D U E o i r ud ig el it i R o e t Riemannian Holonomy. - PowerPoint PPT Presentation

In 1925, ´ E. Cartan made the following assertions: (1) H x is a Lie subgroup of O( T x M ). (2) If H x acts reducibly on T x 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 H x ≃ 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 d R + θ.R = � R ′ , ω � d ω + θ ∧ ω = 0 , d θ + θ ∧ θ = � R, ω ∧ ω � , where ω takes values in R 4 , θ takes values in su (2), R takes values in W 4 , the 5-dimensional real irr. rep. of SU(2), and R ′ ⊂ Hom( R 4 , W 4 ) takes values in V 5 , the 6-dimensional complex irr. rep. of SU(2). The latter is an involutive tableau in Hom( R 4 , W 4 ), with characters (5 , 5 , 2 , 0). QED

Modern Argument: If ( M 4 , g ) has H x ≃ SU(2), then there exist three g -parallel 2-forms on M , say Υ 1 , Υ 2 , and Υ 3 , such that Υ i ∧ Υ j = 2 δ ij d V g . There exist loc. coord. z = ( z 1 , z 2 ) : U → C 2 and φ : z ( U ) → R so that Υ 2 + i Υ 3 = d z 1 ∧ d z 2 Υ 1 = 1 2 i ∂ ¯ and ∂φ, where φ satisfies the elliptic Monge-Amp` ere equation � ∂ 2 φ � ∂ 2 φ � � > 0 and det = 1 . ∂z i ∂ ¯ z j ∂z i ∂ ¯ z j Conversely, such Υ i 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 Λ 2 ( T ∗ M ). Let X 17 ⊂ � 3 Λ 2 ( T ∗ M ) � be the submanifold consisting of triples ( β 1 , β 2 , β 3 ) ∈ Λ 2 ( T ∗ x M ) such that 2 = β 2 2 = β 3 2 � = 0 , β 1 and β 1 ∧ β 2 = β 3 ∧ β 1 = β 2 ∧ β 3 = 0 . The pullbacks Υ i = π ∗ i ( Υ ) define an exterior differential system on X I = { d Υ 1 , d Υ 2 , d Υ 3 } . An integral manifold Y 4 ⊂ X transverse to π : X → M then represents a choice of three closed 2-forms Υ i on an open subset U ⊂ M that satisfy the algebra conditions needed to define an SU(2)-structure on U . Simple calculation shows that I is involutive.

A sharper result: Suppose that ( M 4 , g ) has holonomy SU(2) and let Υ i be three g -parallel 2-forms on M satisfying Υ i ∧ Υ j = 2 δ ij d V g . If N 3 ⊂ M is an oriented hypersurface, with oriented normal n , then there is a coframing η of N defined by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ η 1 n Υ 1 Υ 1 η 2 ∧ η 3 ⎠ = ⎠ = ⎠ = ∗ η η N ∗ η = η 2 n Υ 2 and it satisfies Υ 2 η 3 ∧ η 1 ⎝ ⎝ ⎠ ⎝ ⎝ η 3 n Υ 3 Υ 3 η 1 ∧ η 2 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 d η = − θ ∧ η where θ = − t θ .On N × GL(3 , R ) define ω = g − 1 η γ = g − 1 d g + g − 1 θg, and so that d ω = − γ ∧ ω . On X = N × GL(3 , R ) × R define the three 2-forms ⎛ ⎞ ⎛ ⎞ Υ 1 d t ∧ ω 1 + ω 2 ∧ ω 3 ⎠ = ⎠ = d t ∧ ω + ∗ ω ω . Υ 2 d t ∧ ω 2 + ω 3 ∧ ω 1 ⎝ ⎝ Υ 3 d t ∧ ω 3 + ω 1 ∧ ω 2 Let I be the ideal on X generated by { d Υ 1 , d Υ 2 , d Υ 3 } . One calculates d Υ = � t γ − (tr γ ) I 3 � ∧ ∗ ω ω + γ ∧ ω ∧ d t. Consequently, I is involutive, with characters ( s 1 , s 2 , s 3 , s 4 ) = (0 , 3 , 6 , 0). Since d( ∗ η η ) = 0, the locus L = N × { I 3 } × { 0 } ⊂ X is a regular, real- analytic 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 Υ i pull back to M to be closed and to define the desired SU(2)-structure forms Υ i on M inducing η on N . QED

Proof: Write d η = − θ ∧ η where θ = − t θ . On N × GL(3 , R ) define ω = g − 1 η γ = g − 1 d g + g − 1 θg, and so that d ω = − γ ∧ ω .On X = N × GL(3 , R ) × R define the three 2-forms ⎛ ⎞ ⎛ ⎞ Υ 1 d t ∧ ω 1 + ω 2 ∧ ω 3 ⎠ = ⎠ = d t ∧ ω + ∗ ω ω . Υ 2 d t ∧ ω 2 + ω 3 ∧ ω 1 ⎝ ⎝ Υ 3 d t ∧ ω 3 + ω 1 ∧ ω 2 Let I be the ideal on X generated by { d Υ 1 , d Υ 2 , d Υ 3 } . One calculates d Υ = � t γ − (tr γ ) I 3 � ∧ ∗ ω ω + γ ∧ ω ∧ d t. Consequently, I is involutive, with characters ( s 1 , s 2 , s 3 , s 4 ) = (0 , 3 , 6 , 0). Since d( ∗ η η ) = 0, the locus L = N × { I 3 } × { 0 } ⊂ X is a regular, real- analytic 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 Υ i pull back to M to be closed and to define the desired SU(2)-structure forms Υ i on M inducing η on N . QED

Proof: Write d η = − θ ∧ η where θ = − t θ . On N × GL(3 , R ) define ω = g − 1 η γ = g − 1 d g + g − 1 θg, and so that d ω = − γ ∧ ω . On X = N × GL(3 , R ) × R define the three 2-forms ⎛ ⎞ ⎛ ⎞ Υ 1 d t ∧ ω 1 + ω 2 ∧ ω 3 ⎠ = ⎠ = d t ∧ ω + ∗ ω ω . Υ 2 d t ∧ ω 2 + ω 3 ∧ ω 1 ⎝ ⎝ Υ 3 d t ∧ ω 3 + ω 1 ∧ ω 2 Let I be the ideal on X generated by { d Υ 1 , d Υ 2 , d Υ 3 } . One calculates d Υ = � t γ − (tr γ ) I 3 � ∧ ∗ ω ω + γ ∧ ω ∧ d t. Consequently, I is involutive, with characters ( s 1 , s 2 , s 3 , s 4 ) = (0 , 3 , 6 , 0). Since d( ∗ η η ) = 0, the locus L = N × { I 3 } × { 0 } ⊂ X is a regular, real- analytic 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 Υ i pull back to M to be closed and to define the desired SU(2)-structure forms Υ i on M inducing η on N . QED

Question: Is it necessary to assume that η be real-analytic? The condition d(d t ∧ ω + ∗ ω ) = 0 is an ‘SU(2)-flow’ on coframings of N : d d t ω = ∗ ω (d ω ) − 1 2 ∗ ω ( t ω ∧ d ω ) ω with initial condition ω t =0 = η. 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 ∗ η ( t η ∧ d η ) = 2 C d( ∗ η η ) = 0 and 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(d t ∧ ω + ∗ ω ) = 0 is an ‘SU(2)-flow’ on coframings of N : d d t ω = ∗ ω (d ω ) − 1 2 ∗ ω ( t ω ∧ d ω ) ω with initial condition ω t =0 = η. 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 ∗ η ( t η ∧ d η ) = 2 C d( ∗ η η ) = 0 and 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(d t ∧ ω + ∗ ω ) = 0 is an ‘SU(2)-flow’ on coframings of N : d d t ω = ∗ ω (d ω ) − 1 2 ∗ ω ( t ω ∧ d ω ) ω with initial condition ω t =0 = η. 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 ∗ η ( t η ∧ d η ) = 2 C d( ∗ η η ) = 0 and 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(d t ∧ ω + ∗ ω ) = 0 is an ‘SU(2)-flow’ on coframings of N : d d t ω = ∗ ω (d ω ) − 1 2 ∗ ω ( t ω ∧ d ω ) ω with initial condition ω t =0 = η. 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 ∗ η ( t η ∧ d η ) = 2 C d( ∗ η η ) = 0 and 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 Υ i (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 ∧ d η ) = 2 H 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 ∗ η ( t η ∧ d η ) = 2 C d( ∗ η η ) = 0 and 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 → R 3 satisfying d � ∗ η d x � = 0 . Now, fix a constant C and consider a coframing η = g ( x ) − 1 d x on U ⊂ R 3 where g : U → GL(3 , R ) is a mapping satisfying the first-order, quasi-linear system ∗ η ( t η ∧ d η ) = 2 C, � � d( ∗ η η ) = 0 , d ∗ η d x = 0 . This is an elliptic underdetermined system consisting of 7 equations for 9 unknowns. Standard theory shows that the general solution is not real- analytic.

The G 2 -theory. An analogous situation holds in the case of hypersurfaces in Riemannian manifolds ( M 7 , g ) with holonomy G 2 ⊂ SO(7). In this case, there is a unique g -parallel 3-form σ ∈ Ω 3 ( M ) such that σ ∧ ∗ σ = 7 d V g . Such metrics are Ricci-flat and hence are real-analytic in local g -harmonic coordinate charts. Conversely, there is a open set Ω 3 + ( M 7 ) of definite 3-forms, i.e., σ ∈ Ω 3 + ( M 7 ) if and only if, for all x ∈ M , the stabilizer of σ x in GL( T x M ) is isomorphic to G 2 ⊂ SO(7). These forms are the sections of an open subbundle Λ 3 + ( T ∗ M ) ⊂ Λ 3 ( T ∗ M ). Such a σ ∈ Ω 3 + ( M ) determines a unique metric g σ and orientation ∗ σ and σ is g σ -parallel if and only if d σ = 0 and d( ∗ σ σ ) = 0 . Theorem: (B—) There is an involutive EDS I on Λ 3 + ( T ∗ M ) such that a section σ ∈ Ω 3 + ( M ) is an integral of I iff it is g σ -parallel.

Hypersurfaces. G 2 acts transitively on S 6 ⊂ R 7 , 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 = φ + i ψ = N ∗ σ − i ( n ω = n σ and ∗ σ σ ) . In fact, if one defines f : R × N → M by f ( t, p ) = exp p � t n ( p ) � , then 2 ω 2 − d t ∧ Im(Ω) . f ∗ ( ∗ σ σ ) = 1 f ∗ σ = d t ∧ ω + Re(Ω) and where, now, ω and Ω are forms on N that depend on t . For each fixed t = t 0 , the induced SU(3)-structure on N satisfies t 0 ( ∗ σ σ ) � = 0 , 2 ω 2 ) = d � f ∗ d Re(Ω) = d( f ∗ d( 1 t 0 σ ) = 0 and so these are necessary conditions on the SU(3)-structure on N that it be induced by immersion into a G 2 -holonomy manifold M .

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into a G 2 -holonomy manifold iff its defining forms ω and Ω satisfy 2 ω 2 ) = 0 . d( 1 d Re(Ω) = 0 and Proof: Define a tautological 2-form ω and 3-form Ω on F ( N ) / SU(3) as follows: For a coframe u : T x N → C 3 , define these forms at [ u ] = u · SU(3) ∈ F ( N ) / SU(3) by ω [ u ] = π ∗ � u ∗ ( i 2 ( t d z ∧ d¯ � Ω [ u ] = π ∗ � u ∗ (d z 1 ∧ d z 2 ∧ d z 3 ) � z )) and 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 2 ω 2 − d t ∧ Im( Ω ) . φ = 1 σ = d t ∧ ω + Re( Ω ) and Let I be the EDS generated by the closed 4-form d σ and 5-form d φ . Then I is involutive, with characters ( s 1 , . . ., s 7 ) = (0 , 0 , 1 , 4 , 10 , 13 , 0). = d( 1 2 ω 2 ) = 0, the given SU(3)-structure defines a reg- � � Since d Re(Ω) ular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the ahler Theorem, L lies in a unique I -integral M 7 ⊂ X . QED Cartan-K¨

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into a G 2 -holonomy manifold iff its defining forms ω and Ω satisfy 2 ω 2 ) = 0 . d( 1 d Re(Ω) = 0 and Proof: Define a tautological 2-form ω and 3-form Ω on F ( N ) / SU(3) as follows: For a coframe u : T x N → C 3 , define these forms at [ u ] = u · SU(3) ∈ F ( N ) / SU(3) by ω [ u ] = π ∗ � u ∗ ( i 2 ( t d z ∧ d¯ Ω [ u ] = π ∗ � u ∗ (d z 1 ∧ d z 2 ∧ d z 3 ) � � z )) and 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 2 ω 2 − d t ∧ Im( Ω ) . φ = 1 σ = d t ∧ ω + Re( Ω ) and Let I be the EDS generated by the closed 4-form d σ and 5-form d φ . Then I is involutive, with characters ( s 1 , . . ., s 7 ) = (0 , 0 , 1 , 4 , 10 , 13 , 0). Since d � Re(Ω) � = d( 1 2 ω 2 ) = 0, the given SU(3)-structure defines a regular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the ahler Theorem, L lies in a unique I -integral M 7 ⊂ X . QED Cartan-K¨

Theorem: A real-analytic SU(3)-structure on N 6 is induced by embedding into a G 2 -holonomy manifold iff its defining forms ω and Ω satisfy 2 ω 2 ) = 0 . d( 1 d Re(Ω) = 0 and Proof: Define a tautological 2-form ω and 3-form Ω on F ( N ) / SU(3) as follows: For a coframe u : T x N → C 3 , define these forms at [ u ] = u · SU(3) ∈ F ( N ) / SU(3) by ω [ u ] = π ∗ � u ∗ ( i 2 ( t d z ∧ d¯ � Ω [ u ] = π ∗ � u ∗ (d z 1 ∧ d z 2 ∧ d z 3 ) � z )) and 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 2 ω 2 − d t ∧ Im( Ω ) . φ = 1 σ = d t ∧ ω + Re( Ω ) and Let I be the EDS generated by the closed 4-form d σ and 5-form d φ . Then I is involutive, with characters ( s 1 , . . ., s 7 ) = (0 , 0 , 1 , 4 , 10 , 13 , 0). = d( 1 2 ω 2 ) = 0, the given SU(3)-structure defines a reg- � � Since d Re(Ω) ular integral manifold L ⊂ X of I lying in the hypersurface t = 0. By the ahler Theorem, L lies in a unique I -integral M 7 ⊂ X . QED Cartan-K¨

Theorem: There exist non-real-analytic SU(3)-structures on N 6 whose associated forms satisfy 2 ω 2 ) = 0 � � = d( 1 d Re(Ω) but that are not induced from an immersion into a G 2 -holonomy Riemann- ian 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 G 2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms ( ω, Ω) is induced via a G 2 -immersion N 6 ֒ → M 7 , the mean curvature H of N in M is given � � �� by − 12 H = ∗ 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 2 ω 2 ) = 0 � � = d( 1 d Re(Ω) but that are not induced from an immersion into a G 2 -holonomy Riemann- ian 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 G 2 -immersed.(Such do exist.) Proof: When an SU(3)-structure on N 6 with forms ( ω, Ω) is induced via a G 2 -immersion N 6 ֒ → M 7 , the mean curvature H of N in M is given � � �� by − 12 H = ∗ 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 2 ω 2 ) = 0 � � = d( 1 d Re(Ω) but that are not induced from an immersion into a G 2 -holonomy Riemann- ian 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 G 2 -immersed. (Such do exist.) Proof: When an SU(3)-structure on N 6 with forms ( ω, Ω) is induced via a G 2 -immersion N 6 ֒ → M 7 , the mean curvature H of N in M is given � � �� by − 12 H = ∗ 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( 1 2 ω 2 ) = 0 , � � � � �� d Re(Ω) = 0 , ∗ ω ∧ d Im(Ω) = C constitute 15 + 6 + 1 = 22 equations for the SU(3)-structure. Fix an orientation of N 6 . Say that a 3-form φ ∈ Ω 3 ( N 6 ) is elliptic if, at each point, it is linearly equivalent to Re � d z 1 ∧ d z 2 ∧ d z 3 � . Such a φ defines a unique, orientation-preserving almost-complex structure J φ on N 6 such that Ω φ = φ + i J ∗ φ ( φ ) is of J φ -type (3 , 0). Now assume that φ is also closed. Then dΩ φ is purely imaginary and yet must be a sum of terms of J φ -type (3 , 1) and (2 , 2). Thus, dΩ φ is purely of J φ -type (2 , 2).

So far: φ ∈ Ω 3 e ( N 6 ) defines J φ and Ω φ = φ + i J ∗ φ ( φ ) ∈ Ω 3 , 0 ( N, J φ ). dΩ φ ∈ Ω 2 , 2 ( N, J φ ) . d φ = 0 then yields

So far: φ ∈ Ω 3 e ( N 6 ) defines J φ and Ω φ = φ + i J ∗ φ ( φ ) ∈ Ω 3 , 0 ( N, J φ ). dΩ φ ∈ Ω 2 , 2 ( N, J φ ) . d φ = 0 then yields Fix a constant C � = 0. It is a C 1 -open condition on φ that ω φ = ω φ ∈ Ω 1 , 1 dΩ φ = i 6 C ( ω φ ) 2 for some + ( N, J φ ) .

So far: φ ∈ Ω 3 e ( N 6 ) defines J φ and Ω φ = φ + i J ∗ φ ( φ ) ∈ Ω 3 , 0 ( N, J φ ). dΩ φ ∈ Ω 2 , 2 ( N, J φ ) . d φ = 0 then yields Fix a constant C � = 0. It is a C 1 -open condition on φ that ω φ = ω φ ∈ Ω 1 , 1 dΩ φ = i 6 C ( ω φ ) 2 for some + ( N, J φ ) . Now, the pair ( ω φ , Ω φ ) are the defining forms of an SU(3)-structure on N if and only if 6 ( ω φ ) 3 − 1 1 8 i Ω φ ∧ Ω φ = 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 Ω φ ) = d φ = 0 , and 2 ( ω φ ) 2 � = d � − 3i 1 � = 0 , d � 1 C dΩ φ and, finally ω φ ∧ 1 6 C ( ω φ ) 2 � � � � ∗ φ ω φ ∧ d(Im Ω φ ) = ∗ φ = C.

So far: φ ∈ Ω 3 e ( N 6 ) defines J φ and Ω φ = φ + i J ∗ φ ( φ ) ∈ Ω 3 , 0 ( N, J φ ). dΩ φ ∈ Ω 2 , 2 ( N, J φ ) . d φ = 0 then yields Fix a constant C � = 0. It is a C 1 -open condition on φ that ω φ = ω φ ∈ Ω 1 , 1 dΩ φ = i 6 C ( ω φ ) 2 for some + ( N, J φ ) . Now, the pair ( ω φ , Ω φ ) are the defining forms of an SU(3)-structure on N if and only if 6 ( ω φ ) 3 − 1 1 8 i Ω φ ∧ Ω φ = 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 Ω φ ) = d φ = 0 , and 2 ( ω φ ) 2 � = d � − 3i 1 � = 0 , d � 1 C dΩ φ and, finally ω φ ∧ 1 6 C ( ω φ ) 2 � � � � ∗ φ ω φ ∧ d(Im Ω φ ) = ∗ φ = C.

So far: φ ∈ Ω 3 e ( N 6 ) defines J φ and Ω φ = φ + i J ∗ φ ( φ ) ∈ Ω 3 , 0 ( N, J φ ). dΩ φ ∈ Ω 2 , 2 ( N, J φ ) . d φ = 0 then yields Fix a constant C � = 0. It is a C 1 -open condition on φ that ω φ = ω φ ∈ Ω 1 , 1 dΩ φ = i 6 C ( ω φ ) 2 for some + ( N, J φ ) . Now, the pair ( ω φ , Ω φ ) are the defining forms of an SU(3)-structure on N if and only if 6 ( ω φ ) 3 − 1 1 8 i Ω φ ∧ Ω φ = 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 Ω φ ) = d φ = 0 , and 2 ( ω φ ) 2 � = d � − 3i 1 � = 0 , d � 1 C dΩ φ and, finally ω φ ∧ 1 6 C ( ω φ ) 2 � � � � ∗ φ ω φ ∧ d(Im Ω φ ) = ∗ φ = C.

Interpretation: On N 6 × R , with ( ω, Ω) defining an SU(3)-structure on N 6 depending on t ∈ R , consider the equations � 1 2 ω 2 − d t ∧ Im(Ω) � � � d d t ∧ ω + Re(Ω) = 0 and d = 0 . Think of Ω as φ +i J ∗ φ ( φ ), so the SU(3)-structure is determined by ( ω, φ ) where φ = Re(Ω). The closure conditions for fixed t are d( ω 2 ) = 0 , d φ = 0 and and the G 2 -evolution equations for such ( ω, φ ) are then d d − 1 � � J ∗ �� d t ( φ ) = d ω and d t ( ω ) = − L ω d φ ( φ ) , where L ω : Ω 2 ( N ) → Ω 4 ( N ) is the invertible map L ω ( β ) = ω ∧ β . The discussion shows that this ‘G 2 -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 Φ 0 ∈ Λ 4 ( R 8 ). Thus, a Spin(7)-structure on M 8 is a 4-form Φ ∈ Ω 4 ( M ) that is linearly equivalent to Φ 0 at every point of M . Such a structure Φ determines a metric g Φ and orientation ∗ Φ . Moreover, Φ is g Φ -parallel iff dΦ = 0. Define a 4-form Φ on F ( M ) / Spin(7) by the rule: For u : T x M → R 8 and [ u ] = u · Spin(7) Φ [ u ] = π ∗ � u ∗ Φ 0 � where π : F ( M ) → M is the basepoint projection. Let I be the ideal on F ( M ) / Spin(7) generated by d Φ . 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 g Φ is equal to Spin(7).

Hypersurfaces. Spin(7) acts transitively on S 7 and the stabilizer of a point is G 2 . An oriented hypersurface N 7 ⊂ M 8 inherits a G 2 -structure defined by the rule ∗ σ σ = N ∗ Φ σ = n Φ and satisfies where n is the oriented normal vector field along N . One easily checks that � σ ∧ d σ � = 28 H ∗ σ where H is the mean curvature of N in M . Theorem: If σ ∈ Ω 3 + ( 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 ) / G 2 . It is involutive with characters ( s 1 , . . ., s 8 ) = (0 , 0 , 0 , 1 , 4 , 10 , 20 , 0). The co-closed G 2 - structure σ defines a regular I -integral in the locus t = 0 which, by the aher Theorem, lies in an essentially unique I -integral M 8 . QED Cartan-K¨

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