a higher-dimensional generalisation of the goldberg …a higher-dimensional generalisation of the...
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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)
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)
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)
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)
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)
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
Rdr2
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
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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
Rdr2
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
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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...
Four-dimensional Kerr-NUT-AdS metricI Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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...
Four-dimensional Kerr-NUT-AdS metricI Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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...
Four-dimensional Kerr-NUT-AdS metricI Complexify TM and T∗M: g = 2θ1 � θ1 + 2θ2 � θ2
where
θ1 =
√y2 + r2
2R
(dr +
Ry2 + r2
(dφ+ y2dψ)
),
θ1 =
√y2 + r2
2R
(dr −
Ry2 + r2
(dφ+ y2dψ)
),
θ2 =
√y2 + r2
2Y
(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...
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 ,
where
θµ =
√Uµ2Xµ
(dxµ + i
XµUµ
(σµ(0)
dψ0 + σµ(1)
dψ1)
), 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))
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 ,
where
θµ =
√Uµ2Xµ
(dxµ + i
XµUµ
(σµ(0)
dψ0 + σµ(1)
dψ1)
), 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))
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 ,
where
θµ =
√Uµ2Xµ
(dxµ + i
XµUµ
(σµ(0)
dψ0 + σµ(1)
dψ1)
), 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))
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 ,
where
θµ =
√Uµ2Xµ
(dxµ + i
XµUµ
(σµ(0)
dψ0 + σµ(1)
dψ1)
), 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))
(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 ,
where
θµ =
√Uµ2Xµ
dxµ + iXµUµ
m−1∑k=0
σµ(k)
dψk
, θµ = θµ , e0 =
√cσ(m)
m∑k=0
σ(k)dψk .
with Uµ, Xµ and σµ(k)
are functions of xν .
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
(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 ,
where
θµ =
√Uµ2Xµ
dxµ + iXµUµ
m−1∑k=0
σµ(k)
dψk
, θµ = θµ , e0 =
√cσ(m)
m∑k=0
σ(k)dψk .
with Uµ, Xµ and σµ(k)
are functions of xν .
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
(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 ,
where
θµ =
√Uµ2Xµ
dxµ + iXµUµ
m−1∑k=0
σµ(k)
dψk
, θµ = θµ , e0 =
√cσ(m)
m∑k=0
σ(k)dψk .
with Uµ, Xµ and σµ(k)
are functions of xν .
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
(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 ,
where
θµ =
√Uµ2Xµ
dxµ + iXµUµ
m−1∑k=0
σµ(k)
dψk
, θµ = θµ , e0 =
√cσ(m)
m∑k=0
σ(k)dψk .
with Uµ, Xµ and σµ(k)
are functions of xν .
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
(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 ,
where
θµ =
√Uµ2Xµ
dxµ + iXµUµ
m−1∑k=0
σµ(k)
dψk
, θµ = θµ , e0 =
√cσ(m)
m∑k=0
σ(k)dψk .
with Uµ, Xµ and σµ(k)
are functions of xν .
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
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
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
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
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
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 .
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?
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?
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?
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?
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
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
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
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
free!
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.
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
free!
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.
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
free!
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.
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
free!
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.
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
free!
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.
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;
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;
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;
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;
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 ,
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 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 ,
respectively, for all X ,Y ∈ Γ(N⊥), Z ∈ Γ(N ), and is otherwise generic. Then, N isintegrable.
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 ,
respectively, for all X ,Y ∈ Γ(N⊥), Z ∈ Γ(N ), and is otherwise generic. Then, N isintegrable.
Counterexample to the converse
I Black ring (Emparan, Reall (2002))
g = −e0 ⊗ e0 + 2θ1 � θ1 + 2θ2 � θ2 ,
where
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
),
e0 :=
√F (x)
F (y)
(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−
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
2 .
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
2 .
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.