ,Z ) = 0 , g(∇Z Y ,X) = 0 ,

for all X ,Y ∈ Γ(N ), Z ∈ Γ(N⊥); (∇ Levi-Civita connection)I Conformally invariant.

Pure spinors

I rank-2m spin bundle S over (2m + ε)-dim real or complex pseudo-Riemannianmanifold (M,g)Even dimensions (ε = 0): S = S+ ⊕ S− chiral spin bundles

I Spinor field ξ ∈ Γ(S) (up to scale) defines a distribution Nξ:

Nξp := {X p ∈ C⊗ TpM : X p · ξ = 0} , p ∈M .

Clifford property 0 = X · X · ξ = −g(X ,X )ξ⇒Nξ totally null.I Spinor field ξ is pure if dimNξ = m.

Even dim: ξ pure⇒ ξ chiral.All spinors are pure in dim < 7.

I Spinorial characterisation of the integrability of Nξ:

Y · (∇Xξ) = 0 , X · Y · (∇Zξ) = 0 ,

for all X ,Y ∈ Γ(Nξ) and Z ∈ Γ(N⊥ξ ).

Four dimensions:Use abstract index notation SA′ := S+ and SA := S−.Spinor field ξA′ ∈ Γ(SA′ ) is foliating if and only if

ξA′ξB′∇AA′ξB′ = 0 .

Pure spinors

Nξp := {X p ∈ C⊗ TpM : X p · ξ = 0} , p ∈M .

Y · (∇Xξ) = 0 , X · Y · (∇Zξ) = 0 ,

Pure spinors

Nξp := {X p ∈ C⊗ TpM : X p · ξ = 0} , p ∈M .

Y · (∇Xξ) = 0 , X · Y · (∇Zξ) = 0 ,

Pure spinors

Nξp := {X p ∈ C⊗ TpM : X p · ξ = 0} , p ∈M .

Y · (∇Xξ) = 0 , X · Y · (∇Zξ) = 0 ,

Pure spinors

Nξp := {X p ∈ C⊗ TpM : X p · ξ = 0} , p ∈M .

Y · (∇Xξ) = 0 , X · Y · (∇Zξ) = 0 ,

Integrability conditions

I (M,g) pseudo-Riemannian, N almost null structure. If N is integrable then theWeyl tensor satisfies

C(X ,Y ,Z ,W ) = 0

for all X ,Y ,Z ∈ Γ(N ), W ∈ Γ(N⊥)

Four dimensionsI 4-dim: so(4,C) ∼= sl(2,C)SD × sl(2,C)ASD with standard representations SA′ and SA

respectively.Bundle of tensors with Weyl symmetries: C = SDC ⊕ ASDCWeyl tensor Cabcd ∼ ΨA′B′C′D′ ⊕ΨABCD totally symmetric.

I Suppose ξA′ defines a SD null structure. Then

ΨA′B′C′D′ξA′ξB′ξC′ξD′ = 0 .

How about sufficient curvature conditions?

C(X ,Y ,Z ,W ) = 0

Conformal Killing-Yano 2-form

I Bochner (1948), Yano (1952) introduced a generalisation of conformal Killingvectors:A conformal Killing-Yano (CKY) p-form φa1...ap on a pseudo-Riemannian manifoldsatisfies

∇(aφb)◦c2...cp = 0 .

I Applications: hidden symmetries, separation of variables, etc...I Carter (1968), Kubiznák, Frolov (2007): Kerr-NUT-AdS metric characterised by

conformal Killing-Yano (CKY) 2-form

φ =∑µ

xµθµ ∧ θµ

∇(aφb)◦c2...cp = 0 .

φ =∑µ

xµθµ ∧ θµ

∇(aφb)◦c2...cp = 0 .

φ =∑µ

xµθµ ∧ θµ

Degeneracy of the Weyl tensor in four dimensionsProposition (Walker, Penrose (1970))If ξA′

(i) (i = 1, 2) are the eigenspinors of a generic self-dual CKY 2-form φ, then

ΨA′B′C′D′ξB′(i)ξ

C′(i) ξ

D′(i) = 0 , i = 1, 2 .

I For tensorial approach in Lorentzian sign. Dietz, Rüdiger (1981), Glass, Kress(1999)

Proposition (Walker, Penrose (1970))If ξA′

(i) (i = 1, 2) are the eigenspinors of a generic self-dual CKY 2-form φ, then ξA′(i) is

foliating.

I Algebraic degeneracy of the Weyl tensor↔ integrability of almost null structure.I Converse of Propositions also true: get a CKY 2-form from curvature condition for

DefinitionA SD Weyl tensor is algebraically special along ξA′ ⇔ ΨA′B′C′D′ξ

B′ξC′ξD′ = 0.

Theorem (Goldberg-Sachs (1962) - Plebanski, Hacyan (1975))(M,g) is 4-dimensional Einstein complex Riemannian manifold. Then

ΨA′B′C′D′ξB′ξC′ξD′ = 0⇔ ξA′ is foliating.

C′(i) ξ

D′(i) = 0 , i = 1, 2 .

foliating.

B′ξC′ξD′ = 0.

C′(i) ξ

D′(i) = 0 , i = 1, 2 .

foliating.

B′ξC′ξD′ = 0.

C′(i) ξ

D′(i) = 0 , i = 1, 2 .

foliating.

B′ξC′ξD′ = 0.

C′(i) ξ

D′(i) = 0 , i = 1, 2 .

foliating.

B′ξC′ξD′ = 0.

Degeneracy of the Weyl tensor in higher dimensions

I Semmelmann (2001), Gover, Šilhan (2006):

φ is a CKY 2-form ⇒ [φ,C] = 0

Proposition (Mason, TC (2008))Let φ be a generic CKY 2-form. Then, φ has 2m distinct eigenspinors ξI(I ⊂ {1, . . . ,m}), and

C(X ,Y ,Z , ·) = 0

for all X ,Y ∈ Γ(N⊥ξI), Z ∈ Γ(NξI ), for all I ⊂ {1, . . . ,m}.

Proposition (Mason, TC (2008))Let φ be a generic CKY 2-form, with dφ satisfying some algebraic conditions. Then the2m eigenspinors of φ are foliating.

Next step:I find a direct relation between the algebraic degeneracy of the Weyl tensor and the

integrability of only one almost null structure;I with no reference to any CKY 2-form;

C(X ,Y ,Z , ·) = 0

The complex Goldberg-Sachs theorem in higher dimensions

Theorem (TC 2010, 2011)Let (M, g) be a (2m + ε)-dim complex Riemannian Einstein manifold, N an almostnull structure. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

respectively, for all X ,Y ∈ Γ(N⊥), Z ∈ Γ(N ), and is otherwise generic. Then, N isintegrable.

Proof:I Follows from the Bianchi identity;I Degeneracy of C + Einstein condition⇒ homogeneous system of linear equations

on connection components

g(∇X Y ,Z ) , g(∇Z Y ,X ) ,

for all X ,Y ∈ Γ(N ), Z ∈ Γ(N⊥);I Show that genericity assumption⇒ only solution is trivial.

The complex Goldberg-Sachs theorem in higher dimensions

Theorem (TC 2010, 2011)Let (M, g) be a (2m + ε)-dim complex Riemannian Einstein manifold, N an almostnull structure. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

Proof:I Follows from the Bianchi identity;I Degeneracy of C + Einstein condition⇒ homogeneous system of linear equations

on connection components

g(∇X Y ,Z ) , g(∇Z Y ,X ) ,

for all X ,Y ∈ Γ(N ), Z ∈ Γ(N⊥);I Show that genericity assumption⇒ only solution is trivial.

The multi-Goldberg-Sachs theorem in higher dimensions

Theorem (TC 2011)Let (M, g) be a (2m + ε)-dim complex Riemannian Einstein manifold. Let {NI} be acollection of canonical almost null structures. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

respectively, for all X ,Y ∈ Γ(N⊥I ), Z ∈ Γ(NI), for all {NI} and is otherwise generic.Then the canonical almost null structures {NI} are integrable.

Theorem (TC 2011)Let (M, g) be a (2m + ε)-dim real pseudo-Riemannian Einstein manifold, N an almostnull structure. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

The multi-Goldberg-Sachs theorem in higher dimensions

Theorem (TC 2011)Let (M, g) be a (2m + ε)-dim complex Riemannian Einstein manifold. Let {NI} be acollection of canonical almost null structures. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

respectively, for all X ,Y ∈ Γ(N⊥I ), Z ∈ Γ(NI), for all {NI} and is otherwise generic.Then the canonical almost null structures {NI} are integrable.

Theorem (TC 2011)Let (M, g) be a (2m + ε)-dim real pseudo-Riemannian Einstein manifold, N an almostnull structure. Suppose the Weyl tensor satisfies

C(X ,Y ,Z , ·) = 0 ,

Counterexample to the converse

I Black ring (Emparan, Reall (2002))

g = −e0 ⊗ e0 + 2θ1 � θ1 + 2θ2 � θ2 ,

θ1 :=RF (y)

√G(x)

√2(x − y)

√F (x)

(√F (x)

G(x)dx + idφ

θ2 :=R√−F (x)G(y)√

2(x − y)

(√F (y)

G(y)dy + idψ

√F (x)

(dt + R

√λν(1 + y)dψ

I 5-dim Einstein Lorentzian metric;I 4 null structuresI But the Weyl tensor is not degenerate with respect to any of them.

Classification of the self-dual Weyl tensor in four dimensions

I 4-dim: so(4,C) ∼= sl(2,C)SD × sl(2,C)ASD . acting on chiral spinor representationsSA′ and SA respectively.Bundle C = SDC ⊕ ASDCWeyl tensor Cabcd ∼ ΨA′B′C′D′ ⊕ΨABCD .

I Penrose’s classification of the self-dual Weyl tensor in terms of a spinor field ξA′ :

ΨA′B′C′D′ = 0⇒ ΨA′B′C′D′ξA′ = 0⇒ ΨA′B′C′D′ξ

A′ξB′ = 0

⇒ ΨA′B′C′D′ξA′ξB′ξC′ = 0⇒ ΨA′B′C′D′ξ

A′ξB′ξC′ξD′ = 0

I Can rewrite as a filtration on SDC

M = SDC3 ⊂ SDC2 ⊂ SDC1 ⊂ SDC0 ⊂ SDC−1 ⊂ SDC−2 = SDC .

I Filtration invariant under the stabiliser P of ξA′

P parabolic subgroup of SL(2,C)...

Classification of the curvature tensors

In general,I Idea: existence of almost null structure N ⇔ Frame bundle has structure group

StabN ;I Under StabN , curvature tensors split into irreducible parts;I e.g. Hermitian case StabN = U(m) ⊂ SO(2m): work by Grey, Hervella (1980),

Tricerri, Vanhecke (1981), Falcitelli, Farinola, Salamon (1994)

Here, focus on holomorphic case (i.e. no reality structure) StabN = P ⊂ SO(2m + ε,C)parabolic complex Lie group preserving the flag of vector bundlesM⊂ N ⊂ N⊥ ⊂ TM.In fact, for the purpose of Goldberg-Sachs thm:

I Irreducible decomposition not needed;I Null structure and Weyl tensor conformally invariant. More natural to replace the

Einstein condition by a condition on the Cotton-York tensor(3− 2m − ε)Aabc = ∇d Cdabc .

Classification of the curvature tensors: even dim > 4

I Almost null structure N =: V12

M = V32 ⊂ V

12 ⊂ V−

12 = TM .

I Bundle of tensors with Weyl symmetries C :=∧2 T∗M�◦

∧2 T∗M

M = C3 ⊂ C2 ⊂ C1 ⊂ C0 ⊂ C−1 ⊂ C−2 = C ,

I Bundle of tensors with Cotton-York symmetries A := T∗M�◦∧2 T∗M

M = A52 ⊂ A

32 ⊂ A

12 ⊂ A−

32 = A .

Classification of the curvature tensors: odd dim

I Almost null structure N =: V1, N⊥ =: V0

M = V2 ⊂ V1 ⊂ V0 ⊂ V−1 = TM .

I Bundle of tensors with Weyl symmetries C :=∧2 T∗M�◦

∧2 T∗M

M = C5 ⊂ C4 ⊂ C3 ⊂ . . . ⊂ C−3 ⊂ C−4 = C

I Bundle of tensors with Cotton-York symmetries A := T∗M�◦∧2 T∗M

M = A4 ⊂ A3 ⊂ A2 ⊂ . . . ⊂ A−2 ⊂ A−3 = A .

The obstruction to the generalised complex Goldberg-Sachs theorem

PropositionSuppose that N is integrable and the SD Weyl tensor a section of SDCk for k ≥ 0.Then the SD Cotton-York tensor is a section of SDAk− 1+ε

PropositionSuppose that N is integrable and the Weyl tensor a section of Ck for k ≥ 0. Then theCotton-York tensor is a section of Ak− 1+ε

A partial generalised complex Goldberg-Sachs theorem

TheoremSuppose that the SD Cotton-York tensor is a section of SDAk− 1+ε

2 and the SD Weyltensor a section of SDCk for k ≥ 0. Then N is integrable.

TheoremSuppose that the Cotton-York tensor is a section of Ak− 1+ε

2 and the Weyl tensor asection of Ck for k ≥ 0. Then N is integrable.

Theorem case k = 0Let (M, g) be a (2m + ε)-dim complex Riemannian manifold, N an almost nullstructure. Suppose the Cotton-York tensor and the Weyl tensor satisfy

A(X ,Y ,Z ) = 0 , C(X ,Y ,Z , ·) = 0 ,

respectively, for all X ,Y ∈ Γ(N⊥), Z ∈ Γ(N ). Then, N is integrable.

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