NONEMBEDDING AND NONEXTENSION RESULTS IN SPECIAL HOLONOMY ROBERT L. BRYANT DUKE UNIVERSITY

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Riemannian Holonomy. To a Riemannian manifold (M n , g) associate its Levi-Civita connection ∇, which defines, for a piecewise C 1 γ : [0, 1] → M , a parallel transport Pγ∇ : Tγ(0) M → Tγ(1) M, which is a linear isometry between the two tangent spaces. In 1918, J. Schouten considered the set
 Hx = Pγ∇ γ(0) = γ(1) = x ⊆ O(Tx M ) and called its dimension the number of degrees of freedom of g.  −1 Pγ¯∇ = Pγ∇ and Pγ∇2 ∗γ1 = Pγ∇2 ◦ Pγ∇1 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(Tx M ) is a subgroup and  −1 Hγ(1) = Pγ∇ Hγ(0) Pγ∇ .

Riemannian Holonomy. To a Riemannian manifold (M n , g) associate its Levi-Civita connection ∇, which defines, for a piecewise C 1 γ : [0, 1] → M , a parallel transport Pγ∇ : Tγ(0) M → Tγ(1) M, which is a linear isometry between the two tangent spaces. In 1918, J. Schouten considered the set
 Hx = Pγ∇ γ(0) = γ(1) = x ⊆ O(Tx M ) and called its dimension the number of degrees of freedom of g.  −1 Pγ¯∇ = Pγ∇ and Pγ∇2 ∗γ1 = Pγ∇2 ◦ Pγ∇1 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(Tx M ) is a subgroup and  −1 Hγ(1) = Pγ∇ Hγ(0) Pγ∇ .

Riemannian Holonomy. To a Riemannian manifold (M n , g) associate its Levi-Civita connection ∇, which defines, for a piecewise C 1 γ : [0, 1] → M , a parallel transport Pγ∇ : Tγ(0) M → Tγ(1) M, which is a linear isometry between the two tangent spaces. In 1918, J. Schouten considered the set
 Hx = Pγ∇ γ(0) = γ(1) = x ⊆ O(Tx M ) and called its dimension the number of degrees of freedom of g.  −1 Pγ¯∇ = Pγ∇ and Pγ∇2 ∗γ1 = Pγ∇2 ◦ Pγ∇1 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(Tx M ) is a subgroup and  −1 Hγ(1) = Pγ∇ Hγ(0) Pγ∇ .

Riemannian Holonomy. To a Riemannian manifold (M n , g) associate its Levi-Civita connection ∇, which defines, for a piecewise C 1 γ : [0, 1] → M , a parallel transport Pγ∇ : Tγ(0) M → Tγ(1) M, which is a linear isometry between the two tangent spaces. In 1918, J. Schouten considered the set
 Hx = Pγ∇ γ(0) = γ(1) = x ⊆ O(Tx M ) and called its dimension the number of degrees of freedom of g.  −1 Pγ¯∇ = Pγ∇ and Pγ∇2 ∗γ1 = Pγ∇2 ◦ Pγ∇1 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(Tx M ) is a subgroup and  −1 Hγ(1) = Pγ∇ Hγ(0) Pγ∇ .

Riemannian Holonomy. To a Riemannian manifold (M n , g) associate its Levi-Civita connection ∇, which defines, for a piecewise C 1 γ : [0, 1] → M , a parallel transport Pγ∇ : Tγ(0) M → Tγ(1) M, which is a linear isometry between the two tangent spaces. In 1918, J. Schouten considered the set
 Hx = Pγ∇ γ(0) = γ(1) = x ⊆ O(Tx M ) and called its dimension the number of degrees of freedom of g.  −1 Pγ¯∇ = Pγ∇ and Pγ∇2 ∗γ1 = Pγ∇2 ◦ Pγ∇1 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(Tx M ) is a subgroup and  −1 Hγ(1) = Pγ∇ Hγ(0) Pγ∇ .

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

´ Cartan made the following assertions: In 1925, E. (1) Hx is a Lie subgroup of O(Tx M ). (2) If Hx acts reducibly on Tx M , 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 ) takes values in V5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom(R4 , W4 ), with characters (5, 5, 2, 0). QED

Modern Argument: If (M 4 , g) has Hx  SU(2), then there exist three g-parallel 2-forms on M , say Υ1 , Υ2 , and Υ3 , such that Υi ∧ Υj = 2δij dVg .

There exist loc. coord. z = (z 1 , z 2 ) : U → C2 and φ : z(U ) → R so that ¯ Υ2 + i Υ3 = dz 1 ∧ dz 2 and Υ1 = 1 i ∂ ∂φ, 2

where φ satisfies the elliptic Monge-Amp`ere equation  2   2  ∂ φ ∂ φ >0 and det = 1. ∂z i ∂ z¯j ∂z i ∂ z¯j

Conversely, such Υi uniquely determine (M 4 , g) with holonomy SU(2).

Modern Argument: If (M 4 , g) has Hx  SU(2), then there exist three g-parallel 2-forms on M , say Υ1 , Υ2 , and Υ3 , such that Υi ∧ Υj = 2δij dVg .

There exist loc. coord. z = (z 1 , z 2 ) : U → C2 and φ : z(U ) → R so that ¯ Υ2 + i Υ3 = dz 1 ∧ dz 2 and Υ1 = 1 i ∂ ∂φ, 2

where φ satisfies the elliptic Monge-Amp`ere equation  2   2  ∂ φ ∂ φ >0 and det = 1. ∂z i ∂ z¯j ∂z i ∂ z¯j

Conversely, such Υi uniquely determine (M 4 , g) with holonomy SU(2).

Modern Argument: If (M 4 , g) has Hx  SU(2), then there exist three g-parallel 2-forms on M , say Υ1 , Υ2 , and Υ3 , such that Υi ∧ Υj = 2δij dVg .

There exist loc. coord. z = (z 1 , z 2 ) : U → C2 and φ : z(U ) → R so that ¯ Υ2 + i Υ3 = dz 1 ∧ dz 2 and Υ1 = 1 i ∂ ∂φ, 2

where φ satisfies the elliptic Monge-Amp`ere equation  2   2  ∂ φ ∂ φ >0 and det = 1. ∂z i ∂ z¯j ∂z i ∂ z¯j

Conversely, such Υi uniquely determine (M 4 , g) with holonomy SU(2).

Modern Argument: If (M 4 , g) has Hx  SU(2), then there exist three g-parallel 2-forms on M , say Υ1 , Υ2 , and Υ3 , such that Υi ∧ Υj = 2δij dVg .

There exist loc. coord. z = (z 1 , z 2 ) : U → C2 and φ : z(U ) → R so that ¯ Υ2 + i Υ3 = dz 1 ∧ dz 2 and Υ1 = 1 i ∂ ∂φ, 2

where φ satisfies the elliptic Monge-Amp`ere equation  2   2  ∂ φ ∂ φ >0 and det = 1. ∂z i ∂ z¯j ∂z i ∂ z¯j

Conversely, such Υi uniquely determine (M 4 , g) with holonomy SU(2).

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

Exterior Differential Systems Argument: Let M 4 be an analytic manifold and let Υ be the tautological 2-form on Λ2 (T ∗ M ). Let

 3 X 17 ⊂ Λ2 (T ∗ M ) be the submanifold consisting of triples (β1 , β2 , β3 ) ∈ Λ2 (Tx∗ M ) such that β1 2 = β2 2 = β3 2 = 0,

and

β1 ∧ β2 = β3 ∧ β1 = β2 ∧ β3 = 0.

The pullbacks Υi = πi∗ (Υ) define an exterior differential system on X I = {dΥ1 , dΥ2 , dΥ3 }. An integral manifold Y ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υi on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . 4

Simple calculation shows that I is involutive.

A sharper result:Suppose that (M 4 , g) has holonomy SU(2) and let Υi be three g-parallel 2-forms on M satisfying Υi ∧ Υj = 2δij dVg . If N 3 ⊂ M is an oriented hypersurface, with oriented normal n, then there is a coframing η of N defined by ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ n Υ1 Υ1 η2 ∧η3 η1 η = ⎝η2 ⎠ = ⎝n Υ2 ⎠ and it satisfies N ∗ ⎝Υ2 ⎠ = ⎝η3 ∧η1 ⎠ = ∗η η η3 n Υ3 Υ3 η1 ∧η2 In particular,

d(∗η η) = N ∗ dΥ = 0.

Theorem: If η is a real-analytic coframing of N such that d(∗η η) = 0 then there is an essentially unique embedding of N into a SU(2)-holonomy manifold (M 4 , g) that induces the given coframing η in the above manner.

A sharper result: Suppose that (M 4 , g) has holonomy SU(2) and let Υi be three g-parallel 2-forms on M satisfying Υi ∧ Υj = 2δij dVg . If N 3 ⊂ M is an oriented hypersurface, with oriented normal n, then there is a coframing η of N defined by ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ n Υ1 Υ1 η2 ∧η3 η1 η = ⎝η2 ⎠ = ⎝n Υ2 ⎠ and it satisfies N ∗ ⎝Υ2 ⎠ = ⎝η3 ∧η1 ⎠ = ∗η η η3 n Υ3 Υ3 η1 ∧η2 In particular,

d(∗η η) = N ∗ dΥ = 0.

Theorem: If η is a real-analytic coframing of N such that d(∗η η) = 0 then there is an essentially unique embedding of N into a SU(2)-holonomy manifold (M 4 , g) that induces the given coframing η in the above manner.

A sharper result: Suppose that (M 4 , g) has holonomy SU(2) and let Υi be three g-parallel 2-forms on M satisfying Υi ∧ Υj = 2δij dVg . If N 3 ⊂ M is an oriented hypersurface, with oriented normal n, then there is a coframing η of N defined by ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ n Υ1 Υ1 η2 ∧η3 η1 η = ⎝η2 ⎠ = ⎝n Υ2 ⎠ and it satisfies N ∗ ⎝Υ2 ⎠ = ⎝η3 ∧η1 ⎠ = ∗η η η3 n Υ3 Υ3 η1 ∧η2 In particular,

d(∗η η) = N ∗ dΥ = 0.

Theorem: If η is a real-analytic coframing of N such that d(∗η η) = 0 then there is an essentially unique embedding of N into a SU(2)-holonomy manifold (M 4 , g) that induces the given coframing η in the above manner.

A sharper result: Suppose that (M 4 , g) has holonomy SU(2) and let Υi be three g-parallel 2-forms on M satisfying Υi ∧ Υj = 2δij dVg . If N 3 ⊂ M is an oriented hypersurface, with oriented normal n, then there is a coframing η of N defined by ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ n Υ1 Υ1 η2 ∧η3 η1 η = ⎝η2 ⎠ = ⎝n Υ2 ⎠ and it satisfies N ∗ ⎝Υ2 ⎠ = ⎝η3 ∧η1 ⎠ = ∗η η η3 n Υ3 Υ3 η1 ∧η2 In particular,

d(∗η η) = N ∗ dΥ = 0.

Theorem: If η is a real-analytic coframing of N such that d(∗η η) = 0 then there is an essentially unique embedding of N into a SU(2)-holonomy manifold (M 4 , g) that induces the given coframing η in the above manner.

Proof: Write dη = −θ∧η where θ = −t θ.On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω. On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −t θ. On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω.On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −t θ. On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω. On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −t θ. On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω. On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −t θ. On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω. On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

Proof: Write dη = −θ∧η where θ = −t θ. On N × GL(3, R) define ω = g−1 η

and

γ = g−1 dg + g−1 θg,

so that dω = −γ ∧ω. On X = N × GL(3, R) × R define the three 2-forms ⎞ ⎛ ⎞ ⎛ dt∧ω1 + ω2 ∧ω3 Υ1 ⎝Υ2 ⎠ = ⎝dt∧ω2 + ω3 ∧ω1 ⎠ = dt ∧ ω + ∗ω ω. Υ3 dt∧ω3 + ω1 ∧ω2 Let I be the ideal on X generated by {dΥ1 , dΥ2 , dΥ3 }. One calculates   dΥ = t γ − (tr γ)I3 ∧ ∗ω ω + γ ∧ ω ∧ dt. Consequently, I is involutive, with characters (s1 , s2 , s3 , s4 ) = (0, 3, 6, 0). Since d(∗η η) = 0, the locus L = N × {I3 } × {0} ⊂ X is a regular, realanalytic integral manifold of the real-analytic ideal I. Note that L is just a copy of N . By the Cartan-K¨ahler Theorem, L lies in a unique 4-dimensional integral manifold M ⊂ X. The Υi pull back to M to be closed and to define the desired SU(2)-structure forms Υi on M inducing η on N . QED

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution. In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution. (Such non-real-analytic coframings do exist.)

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution. In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution. (Such non-real-analytic coframings do exist.)

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution. In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution. (Such non-real-analytic coframings do exist.)

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution.In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution. (Such non-real-analytic coframings do exist.)

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution. In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution.(Such non-real-analytic coframings do exist.)

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

When η is real-analytic and satisfies d(∗η η) = 0, this flow with initial condition has a unique solution, which continues to be co-closed. Theorem: There exist η on N 3 with d(∗η η) = 0 for which the SU(2)-flow with initial condition η has no solution. In fact, if η is not real-analytic and d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

for some constant C, then the SU(2)-flow with initial condition η has no solution. (Such non-real-analytic coframings do exist.)

Proof: Suppose that Υi (1 ≤ i ≤ 3) are the parallel 2-forms on an (M 4 , g) with holonomy SU(2) and let N 3 ⊂ M be an oriented hypersurface. Calculation yields that the induced co-closed coframing η satisfies ∗η (t η ∧ dη) = 2H where H is the mean curvature of N in M . Now, (M, g) is real-analytic. If H is constant, then elliptic regularity implies that N must be a real-analytic hypersurface in M and hence η must also be real-analytic. Thus, if η is a non-real-analytic coframing on N 3 that satisfies d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

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

Proof: Suppose that Υi (1 ≤ i ≤ 3) are the parallel 2-forms on an (M 4 , g) with holonomy SU(2) and let N 3 ⊂ M be an oriented hypersurface. Calculation yields that the induced co-closed coframing η satisfies ∗η (t η ∧ dη) = 2H where H is the mean curvature of N in M . Now, (M, g) is real-analytic. If H is constant, then elliptic regularity implies that N must be a real-analytic hypersurface in M and hence η must also be real-analytic. Thus, if η is a non-real-analytic coframing on N 3 that satisfies d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

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

Proof: Suppose that Υi (1 ≤ i ≤ 3) are the parallel 2-forms on an (M 4 , g) with holonomy SU(2) and let N 3 ⊂ M be an oriented hypersurface. Calculation yields that the induced co-closed coframing η satisfies ∗η (t η ∧ dη) = 2H where H is the mean curvature of N in M . Now, (M, g) is real-analytic. If H is constant, then elliptic regularity implies that N must be a real-analytic hypersurface in M and hence η must also be real-analytic. Thus, if η is a non-real-analytic coframing on N 3 that satisfies d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

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

Proof: Suppose that Υi (1 ≤ i ≤ 3) are the parallel 2-forms on an (M 4 , g) with holonomy SU(2) and let N 3 ⊂ M be an oriented hypersurface. Calculation yields that the induced co-closed coframing η satisfies ∗η (t η ∧ dη) = 2H where H is the mean curvature of N in M . Now, (M, g) is real-analytic. If H is constant, then elliptic regularity implies that N must be a real-analytic hypersurface in M and hence η must also be real-analytic. Thus, if η is a non-real-analytic coframing on N 3 that satisfies d(∗η η) = 0

and

∗η (t η ∧ dη) = 2C

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

To finish the proof, note that, if a coframing η on N 3 is real-analytic in any coordinate system at all, it will be real-analytic in harmonic coordinates, i.e., local coordinates x : U → R3 satisfying   d ∗η dx = 0.

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-linear system   d(∗η η) = 0, ∗η (t η ∧ dη) = 2C, d ∗η dx = 0.

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

To finish the proof, note that, if a coframing η on N 3 is real-analytic in any coordinate system at all, it will be real-analytic in harmonic coordinates, i.e., local coordinates x : U → R3 satisfying   d ∗η dx = 0.

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-linear system   d(∗η η) = 0, ∗η (t η ∧ dη) = 2C, d ∗η dx = 0.

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

To finish the proof, note that, if a coframing η on N 3 is real-analytic in any coordinate system at all, it will be real-analytic in harmonic coordinates, i.e., local coordinates x : U → R3 satisfying   d ∗η dx = 0.

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-linear system   d(∗η η) = 0, ∗η (t η ∧ dη) = 2C, d ∗η dx = 0.

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

To finish the proof, note that, if a coframing η on N 3 is real-analytic in any coordinate system at all, it will be real-analytic in harmonic coordinates, i.e., local coordinates x : U → R3 satisfying   d ∗η dx = 0.

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-linear system   d(∗η η) = 0, ∗η (t η ∧ dη) = 2C, d ∗η dx = 0.

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

The G2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds (M 7 , 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-harmonic coordinate charts. Conversely, there is a open set Ω3+ (M 7 ) of definite 3-forms, i.e., σ ∈ if and only if, for all x ∈ M , the stabilizer of σx in GL(Tx M ) is isomorphic to G2 ⊂ SO(7). These forms are the sections of an open subbundle 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 Ω3+ (M 7 )

dσ = 0

and

d(∗σ σ) = 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.

The G2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds (M 7 , 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-harmonic coordinate charts. Conversely, there is a open set Ω3+ (M 7 ) of definite 3-forms, i.e., σ ∈ if and only if, for all x ∈ M , the stabilizer of σx in GL(Tx M ) is isomorphic to G2 ⊂ SO(7). These forms are the sections of an open subbundle 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 Ω3+ (M 7 )

dσ = 0

and

d(∗σ σ) = 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.

The G2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds (M 7 , 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-harmonic coordinate charts. Conversely, there is a open set Ω3+ (M 7 ) of definite 3-forms, i.e., σ ∈ if and only if, for all x ∈ M , the stabilizer of σx in GL(Tx M ) is isomorphic to G2 ⊂ SO(7). These forms are the sections of an open subbundle 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 Ω3+ (M 7 )

dσ = 0

and

d(∗σ σ) = 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.

The G2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds (M 7 , 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-harmonic coordinate charts. Conversely, there is a open set Ω3+ (M 7 ) of definite 3-forms, i.e., σ ∈ if and only if, for all x ∈ M , the stabilizer of σx in GL(Tx M ) is isomorphic to G2 ⊂ SO(7). These forms are the sections of an open subbundle 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 Ω3+ (M 7 )

dσ = 0

and

d(∗σ σ) = 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.

The G2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds (M 7 , 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-harmonic coordinate charts. Conversely, there is a open set Ω3+ (M 7 ) of definite 3-forms, i.e., σ ∈ if and only if, for all x ∈ M , the stabilizer of σx in GL(Tx M ) is isomorphic to G2 ⊂ SO(7). These forms are the sections of an open subbundle 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 Ω3+ (M 7 )

dσ = 0

and

d(∗σ σ) = 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.

Hypersurfaces.G2 acts transitively on S 6 ⊂ R7 , with stabilizer SU(3). Hence, an oriented N 6 ⊂ M inherits a canonical SU(3)-structure, which is determined 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 ∗σ σ).

In fact, if one defines f : R × N → M by   f(t, p) = expp t n(p) , then f ∗ σ = dt ∧ ω + Re(Ω)

and

f ∗ (∗σ σ) =

1 2

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(ft∗0 σ) = 0 and d( 12 ω2 ) = d ft∗0 (∗σ σ) = 0, so these are necessary conditions on the SU(3)-structure on N that it be induced by immersion into a G2 -holonomy manifold M .

Hypersurfaces. G2 acts transitively on S 6 ⊂ R7 , with stabilizer SU(3). Hence, an oriented N 6 ⊂ M inherits a canonical SU(3)-structure, which is determined 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 ∗σ σ).

In fact, if one defines f : R × N → M by   f(t, p) = expp t n(p) , then f ∗ σ = dt ∧ ω + Re(Ω)

and

f ∗ (∗σ σ) =

1 2

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(ft∗0 σ) = 0 and d( 12 ω2 ) = d ft∗0 (∗σ σ) = 0, so these are necessary conditions on the SU(3)-structure on N that it be induced by immersion into a G2 -holonomy manifold M .

Hypersurfaces. G2 acts transitively on S 6 ⊂ R7 , with stabilizer SU(3). Hence, an oriented N 6 ⊂ M inherits a canonical SU(3)-structure, which is determined 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 ∗σ σ).

In fact, if one defines f : R × N → M by   f(t, p) = expp t n(p) , then f ∗ σ = dt ∧ ω + Re(Ω)

and

f ∗ (∗σ σ) =

1 2

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(ft∗0 σ) = 0 and d( 12 ω2 ) = d ft∗0 (∗σ σ) = 0, so these are necessary conditions on the SU(3)-structure on N that it be induced by immersion into a G2 -holonomy manifold M .

Hypersurfaces. G2 acts transitively on S 6 ⊂ R7 , with stabilizer SU(3). Hence, an oriented N 6 ⊂ M inherits a canonical SU(3)-structure, which is determined 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 ∗σ σ).

In fact, if one defines f : R × N → M by   f(t, p) = expp t n(p) , then f ∗ σ = dt ∧ ω + Re(Ω)

and

f ∗ (∗σ σ) =

1 2

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(ft∗0 σ) = 0 and d( 12 ω2 ) = d ft∗0 (∗σ σ) = 0, so these are necessary conditions on the SU(3)-structure on N that it be induced by immersion into a G2 -holonomy manifold M .

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into 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) as follows: For a coframe u : Tx N → C3 , define these forms at [u] = u·SU(3) ∈ F (N )/ SU(3) by     ω[u] = π ∗ u∗ ( 2i (t dz ∧ d¯ z )) and Ω[u] = π ∗ u∗ (dz 1 ∧ dz 2 ∧ dz 3 ) 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=

1 2

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( 12 ω2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the Cartan-K¨ ahler Theorem, L lies in a unique I-integral M 7 ⊂ X. QED

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed.(Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy   d Re(Ω) = d( 12 ω2 ) = 0 but that are not induced from an immersion into a G2 -holonomy Riemannian manifold (M, g). In fact, if such a non-analytic SU(3)-structure satisfies    ∗ ω ∧ d Im(Ω) = C where C is a constant, then it cannot be G2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms (ω, Ω) is induced via 6 7 a G2 -immersion  N → M  , the mean curvature H of N in M is given by −12H = ∗ ω∧d Im(Ω) . Thus, when this latter function is constant it follows, by elliptic regularity, that N 6 is a real-analytic submanifold of the real-analytic (M 7 , g). If the SU(3)-structure is not real-analytic, this is a contradiction. It remains to construct such an example.

  Why it’s somewhat delicate:Since dim GL(6, R)/ SU(3) = 28, a choice of an SU(3)-structure (ω, Ω) on N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

Now assume that φ is also closed. Then dΩφ is purely imaginary and yet 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 N 6 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( 12 ω2 ) = 0,

   ∗ ω ∧ d Im(Ω) = C

constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. 3 6 Fix an orientation of N 6 . Say that a 3-form  1 φ ∈2 Ω (N  ) is elliptic if, at 3 each point, it is linearly equivalent to Re dz ∧dz ∧dz . Such a φ defines a unique, orientation-preserving almost-complex structure Jφ on N 6 such that Ωφ = φ + i Jφ∗(φ) is of Jφ -type (3, 0).

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

So far: φ ∈ Ω3e (N 6 ) defines Jφ and Ωφ = φ + i Jφ∗(φ) ∈ Ω3,0 (N, Jφ ). dφ = 0

then yields

dΩφ ∈ Ω2,2 (N, Jφ ).

So far: φ ∈ Ω3e (N 6 ) defines Jφ and Ωφ = φ + i Jφ∗(φ) ∈ Ω3,0 (N, Jφ ). dφ = 0

then yields

dΩφ ∈ Ω2,2 (N, Jφ ).

Fix a constant C = 0. It is a C 1 -open condition on φ that dΩφ = 6i C (ωφ )2

for some

ωφ = ωφ ∈ Ω1,1 + (N, Jφ ).

So far: φ ∈ Ω3e (N 6 ) defines Jφ and Ωφ = φ + i Jφ∗(φ) ∈ Ω3,0 (N, Jφ ). dφ = 0

then yields

dΩφ ∈ Ω2,2 (N, Jφ ).

Fix a constant C = 0. It is a C 1 -open condition on φ that dΩφ = 6i C (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 N if and only if 1 3 1 6 (ωφ ) − 8 i Ωφ ∧ Ωφ = 0. This is a single, first-order scalar equation on the closed 3-form φ. It is easy to see that there are non-analytic solutions. Assuming this condition is satisfied: d(Re Ωφ ) = dφ = 0, and     d 12 (ωφ )2 = d −3i C1 dΩφ = 0, and, finally     ∗φ ωφ ∧ d(Im Ωφ ) = ∗φ ωφ ∧ 16 C (ωφ )2 = C.

So far: φ ∈ Ω3e (N 6 ) defines Jφ and Ωφ = φ + i Jφ∗(φ) ∈ Ω3,0 (N, Jφ ). dφ = 0

then yields

dΩφ ∈ Ω2,2 (N, Jφ ).

Fix a constant C = 0. It is a C 1 -open condition on φ that dΩφ = 6i C (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 N if and only if 1 3 1 6 (ωφ ) − 8 i Ωφ ∧ Ωφ = 0. This is a single, first-order scalar equation on the closed 3-form φ. It is easy to see that there are non-analytic solutions.Assuming this condition is satisfied: d(Re Ωφ ) = dφ = 0, and     d 12 (ωφ )2 = d −3i C1 dΩφ = 0, and, finally     ∗φ ωφ ∧ d(Im Ωφ ) = ∗φ ωφ ∧ 16 C (ωφ )2 = C.

So far: φ ∈ Ω3e (N 6 ) defines Jφ and Ωφ = φ + i Jφ∗(φ) ∈ Ω3,0 (N, Jφ ). dφ = 0

then yields

dΩφ ∈ Ω2,2 (N, Jφ ).

Fix a constant C = 0. It is a C 1 -open condition on φ that dΩφ = 6i C (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 N if and only if 1 3 1 6 (ωφ ) − 8 i Ωφ ∧ Ωφ = 0. This is a single, first-order scalar equation on the closed 3-form φ. It is easy to see that there are non-analytic solutions. Assuming this condition is satisfied: d(Re Ωφ ) = dφ = 0, and     d 12 (ωφ )2 = d −3i C1 dΩφ = 0, and, finally     ∗φ ωφ ∧ d(Im Ωφ ) = ∗φ ωφ ∧ 16 C (ωφ )2 = C.

Interpretation: On N 6 ×R, with (ω, Ω) defining an SU(3)-structure on N 6 depending on t ∈ R, consider the equations     d dt ∧ ω + Re(Ω) = 0 and d 12 ω2 − dt ∧ Im(Ω) = 0. Think of Ω as φ + iJφ∗ (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 then    d d (φ) = dω and (ω) = −Lω −1 d Jφ∗ (φ) , dt dt 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 initial SU(3)-structures satisfying the closure conditions, but may not exist for non-analytic initial SU(3)-structures satisfying the closure conditions.

Interpretation: On N 6 ×R, with (ω, Ω) defining an SU(3)-structure on N 6 depending on t ∈ R, consider the equations     d dt ∧ ω + Re(Ω) = 0 and d 12 ω2 − dt ∧ Im(Ω) = 0. Think of Ω as φ + iJφ∗ (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 then    d d (φ) = dω and (ω) = −Lω −1 d Jφ∗ (φ) , dt dt 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 initial SU(3)-structures satisfying the closure conditions, but may not exist for non-analytic initial SU(3)-structures satisfying the closure conditions.

Interpretation: On N 6 ×R, with (ω, Ω) defining an SU(3)-structure on N 6 depending on t ∈ R, consider the equations     d dt ∧ ω + Re(Ω) = 0 and d 12 ω2 − dt ∧ Im(Ω) = 0. Think of Ω as φ + iJφ∗ (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 then    d d (φ) = dω and (ω) = −Lω −1 d Jφ∗ (φ) , dt dt 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 initial SU(3)-structures satisfying the closure conditions, but may not exist for non-analytic initial SU(3)-structures satisfying the closure conditions.

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizer of a 4-form Φ0 ∈ Λ4 (R8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω4 (M ) that is linearly equivalent to Φ0 at every point of M . Such a structure Φ determines a metric 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 : Tx M → R8 and [u] = u · Spin(7)   Φ[u] = π ∗ u∗Φ0 where π : F (M ) → M is the basepoint projection. Let I be the ideal on F (M )/ Spin(7) generated by dΦ. Theorem: (B—) I is involutive. Modulo diffeomorphisms, the general I-integral Φ depends on 12 functions of 7 variables and the holonomy of the corresponding metric gΦ is equal to Spin(7).

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizer of a 4-form Φ0 ∈ Λ4 (R8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω4 (M ) that is linearly equivalent to Φ0 at every point of M . Such a structure Φ determines a metric 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 : Tx M → R8 and [u] = u · Spin(7)   Φ[u] = π ∗ u∗Φ0 where π : F (M ) → M is the basepoint projection. Let I be the ideal on F (M )/ Spin(7) generated by dΦ. Theorem: (B—) I is involutive. Modulo diffeomorphisms, the general I-integral Φ depends on 12 functions of 7 variables and the holonomy of the corresponding metric gΦ is equal to Spin(7).

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizer of a 4-form Φ0 ∈ Λ4 (R8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω4 (M ) that is linearly equivalent to Φ0 at every point of M . Such a structure Φ determines a metric 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 : Tx M → R8 and [u] = u · Spin(7)   Φ[u] = π ∗ u∗Φ0 where π : F (M ) → M is the basepoint projection. Let I be the ideal on F (M )/ Spin(7) generated by dΦ. Theorem: (B—) I is involutive. Modulo diffeomorphisms, the general I-integral Φ depends on 12 functions of 7 variables and the holonomy of the corresponding metric gΦ is equal to Spin(7).

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizer of a 4-form Φ0 ∈ Λ4 (R8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω4 (M ) that is linearly equivalent to Φ0 at every point of M . Such a structure Φ determines a metric 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 : Tx M → R8 and [u] = u · Spin(7)   Φ[u] = π ∗ u∗Φ0 where π : F (M ) → M is the basepoint projection. Let I be the ideal on F (M )/ Spin(7) generated by dΦ. Theorem: (B—) I is involutive. Modulo diffeomorphisms, the general I-integral Φ depends on 12 functions of 7 variables and the holonomy of the corresponding metric gΦ is equal to Spin(7).

The Spin(7) case. The group Spin(7) ⊂ SO(8) is the GL(8, R)-stabilizer of a 4-form Φ0 ∈ Λ4 (R8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω4 (M ) that is linearly equivalent to Φ0 at every point of M . Such a structure Φ determines a metric 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 : Tx M → R8 and [u] = u · Spin(7)   Φ[u] = π ∗ u∗Φ0 where π : F (M ) → M is the basepoint projection. Let I be the ideal on F (M )/ Spin(7) generated by dΦ. Theorem: (B—) I is involutive. Modulo diffeomorphisms, the general I-integral Φ depends on 12 functions of 7 variables and the holonomy of the corresponding metric gΦ is equal to Spin(7).

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G2 -structure defined by the rule σ=n Φ and satisfies ∗σ σ = N ∗ Φ where n is the oriented normal vector field along N . One easily checks that   ∗σ σ ∧ dσ = 28H where H is the mean curvature of N in M . Theorem: If σ ∈ Ω3+ (N 7 ) is real-analytic and satisfies d(∗σ σ) = 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 involutive 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 the Cartan-K¨ aher Theorem, lies in an essentially unique I-integral M 8 . QED

Theorem: There exist non-real-analytic G2 -structures σ ∈ Ω3+ (N 7 ) that satisfy d(∗σ σ) = 0 but that are not induced from a Spin(7)-immersion. In fact, if such a nonanalytic G2 -structure satisfies   ∗σ σ ∧ dσ = C 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(∗σ σ) = 0, ∗σ σ ∧ dσ = C, d(∗σ dx) = 0 for σ ∈ Ω3+ (R7 ) are only 21 + 1 + 7 = 29 equations for 35 unknowns. This underdetermined system is not elliptic, but its symbol mapping has constant rank and it can be embedded into the appropriate sequence to show that it has non-real-analytic solutions. QED

Theorem: There exist non-real-analytic G2 -structures σ ∈ Ω3+ (N 7 ) that satisfy d(∗σ σ) = 0 but that are not induced from a Spin(7)-immersion. In fact, if such a nonanalytic G2 -structure satisfies   ∗σ σ ∧ dσ = C 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(∗σ σ) = 0, ∗σ σ ∧ dσ = C, d(∗σ dx) = 0 for σ ∈ Ω3+ (R7 ) are only 21 + 1 + 7 = 29 equations for 35 unknowns. This underdetermined system is not elliptic, but its symbol mapping has constant rank and it can be embedded into the appropriate sequence to show that it has non-real-analytic solutions. QED

Theorem: There exist non-real-analytic G2 -structures σ ∈ Ω3+ (N 7 ) that satisfy d(∗σ σ) = 0 but that are not induced from a Spin(7)-immersion. In fact, if such a nonanalytic G2 -structure satisfies   ∗σ σ ∧ dσ = C 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(∗σ σ) = 0, ∗σ σ ∧ dσ = C, d(∗σ dx) = 0 for σ ∈ Ω3+ (R7 ) are only 21 + 1 + 7 = 29 equations for 35 unknowns. This underdetermined system is not elliptic, but its symbol mapping has constant rank and it can be embedded into the appropriate sequence to show that it has non-real-analytic solutions. QED

Theorem: There exist non-real-analytic G2 -structures σ ∈ Ω3+ (N 7 ) that satisfy d(∗σ σ) = 0 but that are not induced from a Spin(7)-immersion. In fact, if such a nonanalytic G2 -structure satisfies   ∗σ σ ∧ dσ = C 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(∗σ σ) = 0, ∗σ σ ∧ dσ = C, d(∗σ dx) = 0 for σ ∈ Ω3+ (R7 ) are only 21 + 1 + 7 = 29 equations for 35 unknowns. This underdetermined system is not elliptic, but its symbol mapping has constant rank and it can be embedded into the appropriate sequence to show that it has non-real-analytic solutions. QED

Theorem: There exist non-real-analytic G2 -structures σ ∈ Ω3+ (N 7 ) that satisfy d(∗σ σ) = 0 but that are not induced from a Spin(7)-immersion. In fact, if such a nonanalytic G2 -structure satisfies   ∗σ σ ∧ dσ = C 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(∗σ σ) = 0, ∗σ σ ∧ dσ = C, d(∗σ dx) = 0 for σ ∈ Ω3+ (R7 ) are only 21 + 1 + 7 = 29 equations for 35 unknowns. This underdetermined system is not elliptic, but its symbol mapping has constant rank and it can be embedded into the appropriate sequence to show that it has non-real-analytic solutions. QED

The evolution equation. On N 7 × R the condition d (dt ∧ σ + ∗σ σ) = 0 for a t-parametrized family σ ∈ Ω3+ (N 7 ) is equivalent to the condition   that d ∗σ σ = 0 for each fixed t and the evolution equation d (∗σ σ) = dσ. dt In terms of σ directly, this evolution equation is d (σ) = 14 ∗σ (σ ∧ dσ) σ − ∗σ (dσ). dt The above analysis shows that this evolution equation has a (unique) solution for real-analytic initial conditions satisfying d(∗σ σ) = 0, but may not have a solution when the initial condition is not real-analytic.

The evolution equation. On N 7 × R the condition d (dt ∧ σ + ∗σ σ) = 0 for a t-parametrized family σ ∈ Ω3+ (N 7 ) is equivalent to the condition   that d ∗σ σ = 0 for each fixed t and the evolution equation d (∗σ σ) = dσ. dt In terms of σ directly, this evolution equation is d (σ) = 14 ∗σ (σ ∧ dσ) σ − ∗σ (dσ). dt The above analysis shows that this evolution equation has a (unique) solution for real-analytic initial conditions satisfying d(∗σ σ) = 0, but may not have a solution when the initial condition is not real-analytic.

The evolution equation. On N 7 × R the condition d (dt ∧ σ + ∗σ σ) = 0 for a t-parametrized family σ ∈ Ω3+ (N 7 ) is equivalent to the condition   that d ∗σ σ = 0 for each fixed t and the evolution equation d (∗σ σ) = dσ. dt In terms of σ directly, this evolution equation is d (σ) = 14 ∗σ (σ ∧ dσ) σ − ∗σ (dσ). dt The above analysis shows that this evolution equation has a (unique) solution for real-analytic initial conditions satisfying d(∗σ σ) = 0, but may not have a solution when the initial condition is not real-analytic.