Conformal Geometry and Metrics of Holonomy Split …Conformal Geometry and Metrics of Holonomy Split...
Transcript of Conformal Geometry and Metrics of Holonomy Split …Conformal Geometry and Metrics of Holonomy Split...
Conformal Geometryand
Metrics of Holonomy Split G2
Robin GrahamUniversity of Washington
Connections in Geometry and PhysicsFields InstituteMay 14, 2011
Holonomy
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Hol(M, g) is all the linear transformations obtained by paralleltranslation around contractible loops.
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Hol(M, g) is all the linear transformations obtained by paralleltranslation around contractible loops.
Hol(M, g) measures the structure preserved by parallel translation.
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Hol(M, g) is all the linear transformations obtained by paralleltranslation around contractible loops.
Hol(M, g) measures the structure preserved by parallel translation.
Most (M, g) have holonomy SOe(p, q).
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Hol(M, g) is all the linear transformations obtained by paralleltranslation around contractible loops.
Hol(M, g) measures the structure preserved by parallel translation.
Most (M, g) have holonomy SOe(p, q).
Hol(M, g) = {e} if and only if g is flat.
Holonomy
Let (M, g) be a connected pseudo-Riemannian manifold ofsignature (p, q), p + q = n.
Can define Hol(M, g) ⊂ SOe(p, q) (Restricted holonomy group)
Hol(M, g) is all the linear transformations obtained by paralleltranslation around contractible loops.
Hol(M, g) measures the structure preserved by parallel translation.
Most (M, g) have holonomy SOe(p, q).
Hol(M, g) = {e} if and only if g is flat.
Example: Hol(M, g) ⊂ U(n/2) if and only if g is Kahler.
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Say that G ⊂ SOe(p, q) is irreducible if its action on Rn has no
nontrivial invariant subspaces.
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Say that G ⊂ SOe(p, q) is irreducible if its action on Rn has no
nontrivial invariant subspaces.
In 1953 Berger derived list of irreducible subgroups for each p, q, n.
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Say that G ⊂ SOe(p, q) is irreducible if its action on Rn has no
nontrivial invariant subspaces.
In 1953 Berger derived list of irreducible subgroups for each p, q, n.
Every irreducible subgroup of SOe(p, q) which arises as Hol(M, g)for some non-symmetric (M, g) is on the list.
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Say that G ⊂ SOe(p, q) is irreducible if its action on Rn has no
nontrivial invariant subspaces.
In 1953 Berger derived list of irreducible subgroups for each p, q, n.
Every irreducible subgroup of SOe(p, q) which arises as Hol(M, g)for some non-symmetric (M, g) is on the list.
Question becomes: Does every group on Berger’s list arise as aholonomy group?
Holonomy
Question. Which subgroups of SOe(p, q) can arise as Hol(M, g)?
Say that G ⊂ SOe(p, q) is irreducible if its action on Rn has no
nontrivial invariant subspaces.
In 1953 Berger derived list of irreducible subgroups for each p, q, n.
Every irreducible subgroup of SOe(p, q) which arises as Hol(M, g)for some non-symmetric (M, g) is on the list.
Question becomes: Does every group on Berger’s list arise as aholonomy group?
For many, but not all, groups on the list, examples were known of(M, g) with that holonomy.
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
They are the two real forms of G2:
G c2 ⊂ SO(7) and G s
2 ⊂ SO(3, 4).
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
They are the two real forms of G2:
G c2 ⊂ SO(7) and G s
2 ⊂ SO(3, 4).
The existence question for these groups remained open until 1987.
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
They are the two real forms of G2:
G c2 ⊂ SO(7) and G s
2 ⊂ SO(3, 4).
The existence question for these groups remained open until 1987.
Theorem. (R. Bryant, 1987) There exist metrics of holonomyequal to G c
2 and G s2 .
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
They are the two real forms of G2:
G c2 ⊂ SO(7) and G s
2 ⊂ SO(3, 4).
The existence question for these groups remained open until 1987.
Theorem. (R. Bryant, 1987) There exist metrics of holonomyequal to G c
2 and G s2 .
More such metrics are known now, but they are not easy to comeby. New examples are of interest.
Holonomy
Other than SOe(p, q), every group on Berger’s list occurs for neven, with two exceptions.
Both exceptions occur for n = 7.
They are the two real forms of G2:
G c2 ⊂ SO(7) and G s
2 ⊂ SO(3, 4).
The existence question for these groups remained open until 1987.
Theorem. (R. Bryant, 1987) There exist metrics of holonomyequal to G c
2 and G s2 .
More such metrics are known now, but they are not easy to comeby. New examples are of interest.
Manifolds of holonomy G c2 arise in M-theory as an analogue of
Calabi-Yau manifolds.
G2
G2
Let ϕ ∈ Λ3R7∗.
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0)
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc),
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc), ϕ split type if (3, 4)
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc), ϕ split type if (3, 4) (ϕs).
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc), ϕ split type if (3, 4) (ϕs).
Fact. ϕc and ϕs are unique up to GL(7,R).
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc), ϕ split type if (3, 4) (ϕs).
Fact. ϕc and ϕs are unique up to GL(7,R).
Definition. G c2 = {A ∈ GL(7,R) : A∗ϕc = ϕc} ⊂ SO(7)
G s2 = {A ∈ GL(7,R) : A∗ϕs = ϕs} ⊂ SO(3, 4).
G2
Let ϕ ∈ Λ3R7∗. Define 〈·, ·〉ϕ by
(X ϕ) ∧ (Y ϕ) ∧ ϕ = 〈X ,Y 〉ϕ e∗1 ∧ . . . ∧ e∗7
Definition. ϕ is nondegenerate if 〈X ,Y 〉ϕ is nondegenerate.
Theorem. ϕ nondegenerate =⇒±〈·, ·〉ϕ has signature (7, 0) or (3, 4).
Say ϕ is compact type if (7, 0) (ϕc), ϕ split type if (3, 4) (ϕs).
Fact. ϕc and ϕs are unique up to GL(7,R).
Definition. G c2 = {A ∈ GL(7,R) : A∗ϕc = ϕc} ⊂ SO(7)
G s2 = {A ∈ GL(7,R) : A∗ϕs = ϕs} ⊂ SO(3, 4).
From now on, G2 = G s2 .
2-plane Fields in Dimension 5
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2.
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame.
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
E. Cartan (1910) solved the equivalence problem for such D.Constructed a principal bundle and Cartan connection.
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
E. Cartan (1910) solved the equivalence problem for such D.Constructed a principal bundle and Cartan connection.
The model is G2/P ∼= S2 × S3, P ⊂ G2 parabolic subgroup.
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
E. Cartan (1910) solved the equivalence problem for such D.Constructed a principal bundle and Cartan connection.
The model is G2/P ∼= S2 × S3, P ⊂ G2 parabolic subgroup.
So G2 acts on S2 × S3 preserving the model D ⊂ T (S2 × S3).
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
E. Cartan (1910) solved the equivalence problem for such D.Constructed a principal bundle and Cartan connection.
The model is G2/P ∼= S2 × S3, P ⊂ G2 parabolic subgroup.
So G2 acts on S2 × S3 preserving the model D ⊂ T (S2 × S3).
The model D ⊂ T (S2 × S3) can be defined algebraically using thealgebraic structure of the imaginary split octonians,
2-plane Fields in Dimension 5
Let D ⊂ TM5 , dimDx = 2. X , Y local frame. Set Z = [X ,Y ].
Definition. D is generic if X , Y , Z , [X ,Z ], [Y ,Z ] are everywherelinearly independent.
E. Cartan (1910) solved the equivalence problem for such D.Constructed a principal bundle and Cartan connection.
The model is G2/P ∼= S2 × S3, P ⊂ G2 parabolic subgroup.
So G2 acts on S2 × S3 preserving the model D ⊂ T (S2 × S3).
The model D ⊂ T (S2 × S3) can be defined algebraically using thealgebraic structure of the imaginary split octonians,
or as a nonholonomic constraint in a classical mechanical system.
Rolling Balls
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
The group of local diffeomorphisms of S2 × SO(3) preserving Dcontains SO(3)× SO(3)
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
The group of local diffeomorphisms of S2 × SO(3) preserving Dcontains SO(3)× SO(3) with equality if (r1/r2)
±1 6=√3.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
The group of local diffeomorphisms of S2 × SO(3) preserving Dcontains SO(3)× SO(3) with equality if (r1/r2)
±1 6=√3.
But if r1/r2 =√3, this local symmetry group is G2.
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
The group of local diffeomorphisms of S2 × SO(3) preserving Dcontains SO(3)× SO(3) with equality if (r1/r2)
±1 6=√3.
But if r1/r2 =√3, this local symmetry group is G2.
Lift D via the double cover S2 × S3 → S2 × SO(3).
Rolling Balls
Consider 2 balls in R3 of radii r1, r2 rolling on one another without
slipping or spinning.
Identify configurations equivalent under Euclidean motions.The configuration space is then C = S2 × SO(3).
The no-slip, no-spin constraint defines a distribution D ⊂ TC.If r1 = r2, then D is integrable. Otherwise D is generic.
The group of local diffeomorphisms of S2 × SO(3) preserving Dcontains SO(3)× SO(3) with equality if (r1/r2)
±1 6=√3.
But if r1/r2 =√3, this local symmetry group is G2.
Lift D via the double cover S2 × S3 → S2 × SO(3).
This gives the model D ⊂ T (S2 × S3).
Conformal Group
Conformal Group
How does G2 act on S2 × S3?
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1}
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ;
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Now recall G2 ⊂ SO(3, 4)
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Now recall G2 ⊂ SO(3, 4) = conformal group of S2 × S3.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Now recall G2 ⊂ SO(3, 4) = conformal group of S2 × S3.
This conformal action of G2 is the action preserving D.
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Now recall G2 ⊂ SO(3, 4) = conformal group of S2 × S3.
This conformal action of G2 is the action preserving D.
So any diffeomorphism preserving D also preserves the (2, 3)conformal structure on S2 × S3!
Conformal Group
How does G2 act on S2 × S3? By conformal transformations!
Consider Rp+q+2 = {(x , y) : x ∈ Rp+1, y ∈ R
q+1} with quadraticform |x |2 − |y |2 and null cone N = {|x |2 = |y |2}.Then N/R+ = {|x |2 = |y |2 = 1} = Sp × Sq.
SO(p + 1, q + 1) acts linearly on Rp+q+2 preserving N ; this
induces an action on Sp × Sq.
This action preserves the (p, q) metric gSp − gSq up to scale andrealizes SO(p + 1, q + 1) as the conformal group of Sp × Sq.
Now recall G2 ⊂ SO(3, 4) = conformal group of S2 × S3.
This conformal action of G2 is the action preserving D.
So any diffeomorphism preserving D also preserves the (2, 3)conformal structure on S2 × S3! True locally too.
Nurowski’s Conformal Structures
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Follows immediately from the existence of the Cartan connection.
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Follows immediately from the existence of the Cartan connection.
For any D, can choose local coordinates (x , y , z , p, q) on M so that
D = span{∂q, ∂x + p∂y + q∂p + F∂z}
where F = F (x , y , z , p, q) and Fqq is nonvanishing.
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Follows immediately from the existence of the Cartan connection.
For any D, can choose local coordinates (x , y , z , p, q) on M so that
D = span{∂q, ∂x + p∂y + q∂p + F∂z}
where F = F (x , y , z , p, q) and Fqq is nonvanishing.
F = q2 for the model.
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Follows immediately from the existence of the Cartan connection.
For any D, can choose local coordinates (x , y , z , p, q) on M so that
D = span{∂q, ∂x + p∂y + q∂p + F∂z}
where F = F (x , y , z , p, q) and Fqq is nonvanishing.
F = q2 for the model.
Nurowski gives a formula for g in terms of F and its derivatives oforders ≤ 4.
Nurowski’s Conformal Structures
Theorem. (Nurowski, 2005) Any D ⊂ TM5 generic. There is aconformal class [g ] on M of signature (2, 3) associated to D.
Follows immediately from the existence of the Cartan connection.
For any D, can choose local coordinates (x , y , z , p, q) on M so that
D = span{∂q, ∂x + p∂y + q∂p + F∂z}
where F = F (x , y , z , p, q) and Fqq is nonvanishing.
F = q2 for the model.
Nurowski gives a formula for g in terms of F and its derivatives oforders ≤ 4.
Approximately 70 terms. Very nasty.
Ambient Metric
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Let G = {(x , gx) : x ∈ M, g ∈ [g ]} ⊂ S2T ∗M. Metric bundle.
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Let G = {(x , gx) : x ∈ M, g ∈ [g ]} ⊂ S2T ∗M. Metric bundle.
Dilations δs : G → G
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Let G = {(x , gx) : x ∈ M, g ∈ [g ]} ⊂ S2T ∗M. Metric bundle.
Dilations δs : G → G δs(x , gx) = (x , s2gx)
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Let G = {(x , gx) : x ∈ M, g ∈ [g ]} ⊂ S2T ∗M. Metric bundle.
Dilations δs : G → G δs(x , gx) = (x , s2gx)
Set G = G × (−1, 1).
Ambient Metric
Given conformal manifold (M, [g ]) of signature (p, q), p + q = n.
Obtain a manifold G of dimension n + 2 with an embeddedhypersurface G ⊂ G and a formal expansion along G for a metric g
on G of signature (p + 1, q + 1).
Conformally flat case: for M = Sp × Sq, g = gSp − gSq ,
Obtain G = Rp+q+2,
G = N = null cone of |x |2 − |y |2
The ambient metric is the flat metric g = |dx |2 − |dy |2.
General case:
Let G = {(x , gx) : x ∈ M, g ∈ [g ]} ⊂ S2T ∗M. Metric bundle.
Dilations δs : G → G δs(x , gx) = (x , s2gx)
Set G = G × (−1, 1). Inclusion: ι : G → G, ι(z) = (z , 0).
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Initial condition on G determined by the conformal structure
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Initial condition on G determined by the conformal structure
Ric(g) = 0 to infinite order on G.
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Initial condition on G determined by the conformal structure
Ric(g) = 0 to infinite order on G.
Theorem (C. Fefferman-G., 1985) If n is odd, there exists such g ,unique to infinite order up to diffeomorphism.
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Initial condition on G determined by the conformal structure
Ric(g) = 0 to infinite order on G.
Theorem (C. Fefferman-G., 1985) If n is odd, there exists such g ,unique to infinite order up to diffeomorphism.
If (M, g) is real-analytic, then series for g converges.
Ambient Metric
The ambient metric g is a metric on G of signature (p + 1, q + 1).Required to satisfy:
δ∗s g = s2g Homogeneous
Initial condition on G determined by the conformal structure
Ric(g) = 0 to infinite order on G.
Theorem (C. Fefferman-G., 1985) If n is odd, there exists such g ,unique to infinite order up to diffeomorphism.
If (M, g) is real-analytic, then series for g converges.
If n is even, there is a formal obstruction at order n/2.
Leistner-Nurowski Holonomy Result
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Nurowski (2007). Consider D ⊂ TR5 given by
F = q2 +6∑
k=0
akpk + bz , ak , b ∈ R
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Nurowski (2007). Consider D ⊂ TR5 given by
F = q2 +6∑
k=0
akpk + bz , ak , b ∈ R
Then can write g explicitly.
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Nurowski (2007). Consider D ⊂ TR5 given by
F = q2 +6∑
k=0
akpk + bz , ak , b ∈ R
Then can write g explicitly. Expansion terminates at order 2.
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Nurowski (2007). Consider D ⊂ TR5 given by
F = q2 +6∑
k=0
akpk + bz , ak , b ∈ R
Then can write g explicitly. Expansion terminates at order 2.
Theorem. (Leistner-Nurowski, 2009) F as above.
For all ak , b, have Hol(G, g) ⊂ G2
Leistner-Nurowski Holonomy Result
Put these together:
D ⊂ TM5 Nurowski→ (M, [g ])Ambientmetric→ (G7, g)
Produces a metric g of signature (3, 4) from D.
Nurowski (2007). Consider D ⊂ TR5 given by
F = q2 +6∑
k=0
akpk + bz , ak , b ∈ R
Then can write g explicitly. Expansion terminates at order 2.
Theorem. (Leistner-Nurowski, 2009) F as above.
For all ak , b, have Hol(G, g) ⊂ G2
If one of a3, a4, a5, a6 6= 0, then Hol(G, g) = G2.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
For model D on S2 × S3 = G2/P , have
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
For model D on S2 × S3 = G2/P , have
g = flat metric of signature (3, 4) on R7.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
For model D on S2 × S3 = G2/P , have
g = flat metric of signature (3, 4) on R7.
Take ϕ = the three form on R7 defining G2.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
For model D on S2 × S3 = G2/P , have
g = flat metric of signature (3, 4) on R7.
Take ϕ = the three form on R7 defining G2.
Leistner-Nurowski write down ϕ for their F ’s explicitly.
Leistner-Nurowski Holonomy Result
Gives a completely explicit 8-parameter family of metrics ofholonomy G2.
To show a metric in dimension 7 has holonomy ⊂ G2, need toconstruct a parallel 3-form ϕ compatible with the metric.
For model D on S2 × S3 = G2/P , have
g = flat metric of signature (3, 4) on R7.
Take ϕ = the three form on R7 defining G2.
Leistner-Nurowski write down ϕ for their F ’s explicitly.
But what about g for other D?
General Generic Distributions D
General Generic Distributions D
Work with Travis Willse.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
So we obtain an infinite-dimensional space of metrics g ofholonomy G2, parametrized by an almost arbitrary generic 2-planefield D.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
So we obtain an infinite-dimensional space of metrics g ofholonomy G2, parametrized by an almost arbitrary generic 2-planefield D.
In the remaining time:
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
So we obtain an infinite-dimensional space of metrics g ofholonomy G2, parametrized by an almost arbitrary generic 2-planefield D.
In the remaining time:
1. Formulate conditions for Hol(G, g) = G2.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
So we obtain an infinite-dimensional space of metrics g ofholonomy G2, parametrized by an almost arbitrary generic 2-planefield D.
In the remaining time:
1. Formulate conditions for Hol(G, g) = G2.
2. Outline the proof that Hol(G, g) ⊂ G2.
General Generic Distributions D
Work with Travis Willse.
Theorem. Let D ⊂ TM5 be generic and real-analytic.
Hol(G, g) ⊂ G2 always.
Hol(G, g) = G2 for an explicit open dense set of D.
So we obtain an infinite-dimensional space of metrics g ofholonomy G2, parametrized by an almost arbitrary generic 2-planefield D.
In the remaining time:
1. Formulate conditions for Hol(G, g) = G2.
2. Outline the proof that Hol(G, g) ⊂ G2.
3. Describe associated Poincare-Einstein metrics.
Conditions for Hol(G, g) = G2
.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective. Nondegeneracy condition on (W ,C ).
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective. Nondegeneracy condition on (W ,C ).
Let A ∈ Γ(S4D∗) be Cartan’s fundamental curvature invariant forgeneric distributions D ⊂ TM5.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective. Nondegeneracy condition on (W ,C ).
Let A ∈ Γ(S4D∗) be Cartan’s fundamental curvature invariant forgeneric distributions D ⊂ TM5. A is a binary quartic.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective. Nondegeneracy condition on (W ,C ).
Let A ∈ Γ(S4D∗) be Cartan’s fundamental curvature invariant forgeneric distributions D ⊂ TM5. A is a binary quartic.
Say that A is 3-nondegenerate at p if the only vector X ∈ Dp suchthat A(X ,Y ,Y ,Y ) = 0 for all Y ∈ Dp is X = 0.
Conditions for Hol(G, g) = G2
.Let Wijkl = Weyl tensor, Cjkl = Cotton tensor of Nurowski’s g .
Define Lp : TpM × R → ⊗3T ∗pM by
L(v , λ) = Wijklvi + Cjklλ.
Impose: Lp is injective. Nondegeneracy condition on (W ,C ).
Let A ∈ Γ(S4D∗) be Cartan’s fundamental curvature invariant forgeneric distributions D ⊂ TM5. A is a binary quartic.
Say that A is 3-nondegenerate at p if the only vector X ∈ Dp suchthat A(X ,Y ,Y ,Y ) = 0 for all Y ∈ Dp is X = 0.
Theorem. Given (M,D) real analytic. If there are p, q ∈ M sothat Lp is injective and Aq is 3-nondegenerate, then g hasholonomy = G2.
Conditions for Hol(G, g) = G2
In particular, have Hol(G, g) = G2 if there is p ∈ M so that Lp isinjective and Ap is 3-nondegenerate.
Conditions for Hol(G, g) = G2
In particular, have Hol(G, g) = G2 if there is p ∈ M so that Lp isinjective and Ap is 3-nondegenerate.
Each condition is an algebraic condition on the 7-jet of F at p.
Conditions for Hol(G, g) = G2
In particular, have Hol(G, g) = G2 if there is p ∈ M so that Lp isinjective and Ap is 3-nondegenerate.
Each condition is an algebraic condition on the 7-jet of F at p.
So if the 7-jet of F avoids a particular algebraic set at a singlepoint, then Hol(G, g) = G2.
Conditions for Hol(G, g) = G2
In particular, have Hol(G, g) = G2 if there is p ∈ M so that Lp isinjective and Ap is 3-nondegenerate.
Each condition is an algebraic condition on the 7-jet of F at p.
So if the 7-jet of F avoids a particular algebraic set at a singlepoint, then Hol(G, g) = G2.
This is a weak condition, explicitly checkable.
Proof that Hol(G, g) ⊂ G2
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5,
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s,
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
1. Construct ϕ|G . Should be homogeneous of degree 3.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
1. Construct ϕ|G . Should be homogeneous of degree 3.
2. Extend ϕ|G to G to be parallel.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
1. Construct ϕ|G . Should be homogeneous of degree 3.
2. Extend ϕ|G to G to be parallel.
Step 1. Construct ϕ|G
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
1. Construct ϕ|G . Should be homogeneous of degree 3.
2. Extend ϕ|G to G to be parallel.
Step 1. Construct ϕ|G
This reduces to a problem just involving conformal geometry of(M, [g ])–no ambient considerations.
Proof that Hol(G, g) ⊂ G2
Let D ⊂ TM5, [g ] = Nurowski’s, g = ambient metric.
Theorem. Hol(G, g) ⊂ G2.
Proof. Construct parallel ϕ. There are 2 steps:
1. Construct ϕ|G . Should be homogeneous of degree 3.
2. Extend ϕ|G to G to be parallel.
Step 1. Construct ϕ|G
This reduces to a problem just involving conformal geometry of(M, [g ])–no ambient considerations.
Reinterpret ϕ|G in terms of the tractor bundle of (M, [g ]).
Tractor bundle and connection
Tractor bundle and connection
(M, [g ]) conformal manifold.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M. If p ∈ M,
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M. If p ∈ M,
Tp ={U ∈ Γ(T G
∣∣π−1(p)
) : (δs)∗U = sU, s > 0}.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M. If p ∈ M,
Tp ={U ∈ Γ(T G
∣∣π−1(p)
) : (δs)∗U = sU, s > 0}.
The tractor metric and connection are induced from g and ∇.
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M. If p ∈ M,
Tp ={U ∈ Γ(T G
∣∣π−1(p)
) : (δs)∗U = sU, s > 0}.
The tractor metric and connection are induced from g and ∇.
Conclusion. ϕ|G can be viewed as a section of Λ3T ∗:
Tractor bundle and connection
(M, [g ]) conformal manifold. The tractor connection is theconformal analogue of the Levi-Civita connection:
There is a rank n + 2 vector bundle T → M with fiber metric ofsignature (p + 1, q + 1) and connection ∇.
T is an associated bundle to the Cartan principal structure bundle.
Can alternately realize T as homogeneous sections of T G|G :Recall metric bundle G, with π : G → M. If p ∈ M,
Tp ={U ∈ Γ(T G
∣∣π−1(p)
) : (δs)∗U = sU, s > 0}.
The tractor metric and connection are induced from g and ∇.
Conclusion. ϕ|G can be viewed as a section of Λ3T ∗: a tractor3-form.
Step 1: Construct ϕ|G.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
So this solves Step 1: construct ϕ|G such that ∇X
(ϕ|G
)= 0 for
X ∈ TG.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
So this solves Step 1: construct ϕ|G such that ∇X
(ϕ|G
)= 0 for
X ∈ TG.
Remark. Other direction is true as well:
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
So this solves Step 1: construct ϕ|G such that ∇X
(ϕ|G
)= 0 for
X ∈ TG.
Remark. Other direction is true as well:
Theorem. (Hammerl-Sagerschnig, 2009) Nurowski’s conformalstructures (M, [g ]) associated to generic D are characterized bythe existence of a compatible parallel tractor 3-form.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
So this solves Step 1: construct ϕ|G such that ∇X
(ϕ|G
)= 0 for
X ∈ TG.
Remark. Other direction is true as well:
Theorem. (Hammerl-Sagerschnig, 2009) Nurowski’s conformalstructures (M, [g ]) associated to generic D are characterized bythe existence of a compatible parallel tractor 3-form.
This is a conformal holonomy characterization.
Step 1: Construct ϕ|G.If ∇ϕ = 0, in particular must have ∇X
(ϕ|G
)= 0 for X ∈ TG.
Equivalent to saying that ϕ|G is a parallel tractor 3-form associatedto (M, [g ]).
Existence of a parallel tractor 3-form follows from the Cartanconnection and the fact that G2 fixes a 3-form on R
7.
So this solves Step 1: construct ϕ|G such that ∇X
(ϕ|G
)= 0 for
X ∈ TG.
Remark. Other direction is true as well:
Theorem. (Hammerl-Sagerschnig, 2009) Nurowski’s conformalstructures (M, [g ]) associated to generic D are characterized bythe existence of a compatible parallel tractor 3-form.
This is a conformal holonomy characterization.(Conformal holonomy = holonomy of tractor connection.)
Parallel Extension Theorem
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Theorem. Let (M, [g ]) be a conformal manifold, with ambientmetric g . Suppose ϕ is a parallel tractor-tensor of rank r ∈ N.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Theorem. Let (M, [g ]) be a conformal manifold, with ambientmetric g . Suppose ϕ is a parallel tractor-tensor of rank r ∈ N.
If n is odd, then ϕ has an ambient extension ϕ such that ∇ϕvanishes to infinite order along G.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Theorem. Let (M, [g ]) be a conformal manifold, with ambientmetric g . Suppose ϕ is a parallel tractor-tensor of rank r ∈ N.
If n is odd, then ϕ has an ambient extension ϕ such that ∇ϕvanishes to infinite order along G.If n is even, then ϕ has an ambient extension such that ∇ϕvanishes to order n/2− 1.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Theorem. Let (M, [g ]) be a conformal manifold, with ambientmetric g . Suppose ϕ is a parallel tractor-tensor of rank r ∈ N.
If n is odd, then ϕ has an ambient extension ϕ such that ∇ϕvanishes to infinite order along G.If n is even, then ϕ has an ambient extension such that ∇ϕvanishes to order n/2− 1.
This had been previously proved by Gover for r = 1, different proof.
Parallel Extension Theorem
Step 2. Extend ϕ|G to G to be parallel
We prove an ambient extension theorem for parallel tractor-tensors.
Holds for general conformal structures in any signature anddimension and for parallel tractors having arbitrary symmetry.
Let T = tractor bundle. Tractor-tensor means a section of ⊗rT ∗.
Theorem. Let (M, [g ]) be a conformal manifold, with ambientmetric g . Suppose ϕ is a parallel tractor-tensor of rank r ∈ N.
If n is odd, then ϕ has an ambient extension ϕ such that ∇ϕvanishes to infinite order along G.If n is even, then ϕ has an ambient extension such that ∇ϕvanishes to order n/2− 1.
This had been previously proved by Gover for r = 1, different proof.
Immediately conclude Hol(G, g) ⊂ G2.
Associated Poincare Metrics
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1}
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1)
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1 Infinitesimal dilation
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1 Infinitesimal dilation
Set H+ = {g(T ,T ) = 1}
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1 Infinitesimal dilation
Set H+ = {g(T ,T ) = 1} and H− = {g(T ,T ) = −1}.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1 Infinitesimal dilation
Set H+ = {g(T ,T ) = 1} and H− = {g(T ,T ) = −1}.Then g+ = g |TH+ has signature (p, q + 1) and Ric(g+) = ng+.
Associated Poincare Metrics
Recall: ambient metric for M = Sp × Sq, g = gSp − gSq isg = |dx |2 − |dy |2 on R
n+2 and Sp × Sq = N/R+.
Let H+ = {|x |2 − |y |2 = 1} and H− = {|x |2 − |y |2 = −1}.Then g+ = g |TH+ has signature (p, q + 1) and curvature +1.
Likewise g− = g |TH−
has signature (p + 1, q) and curvature −1.
Analogous construction for general (M, [g ]):
Let T = ddsδs |s=1 Infinitesimal dilation
Set H+ = {g(T ,T ) = 1} and H− = {g(T ,T ) = −1}.Then g+ = g |TH+ has signature (p, q + 1) and Ric(g+) = ng+.
And g− = g |TH−
has signature (p + 1, q) and Ric(g−) = −ng−.
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity.
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ)
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ) and
(r2g±)|TM = g .
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ) and
(r2g±)|TM = g .
Moreover, g is a cone metric over g±:
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ) and
(r2g±)|TM = g .
Moreover, g is a cone metric over g±:
g = s2g+ + ds2 on {g(T ,T ) > 0}
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ) and
(r2g±)|TM = g .
Moreover, g is a cone metric over g±:
g = s2g+ + ds2 on {g(T ,T ) > 0}
g = s2g− − ds2 on {g(T ,T ) < 0}
Associated Poincare Metrics
g+ and g− are asymptotically hyperbolic with (M, [g ]) asconformal infinity. g± are defined on M × (0, ǫ) and
(r2g±)|TM = g .
Moreover, g is a cone metric over g±:
g = s2g+ + ds2 on {g(T ,T ) > 0}
g = s2g− − ds2 on {g(T ,T ) < 0}
Constructing g is equivalent to constructing g±.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Proposition. g+ is nearly-Kahler of constant type 1.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Proposition. g+ is nearly-Kahler of constant type 1.
Definition. A metric is nearly Kahler if there exists an orthogonalalmost complex structure J such that (∇X J)(X ) = 0 for all X .
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Proposition. g+ is nearly-Kahler of constant type 1.
Definition. A metric is nearly Kahler if there exists an orthogonalalmost complex structure J such that (∇X J)(X ) = 0 for all X .
Definition. g is nearly Kahler of constant type 1 if
|(∇X J)(Y )|2 = |X |2|Y |2 − 〈X ,Y 〉2 − 〈JX ,Y 〉2.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Proposition. g+ is nearly-Kahler of constant type 1.
Definition. A metric is nearly Kahler if there exists an orthogonalalmost complex structure J such that (∇X J)(X ) = 0 for all X .
Definition. g is nearly Kahler of constant type 1 if
|(∇X J)(Y )|2 = |X |2|Y |2 − 〈X ,Y 〉2 − 〈JX ,Y 〉2.
Similarly, g− is a signature (3, 3) metric with Ric(g−) = −5g−which is “nearly-para-Kahler of constant type 1”.
Associated Poincare Metrics
Suppose now that [g ] arises from D ⊂ TM5.
So g+ has signature (2, 4) and Ric(g+) = 5g+.
Proposition. g+ is nearly-Kahler of constant type 1.
Definition. A metric is nearly Kahler if there exists an orthogonalalmost complex structure J such that (∇X J)(X ) = 0 for all X .
Definition. g is nearly Kahler of constant type 1 if
|(∇X J)(Y )|2 = |X |2|Y |2 − 〈X ,Y 〉2 − 〈JX ,Y 〉2.
Similarly, g− is a signature (3, 3) metric with Ric(g−) = −5g−which is “nearly-para-Kahler of constant type 1”.
So we obtain new examples of metrics of these types parametrizedby D ⊂ TM5.