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.