· In 1925, E. Cartan made the following assertions:´ (1) H x is a Lie subgroup of O(T xM). (2)...

NONEMBEDDING AND NONEXTENSION RESULTSIN SPECIAL HOLONOMY

ROBERT L. BRYANT

DUKE UNIVERSITY

Eruditio et Relig

Riemannian Holonomy. To a Riemannian manifold (Mn, g) associate itsLevi-Civita connection ∇, which defines, for a piecewise C1 γ : [0, 1] → M ,a parallel transport

P∇γ : Tγ(0)M → Tγ(1)M,

which is a linear isometry between the two tangent spaces.

In 1918, J. Schouten considered the set

Hx =P∇

γ γ(0) = γ(1) = x⊆ O(TxM)

and called its dimension the number of degrees of freedom of g.

P∇γ =

)−1and P∇

γ2∗γ1= P∇

γ2 P∇

where γ is the reverse of γ and γ2∗γ1 is the concatenation of paths γ1 and γ2

satisfying γ1(1) = γ2(0).

In particular, Hx ⊂ O(TxM) is a subgroup and

Hγ(1) = P∇γ Hγ(0)

)−1.

P∇γ : Tγ(0)M → Tγ(1)M,

Hx =P∇

γ γ(0) = γ(1) = x⊆ O(TxM)

P∇γ =

)−1and P∇

γ2∗γ1= P∇

γ2 P∇

Hγ(1) = P∇γ Hγ(0)

)−1.

P∇γ : Tγ(0)M → Tγ(1)M,

Hx =P∇

γ γ(0) = γ(1) = x⊆ O(TxM)

P∇γ =

)−1and P∇

γ2∗γ1= P∇

γ2 P∇

Hγ(1) = P∇γ Hγ(0)

)−1.

P∇γ : Tγ(0)M → Tγ(1)M,

Hx =P∇

γ γ(0) = γ(1) = x⊆ O(TxM)

P∇γ =

)−1and P∇

γ2∗γ1= P∇

γ2 P∇

Hγ(1) = P∇γ Hγ(0)

)−1.

P∇γ : Tγ(0)M → Tγ(1)M,

Hx =P∇

γ γ(0) = γ(1) = x⊆ O(TxM)

P∇γ =

)−1and P∇

γ2∗γ1= P∇

γ2 P∇

Hγ(1) = P∇γ Hγ(0)

)−1.

In 1925, E. Cartan made the following assertions:(1) Hx is a Lie subgroup of O(TxM).(2) If Hx acts reducibly on TxM , then g is a product metric.

In dimensions 2 and 3, this determines the possible holonomy groups.

He considered the case n = 4 and Hx SU(2) and stated:(1) Such metrics g have vanishing Ricci tensor.(2) Such metrics g are what we now call ‘self-dual’.(3) Such metrics g depend on 2 functions of 3 variables.

Cartan gave no indication of proof and never mentioned the subject again.

Probable argument: The SU(2)-frame bundle of such a metric satisfies

dω + θ ∧ω = 0, dθ + θ ∧ θ = 〈R, ω ∧ω〉, dR + θ.R = 〈R′, ω〉where ω takes values in R4, θ takes values in su(2), R takes values in W4,the 5-dimensional real irr. rep. of SU(2), and R′ ⊂ Hom(R4, W4) takesvalues in V5, the 6-dimensional complex irr. rep. of SU(2). The latter is aninvolutive tableau in Hom(R4, W4), with characters (5, 5, 2, 0). QED

Modern Argument: If (M4, g) has Hx SU(2), then there exist threeg-parallel 2-forms on M , say Υ1, Υ2, and Υ3, such that

Υi ∧Υj = 2δij dVg .

There exist loc. coord. z = (z1, z2) : U → C2 and φ : z(U) → R so that

Υ2 + i Υ3 = dz1 ∧ dz2 and Υ1 = 12 i ∂∂φ,

where φ satisfies the elliptic Monge-Ampere equation(∂2φ

∂zi∂zj

)> 0 and det

(∂2φ

∂zi∂zj

Conversely, such Υi uniquely determine (M4, g) with holonomy SU(2).

Υ2 + i Υ3 = dz1 ∧ dz2 and Υ1 = 12 i ∂∂φ,

∂zi∂zj

)> 0 and det

(∂2φ

∂zi∂zj

Υ2 + i Υ3 = dz1 ∧ dz2 and Υ1 = 12 i ∂∂φ,

∂zi∂zj

)> 0 and det

(∂2φ

∂zi∂zj

Υ2 + i Υ3 = dz1 ∧ dz2 and Υ1 = 12 i ∂∂φ,

∂zi∂zj

)> 0 and det

(∂2φ

∂zi∂zj

Exterior Differential Systems Argument:

Let M4 be an analytic manifold and let Υ be the tautological 2-formon Λ2(T ∗M).

LetX17 ⊂

(Λ2(T ∗M)

be the submanifold consisting of triples (β1, β2, β3) ∈ Λ2(T ∗x M) such that

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

The pullbacks Υi = π∗i (Υ) define an exterior differential system on X

I = dΥ1, dΥ2, dΥ3.An integral manifold Y 4 ⊂ X transverse to π : X → M then represents achoice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy thealgebra conditions needed to define an SU(2)-structure on U .

Simple calculation shows that I is involutive.

LetX17 ⊂

(Λ2(T ∗M)

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

LetX17 ⊂

(Λ2(T ∗M)

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

LetX17 ⊂

(Λ2(T ∗M)

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

LetX17 ⊂

(Λ2(T ∗M)

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

LetX17 ⊂

(Λ2(T ∗M)

β12 = β2

2 = β32 = 0, and β1 ∧β2 = β3 ∧β1 = β2 ∧ β3 = 0.

A sharper result:Suppose that (M4, g) has holonomy SU(2) and let Υi

be three g-parallel 2-forms on M satisfying

If N3 ⊂ M is an oriented hypersurface, with oriented normal n, thenthere is a coframing η of N defined by

⎛⎝η1

⎞⎠ =

⎛⎝n Υ1

⎞⎠ and it satisfies N∗

⎛⎝Υ1

⎞⎠ =

⎛⎝η2∧η3

η3∧η1

η1∧η2

⎞⎠ = ∗ηη

In particular,d(∗ηη) = N∗dΥ = 0.

Theorem: If η is a real-analytic coframing of N such that d(∗ηη) = 0then there is an essentially unique embedding of N into a SU(2)-holonomymanifold (M4, g) that induces the given coframing η in the above manner.

A sharper result: Suppose that (M4, g) has holonomy SU(2) and let Υi

⎛⎝η1

⎞⎠ =

⎛⎝n Υ1

⎛⎝Υ1

⎞⎠ =

⎛⎝η2∧η3

η3∧η1

η1∧η2

⎞⎠ = ∗ηη

⎛⎝η1

⎞⎠ =

⎛⎝n Υ1

⎛⎝Υ1

⎞⎠ =

⎛⎝η2∧η3

η3∧η1

η1∧η2

⎞⎠ = ∗ηη

⎛⎝η1

⎞⎠ =

⎛⎝n Υ1

⎛⎝Υ1

⎞⎠ =

⎛⎝η2∧η3

η3∧η1

η1∧η2

⎞⎠ = ∗ηη

Proof: Write dη = −θ∧η where θ = −tθ.On N × GL(3, R) define

ω = g−1 η and γ = g−1dg + g−1θg,

so that dω = −γ∧ω. On X = N × GL(3, R) × R define the three 2-forms⎛⎝Υ1

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

Let I be the ideal on X generated by dΥ1, dΥ2, dΥ3. One calculates

dΥ =(tγ − (tr γ)I3

)∧ ∗ωω + γ ∧ω∧dt.

Consequently, I is involutive, with characters (s1, s2, s3, s4) = (0, 3, 6, 0).Since d(∗ηη) = 0, the locus L = N × I3 × 0 ⊂ X is a regular, real-

analytic integral manifold of the real-analytic ideal I. Note that L is justa copy of N .

By the Cartan-Kahler Theorem, L lies in a unique 4-dimensional integralmanifold M ⊂ X. The Υi pull back to M to be closed and to define thedesired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −tθ. On N × GL(3, R) define

so that dω = −γ∧ω.On X = N × GL(3, R) × R define the three 2-forms⎛⎝Υ1

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

)∧ ∗ωω + γ ∧ω∧dt.

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

)∧ ∗ωω + γ ∧ω∧dt.

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

)∧ ∗ωω + γ ∧ω∧dt.

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

)∧ ∗ωω + γ ∧ω∧dt.

⎞⎠ =

⎛⎝dt∧ω1 + ω2∧ω3

dt∧ω2 + ω3∧ω1

dt∧ω3 + ω1∧ω2

⎞⎠ = dt ∧ω + ∗ωω.

)∧ ∗ωω + γ ∧ω∧dt.

Question: Is it necessary to assume that η be real-analytic?

The condition d(dt∧ω + ∗ω) = 0 is an ‘SU(2)-flow’ on coframings of N :ddt

ω = ∗ω(dω) − 12 ∗ω(tω ∧dω) ω with initial condition ω t=0 = η.

When η is real-analytic and satisfies d(∗ηη) = 0, this flow with initialcondition has a unique solution, which continues to be co-closed.

Theorem: There exist η on N3 with d(∗ηη) = 0 for which the SU(2)-flowwith initial condition η has no solution. In fact, if η is not real-analytic and

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has nosolution. (Such non-real-analytic coframings do exist.)

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

Theorem: There exist η on N3 with d(∗ηη) = 0 for which the SU(2)-flowwith initial condition η has no solution.In fact, if η is not real-analytic and

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has nosolution.(Such non-real-analytic coframings do exist.)

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

Proof: Suppose that Υi (1 ≤ i ≤ 3) are the parallel 2-forms on an (M4, g)with holonomy SU(2) and let N3 ⊂ M be an oriented hypersurface.

Calculation yields that the induced co-closed coframing η satisfies

∗η(tη ∧dη) = 2H

where H is the mean curvature of N in M .

Now, (M, g) is real-analytic. If H is constant, then elliptic regularityimplies that N must be a real-analytic hypersurface in M and hence ηmust also be real-analytic.

Thus, if η is a non-real-analytic coframing on N3 that satisfies

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

for some constant C, then η cannot be induced on N by an embedding intoan SU(2)-holonomy 4-manifold.

d(∗ηη) = 0 and ∗η(tη ∧ dη) = 2C

To finish the proof, note that, if a coframing η on N3 is real-analytic in anycoordinate system at all, it will be real-analytic in harmonic coordinates,i.e., local coordinates x : U → R3 satisfying

d(∗ηdx

Now, fix a constant C and consider a coframing η = g(x)−1 dx on U ⊂ R3

where g : U → GL(3, R) is a mapping satisfying the first-order, quasi-linearsystem

d(∗ηη) = 0, ∗η(tη ∧dη) = 2C, d(∗ηdx

This is an elliptic underdetermined system consisting of 7 equations for 9unknowns. Standard theory shows that the general solution is not real-analytic.

d(∗ηdx

d(∗ηη) = 0, ∗η(tη ∧dη) = 2C, d(∗ηdx

d(∗ηdx

d(∗ηη) = 0, ∗η(tη ∧dη) = 2C, d(∗ηdx

d(∗ηdx

d(∗ηη) = 0, ∗η(tη ∧dη) = 2C, d(∗ηdx

The G2-theory. An analogous situation holds in the case of hypersurfacesin Riemannian manifolds (M7, g) with holonomy G2 ⊂ SO(7). In this case,there is a unique g-parallel 3-form σ ∈ Ω3(M) such that

σ ∧ ∗σ = 7 dVg .

Such metrics are Ricci-flat and hence are real-analytic in local g-harmoniccoordinate charts.

Conversely, there is a open set Ω3+(M7) of definite 3-forms, i.e., σ ∈

Ω3+(M7) if and only if, for all x ∈ M , the stabilizer of σx in GL(TxM)

is isomorphic to G2 ⊂ SO(7). These forms are the sections of an opensubbundle Λ3

+(T ∗M) ⊂ Λ3(T ∗M).Such a σ ∈ Ω3

+(M) determines a unique metric gσ and orientation ∗σ

and σ is gσ-parallel if and only if

dσ = 0 and d(∗σσ) = 0.

Theorem: (B—) There is an involutive EDS I on Λ3+(T ∗M) such that a

section σ ∈ Ω3+(M) is an integral of I iff it is gσ-parallel.

σ ∧ ∗σ = 7 dVg .

+(T ∗M) ⊂ Λ3(T ∗M).Such a σ ∈ Ω3

dσ = 0 and d(∗σσ) = 0.

σ ∧ ∗σ = 7 dVg .

+(T ∗M) ⊂ Λ3(T ∗M).Such a σ ∈ Ω3

dσ = 0 and d(∗σσ) = 0.

σ ∧ ∗σ = 7 dVg .

+(T ∗M) ⊂ Λ3(T ∗M).Such a σ ∈ Ω3

dσ = 0 and d(∗σσ) = 0.

σ ∧ ∗σ = 7 dVg .

+(T ∗M) ⊂ Λ3(T ∗M).Such a σ ∈ Ω3

dσ = 0 and d(∗σσ) = 0.

Hypersurfaces.G2 acts transitively on S6 ⊂ R7, with stabilizer SU(3).

Hence, an oriented N6 ⊂ M inherits a canonical SU(3)-structure, which isdetermined by the (1, 1)-form ω and (3, 0)-form Ω = φ + i ψ defined by

ω = n σ and Ω = φ + i ψ = N∗σ − i (n ∗σσ).

In fact, if one defines f : R × N → M by

f(t, p) = expp

(tn(p)

f∗σ = dt ∧ω + Re(Ω) and f∗(∗σσ) = 12 ω2 − dt ∧ Im(Ω).

where, now, ω and Ω are forms on N that depend on t.

For each fixed t = t0, the induced SU(3)-structure on N satisfies

d Re(Ω) = d(f∗t0

σ) = 0 and d(12

ω2) = d(f∗

t0(∗σσ)

so these are necessary conditions on the SU(3)-structure on N that it beinduced by immersion into a G2-holonomy manifold M .

Hypersurfaces. G2 acts transitively on S6 ⊂ R7, with stabilizer SU(3).

f(t, p) = expp

(tn(p)

d Re(Ω) = d(f∗t0

σ) = 0 and d(12

ω2) = d(f∗

t0(∗σσ)

f(t, p) = expp

(tn(p)

d Re(Ω) = d(f∗t0

σ) = 0 and d(12

ω2) = d(f∗

t0(∗σσ)

f(t, p) = expp

(tn(p)

d Re(Ω) = d(f∗t0

σ) = 0 and d(12

ω2) = d(f∗

t0(∗σσ)

Theorem: A real-analytic SU(3)-structure on N6 is induced by embeddinginto a G2-holonomy manifold iff its defining forms ω and Ω satisfy

d Re(Ω) = 0 and d(12 ω2) = 0.

Proof: Define a tautological 2-form ω and 3-form Ω on F(N)/ SU(3) asfollows: For a coframe u : TxN → C3, define these forms at [u] = u·SU(3) ∈F(N)/ SU(3) by

ω[u] = π∗(u∗( i2 (tdz ∧dz))

)and Ω[u] = π∗(u∗(dz1 ∧dz2 ∧dz3)

)where π : F(N)/ SU(3) → N is the basepoint projection.

On X = R × F(N)/ SU(3), consider the 3-form and 4-form

σ = dt ∧ω + Re(Ω) and φ = 12 ω2 − dt ∧ Im(Ω).

Let I be the EDS generated by the closed 4-form dσ and 5-form dφ.Then I is involutive, with characters (s1, . . . , s7) = (0, 0, 1, 4, 10, 13, 0).

Since d(Re(Ω)

)= d(1

2 ω2) = 0, the given SU(3)-structure defines a reg-ular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By theCartan-Kahler Theorem, L lies in a unique I-integral M7 ⊂ X. QED

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

2ω2) = 0, the given SU(3)-structure defines a reg-

ular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By theCartan-Kahler Theorem, L lies in a unique I-integral M7 ⊂ X. QED

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

d Re(Ω) = 0 and d(12 ω2) = 0.

ω[u] = π∗(u∗( i2 (tdz ∧dz))

Since d(Re(Ω)

)= d(1

Theorem: There exist non-real-analytic SU(3)-structures on N6 whoseassociated forms satisfy

d(Re(Ω)

)= d(1

2 ω2) = 0

but that are not induced from an immersion into a G2-holonomy Riemann-ian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies

∗(ω ∧ d

(Im(Ω)

where C is a constant, then it cannot be G2-immersed. (Such do exist.)

Proof: When an SU(3)-structure on N6 with forms (ω, Ω) is induced viaa G2-immersion N6 → M7, the mean curvature H of N in M is givenby −12H = ∗

(ω∧d

(Im(Ω)

)). Thus, when this latter function is constant it

follows, by elliptic regularity, that N6 is a real-analytic submanifold of thereal-analytic (M7, g). If the SU(3)-structure is not real-analytic, this is acontradiction.

It remains to construct such an example.

d(Re(Ω)

)= d(1

2 ω2) = 0

∗(ω ∧ d

(Im(Ω)

where C is a constant, then it cannot be G2-immersed.(Such do exist.)

(ω∧d

(Im(Ω)

d(Re(Ω)

)= d(1

2 ω2) = 0

∗(ω ∧ d

(Im(Ω)

(ω∧d

(Im(Ω)

d(Re(Ω)

)= d(1

2 ω2) = 0

∗(ω ∧ d

(Im(Ω)

(ω∧d

(Im(Ω)

d(Re(Ω)

)= d(1

2 ω2) = 0

∗(ω ∧ d

(Im(Ω)

(ω∧d

(Im(Ω)

d(Re(Ω)

)= d(1

2 ω2) = 0

∗(ω ∧ d

(Im(Ω)

(ω∧d

(Im(Ω)

Why it’s somewhat delicate:Since dim(GL(6, R)/ SU(3)

)= 28, a choice

of an SU(3)-structure (ω, Ω) on N6 depends on 28 functions of 6 variables.Modulo diffeomorphisms, this leaves 22 functions of 6 variables.

On the other hand, the equations

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure.

Fix an orientation of N6. Say that a 3-form φ ∈ Ω3(N6) is elliptic if, ateach point, it is linearly equivalent to Re

(dz1∧dz2∧dz3

). Such a φ defines

a unique, orientation-preserving almost-complex structure Jφ on N6 suchthat

Ωφ = φ + i J∗φ(φ)

is of Jφ-type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary andyet must be a sum of terms of Jφ-type (3, 1) and (2, 2).

Thus, dΩφ is purely of Jφ-type (2, 2).

Why it’s somewhat delicate: Since dim(GL(6, R)/ SU(3)

)= 28, a

choice of an SU(3)-structure (ω, Ω) on N6 depends on 28 functions of 6variables. Modulo diffeomorphisms, this leaves 22 functions of 6 variables.

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

)= 28, a

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

)= 28, a

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

)= 28, a

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

)= 28, a

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

)= 28, a

d(Re(Ω)

)= 0, d(1

2 ω2) = 0, ∗(ω ∧d

(Im(Ω)

(dz1∧dz2∧dz3

So far: φ ∈ Ω3e(N

6) defines Jφ and Ωφ = φ + i J∗φ(φ) ∈ Ω3,0(N, Jφ).

dφ = 0 then yields dΩφ ∈ Ω2,2(N, Jφ).

Fix a constant C = 0. It is a C1-open condition on φ that

dΩφ = i6C (ωφ)2 for some ωφ = ωφ ∈ Ω1,1

+ (N, Jφ).

Now, the pair (ωφ, Ωφ) are the defining forms of an SU(3)-structure on Nif and only if

16(ωφ)3 − 1

8 i Ωφ ∧ Ωφ = 0.

This is a single, first-order scalar equation on the closed 3-form φ. It iseasy to see that there are non-analytic solutions. Assuming this conditionis satisfied:

d(ReΩφ) = dφ = 0,

12 (ωφ)2

(−3i 1

C dΩφ

and, finally

(ωφ ∧d(Im Ωφ)

)= ∗φ

(ωφ ∧ 1

6C (ωφ)2)

+ (N, Jφ).

16(ωφ)3 − 1

8 i Ωφ ∧ Ωφ = 0.

This is a single, first-order scalar equation on the closed 3-form φ. It iseasy to see that there are non-analytic solutions.Assuming this condition issatisfied:

12 (ωφ)2

(−3i 1

C dΩφ

and, finally

(ωφ ∧d(Im Ωφ)

)= ∗φ

(ωφ ∧ 1

6C (ωφ)2)

+ (N, Jφ).

16(ωφ)3 − 1

8 i Ωφ ∧ Ωφ = 0.

This is a single, first-order scalar equation on the closed 3-form φ. It iseasy to see that there are non-analytic solutions. Assuming this conditionis satisfied:

12 (ωφ)2

(−3i 1

C dΩφ

and, finally

(ωφ ∧d(Im Ωφ)

)= ∗φ

(ωφ ∧ 1

6C (ωφ)2)

Interpretation: On N6×R, with (ω, Ω) defining an SU(3)-structure on N6

depending on t ∈ R, consider the equations

d(dt ∧ ω + Re(Ω)

)= 0 and d

(12 ω2 − dt ∧ Im(Ω)

Think of Ω as φ+iJ∗φ(φ), so the SU(3)-structure is determined by (ω, φ)

where φ = Re(Ω). The closure conditions for fixed t are

dφ = 0 and d(ω2) = 0,

and the G2-evolution equations for such (ω, φ) are thenddt

(φ) = dω andddt

(ω) = −Lω−1

(d(J∗

φ(φ)))

where Lω : Ω2(N) → Ω4(N) is the invertible map Lω(β) = ω∧β.

The discussion shows that this ‘G2-flow’ does exist for analytic initialSU(3)-structures satisfying the closure conditions, but may not exist fornon-analytic initial SU(3)-structures satisfying the closure conditions.

)= 0 and d

(12 ω2 − dt ∧ Im(Ω)

dφ = 0 and d(ω2) = 0,

(φ) = dω andddt

(ω) = −Lω−1

(d(J∗

φ(φ)))

)= 0 and d

(12 ω2 − dt ∧ Im(Ω)

dφ = 0 and d(ω2) = 0,

(φ) = dω andddt

(ω) = −Lω−1

(d(J∗

φ(φ)))

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizerof a 4-form Φ0 ∈ Λ4(R8).

Thus, a Spin(7)-structure on M8 is a 4-form Φ ∈ Ω4(M) that is linearlyequivalent to Φ0 at every point of M . Such a structure Φ determines ametric gΦ and orientation ∗Φ. Moreover, Φ is gΦ-parallel iff dΦ = 0.

Define a 4-form Φ on F(M)/ Spin(7) by the rule: For u : TxM → R8

and [u] = u · Spin(7)Φ[u] = π∗(u∗Φ0

)where π : F(M) → M is the basepoint projection. Let I be the idealon F(M)/ Spin(7) generated by dΦ.

Theorem: (B—) I is involutive. Modulo diffeomorphisms, the generalI-integral Φ depends on 12 functions of 7 variables and the holonomy ofthe corresponding metric gΦ is equal to Spin(7).

Hypersurfaces. Spin(7) acts transitively on S7 and the stabilizer of apoint is G2.

An oriented hypersurface N7 ⊂ M8 inherits a G2-structure defined bythe rule

σ = n Φ and satisfies ∗σσ = N∗Φwhere n is the oriented normal vector field along N . One easily checks that

(σ ∧ dσ

)= 28H

Theorem: If σ ∈ Ω3+(N7) is real-analytic and satisfies d(∗σσ) = 0, then σ

is induced by an essentially unique Spin(7)-immersion.

Proof: Construct the obvious ideal I on R × F(N)/G2. It is involu-tive with characters (s1, . . . , s8) = (0, 0, 0, 1, 4, 10, 20, 0). The co-closed G2-structure σ defines a regular I-integral in the locus t = 0 which, by theCartan-Kaher Theorem, lies in an essentially unique I-integral M8. QED

(σ ∧ dσ

)= 28H

(σ ∧ dσ

)= 28H

(σ ∧ dσ

)= 28H

(σ ∧ dσ

)= 28H

(σ ∧ dσ

)= 28H

Theorem: There exist non-real-analytic G2-structures σ ∈ Ω3+(N7) that

satisfyd(∗σσ) = 0

but that are not induced from a Spin(7)-immersion. In fact, if such a non-analytic G2-structure satisfies

(σ ∧dσ

where C is a constant, then it cannot be Spin(7)-immersed. (Such do exist.)

Proof: Same idea as for SU(2). The system of first-order equations

d(∗σσ) = 0, ∗σ

(σ ∧ dσ

)= C, d(∗σdx) = 0

for σ ∈ Ω3+(R7) are only 21 + 1 + 7 = 29 equations for 35 unknowns.

This underdetermined system is not elliptic, but its symbol mappinghas constant rank and it can be embedded into the appropriate sequenceto show that it has non-real-analytic solutions. QED

(σ ∧dσ

d(∗σσ) = 0, ∗σ

(σ ∧ dσ

)= C, d(∗σdx) = 0

(σ ∧dσ

d(∗σσ) = 0, ∗σ

(σ ∧ dσ

)= C, d(∗σdx) = 0

(σ ∧dσ

d(∗σσ) = 0, ∗σ

(σ ∧ dσ

)= C, d(∗σdx) = 0

(σ ∧dσ

d(∗σσ) = 0, ∗σ

(σ ∧ dσ

)= C, d(∗σdx) = 0

The evolution equation. On N7 × R the condition

d (dt ∧σ + ∗σσ) = 0

for a t-parametrized family σ ∈ Ω3+(N7) is equivalent to the condition

that d(∗σσ

)= 0 for each fixed t and the evolution equation

(∗σσ) = dσ.

In terms of σ directly, this evolution equation isddt

(σ) = 14 ∗σ(σ ∧ dσ)σ − ∗σ(dσ).

The above analysis shows that this evolution equation has a (unique)solution for real-analytic initial conditions satisfying d(∗σσ) = 0, but maynot have a solution when the initial condition is not real-analytic.

d (dt ∧σ + ∗σσ) = 0

that d(∗σσ

(∗σσ) = dσ.

(σ) = 14 ∗σ(σ ∧ dσ)σ − ∗σ(dσ).

d (dt ∧σ + ∗σσ) = 0

that d(∗σσ

(∗σσ) = dσ.

(σ) = 14 ∗σ(σ ∧ dσ)σ − ∗σ(dσ).

· In 1925, E. Cartan made the following assertions:´ (1) H x is a Lie subgroup of O(T xM). (2)...

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Transcript of · In 1925, E. Cartan made the following assertions:´ (1) H x is a Lie subgroup of O(T xM). (2)...