Eric Loubeau

Lázaro Rodríguez

Henrique Sá Earp

Parallelisms on the 7-sphere

We survey Kirchhoff's classical construction of parallelisms on spheres, induced by almost complex structures on their equator. Motivated by the Hopf problem, we focus on the 7-dimensional context, highlighting the construction's proximity to notions of Lie group structure and integrability of almost-complex structures on S 6 . We explain how integrability of such almost complex structures and parallelisms are mediated by a particular notion of torsion, a fact which has interesting algebraic and geometric repercussions.

Introduction

The existence of special algebraic and geometric structures on spheres is a long-standing matter of mathematical inquiry, arguably dating back to Cartan's seminal proof that high-dimensional spheres cannot be Lie groups. In particular, the existence of global frame fields, or parallelisms, on odd-dimensional spheres has been shown by [START_REF] Kirchhoff | Sur l'existence de certains champs tensoriels sur les sphères à n dimensions[END_REF] to bear relation to almost complex structures on their even-dimensional equatorial sphere. Since then, a fruitful classification programme spanned over almost two decades, to which Borel-Serre, Kervaire and Bott-Milnor (independently), and Adams added crucial contributions:

(1) [Car1936] S n+1 is a Lie group =⇒ n = 0, 2.

(2) [Kir1947] S n is almost complex =⇒ S n+1 is parallelisable.

(3) [BS1951] S n is almost complex =⇒ n = 0, 2, 6.

(4) [Ker1958, BM1958] S n+1 is parallelisable =⇒ n = 0, 2, 6.

(5) [Ada1960] S n+1 is an H-space ⇐⇒ n = 0, 2, 6 (Adams) NB.: Of course (2)&(4) =⇒ (3), but the historical timeline worked out differently.

This successful line of investigation culminates at the specificity of the problem of parallelisation on the spheres S 3 and S 7 , which correlates to the study of almost complex structures (ACSs) respectively on S 2 and S 6 . Moreover, it acquires additional texture from the perspective of differential topology, once we pose the integrability question about those structures, even if only to low-order obstructions. For a parallelism, it takes the form of global constancy of its structure functions, ie. of constant torsion, whereas for an almost complex structure it amounts to the vanishing of its Nijenhuis tensor. The situation on S 3 SU(2) is completely understood, and a smooth constant parallelism can be explicitly gathered from any three independent left-invariant vector fields given by its Lie group structure, as well as being directly associated to the canonical (integrable) complex structure on S 2 . Meanwhile, a similar narrative is far from clear on S 7 , where indeed it stumbles upon the famous Hopf problem: whether a(n integrable) complex structure exists at all on S 6 . If such a structure could exist, then it makes sense to ponder what consequences that would bear upon the torsion of the corresponding parallelism on S 7 , since after all the latter does not admit a Lie group structure.

For the history of the Hopf problem, we refer the reader to the excellent survey [ABG + 2018]. Our central motivation here is the fact that the construction of Kirchhoff frames is an example of simple mathematics yielding thought-provoking conclusions, intertwining geometry, algebra and topology. In particular, we feel that a number of interesting connections in this story have been somewhat estranged by maths curricula over the past decades, and it might be a good moment to refresh the topic under a modern light. For example, although the relation between the associator and the Nijenhuis tensor in the case of the classical almost complex structure induced by the octonions is probably well-known to experts, it is not well-documented, cf. [KP2018, Remark 2.3]. We should also mention that a contemporary perspective on {e}-structures, seen as homogeneous sections defining a (trivial) reduction of the frame bundle, as well as their torsion and their infinitesimal deformations, can be found in [LSE2019, FLMSE2022].

The material is organised as follows. In §2 we review the relations between parallelisms and algebraic structures on spheres, as multiplicative spaces. In particular, we explain how the non-associativity of the octonions is in a precise sense 'to blame' for the non-constancy of the canonical parallelism on S 7 and for the non-integrability of the standard ACS on S 6 . In §3, we review Kirchhoff's construction of global frames on a sphere from an ACS on the equator, with a few additions such as a class of spherical metrics on S 7 which are particularly compatible with Kirchhoff frames, and a computation of their torsion therewith. Finally, in §4 we address the Hopf problem, explaining how the hypothetical integrability of an ACS on S 6 could lead to a contradiction, by way of an analytical argument yielding an integrable parallelism on S 7 .

2 Parallelisms on spheres and algebraic structures 2.1 When does a parallelism come from a Lie group structure?

Each smooth global frame σ = {X 1 , • • • , X n } on a manifold M induces a flat (zero curvature) connection Γ, corresponding to the covariant derivative defined by

∇ Z f i X i := Z(f i )X i , for Z ∈ X (M ).
The global structure equations of Γ in the frame σ are:

dω i = 1 2 T i jk ω j ∧ ω k , (1)
where

{ω 1 , • • • , ω n } is the dual coframe of {X 1 , • • • , X n }. The torsion tensor of Γ is given by T (X j , X k ) = n i=1 T i jk X i = -[X j , X k ]. (2)
Furthermore the torsion tensor of Γ is parallel if and only if the structure functions T i jk : M → R are constant. More generally, a tensor field on M is parallel with respect to Γ if and only if it has constant components with respect to the frame field {X 1 , • • • , X n }. Equation (2) resembles the way in which one defines the structure constants of a Lie algebra, i.e., (1) looks like the Maurer-Cartan equation; this is not a coincidence, since Lie groups are always parallelisable. Moreover, the following converse states which parallelisms come from a Lie group structure on M : Theorem 1 ([Hic1959], Theorem 5). Let M be a simply connected manifold admitting a complete flat linear connection with torsion invariant under parallel translation. Then M admits a Lie group structure such that left-translations induce the original connection.

The above theorem was proved by Chern [Che1953, Section 5, page 128], in terms of an {e}-structure on M . Generalisations of this result to not necessarily simply-connected manifolds have been proved many times in the literature, e.g., Wolf [Wol1971, Proposition 2.5] within the context of absolute parallelisms. All of those are rooted, one way or another, on Cartan's local equivalence method, see Sternberg [Ste1964, Theorem 2.4, Chapter V].

Proposition 1 ([Wol1971], Proposition 2.5). Let σ be a smooth parallelism on a connected manifold M . The following statements are equivalent:

(i) σ has complete associated connection and parallel torsion;

(ii) M has the structure of the coset space G/D = {Dg : g ∈ G}, for a connected Lie group G and a discrete subgroup D ⊂ G, such that σ is induced by left-translations of G.

In general, a linear connection on a compact manifold is not necessarily complete. However, as the geodesics of the associated connection Γ consist of the integral curves of the vector fields

{X 1 , • • • , X n }, compactness of M implies that Γ is complete. In a local coordinate patch (U, x 1 , • • • , x n ),
we can write the vector fields X j = X l j ∂ ∂x l and the dual forms ω j = ω j l dx l , where (ω j l ) is the inverse matrix of X l j ) in terms of the local basis of vectors fields and differential forms. The structure functions may then be computed by the formula:

T i jk = n r,s=1 X r j X s k ∂ω i s ∂x r - ∂ω i r ∂x s .
(3)

The octonions and the classical parallelism of the 7-sphere

Let us briefly discuss the parallelism of S 7 induced by the octonions, in the light of Hicks' Theorem 1. The next proposition explains how orthogonal multiplications define vector fields on spheres.

Proposition 2 ([BC1970], Proposition 7.3.1). Suppose we have a map ν : R k+1 × R n+1 → R n+1 , linear in the first factor and continuous in the second factor, satisfying:

i) ν(v, z) = 0 implies z = 0 or v = 0,
ii) there exists e ∈ R k+1 such that ν(e, z) = z for all z ∈ R n+1 , then S n admits k independent vector fields.

The proof can be found in [BC1970] and, though stated for bilinear maps ν, it only uses linearity in the first factor. As a corollary of the above Proposition, the multiplications in the complex, quaternion, and octonion numbers induce parallelisms on S 1 , S 3 and S 7 , respectively. For instance, fix the canonical basis of O given by the identity 1 and seven imaginary octonions e i , i = 1, • • • , 7, satisfying the multiplication rule e i e j = -δ ij + a ijk e k ,

where the structure constants a ijk are totally skew-symmetric. Using Proposition 2, we construct seven linearly independent vector fields X i on the sphere S 7 ⊂ O of unit octonions as follows:

X i (x) = e i x for x ∈ S 7 , i = 1, • • • , 7.
Let us compute the structure functions of this global frame. Note that multiplication in this particular case is linear in both factors, therefore the Lie brackets [X i , X j ] can be computed by the commutator of the corresponding linear maps.

[X i , X j ](x) = e i (e j x) -e j (e i x)

(4) = 2a ijk e k x -2[e i , e j , x] = 2 (a ijk -[e i , e j , x], e k x ) X k (x),
where [a, b, c] := (ab)c -a(bc) is the associator, a, b := 1 2 a b + bā is the standard inner product, and conjugation is defined by 1 = 1, ēi = -e i and ab = bā.

Remark 3. While the non-commutativity of the octonions causes the non-vanishing of the torsion, their non-associativity causes the non-constancy of the structure functions of the classical parallelism of S 7 . Compare to Remark 6, below.

Remark 4. We used the alternativity of the octonionic product to prove the second equality in (4). Compare to Remark 7, below.

Remark 5. Note the structure functions coincide with the structure constants of the algebra at the North and South poles, i.e., at 1 and -1 in O. Compare to Remark 14, below.

The octonions and the induced almost complex structure on the 6-sphere

Let us briefly recall how the multiplication in the octonions induces an almost complex structure on S 6 . As we will see in §3, Kirchhoff's construction is modeled on this, in fact its proof reverses this process, by reconstructing the 'multiplication' of R 8 from the almost complex structure, see also Remark 15 and Section 4.2.

Let Im O ⊂ O denotes the hyperplane of imaginary octonions orthogonal to 1 ∈ O, and let S 6 ⊂ Im O be the sphere of unit imaginary octonions. Right-multiplication by y ∈ S 6 induces an orthogonal linear map R y : O → O such that (R y ) 2 = -1. Moreover, R y maps 1 → y and y → -1, hence it preserves the 2-plane spanned by 1 and y, as well as its orthogonal 6-plane, which can be identified with T y S 6 ⊂ O. It follows that R y induces an almost complex structure on S 6 . Now we are going to show that its Nijenhuis tensor can be expressed in terms of the associator of the octonions O.

The Nijenhuis tensor can be computed by:

N (X, Y ) = d(JY )(JX) -d(JX)(JY ) -dY (X) + dX(Y ) -J (d(JY )(X) -dX(JY )) -J (dY (JX) -d(JX)(Y )) .
To see this, note that in Euclidean space we can compute the Lie bracket of vector fields X, Y :

S 6 → R 7 by [X, Y ] = dY (X) -dX(Y ),
where dX and dY denote respectively the differentials of X and Y , as maps. By definition J a Y a = Y a • a, where a ∈ S 6 and Y is a vector field on S 6 . Differentiating, we get

d(JY )(JX) = J (dY (JX)) + Y • JX, J (d(JY )(X)) = (-1)dY (X) + J(Y • X). Then N (X, Y ) = Y • JX -X • JY -J(Y • X) + J(X • Y ).
For b, c ∈ T a S 6 we get:

N a (b, c) = c • (b • a) -b • (c • a) -(c • b) • a + (b • c) • a = 2[a, b, c]. (5)
Remark 6. Therefore the non-associativity of the octonions is responsible for the non-integrability of the almost complex structure J. Compare to Remark 3.

Remark 7. To establish the last equality in (5) we used the fact that the algebra of octonions is alternative.

Compare to Remark 4.

3 Kirchhoff's theorem: old and new

Statement and proof

Now we state Kirchhoff's theorem and then survey its proof, which is fairly straightforward but maybe not so widely remembered nowadays, cf. [Kir1947], [Kir1953, Theorem V], see also Kobayashi Theorem 2 ([Kir1947], Theorem 4). If the sphere S n admits an almost complex structure, then S n+1 is parallelisable.

We need to exhibit a field σ of linear frames on S n+1 . Let J be an almost complex structure on the equatorial sphere S n . Fix a subspace R n+1 ⊂ R n+2 in the ambient vector space and a unit vector e := e 0 ∈ R n+2 perpendicular to R n+1 , in the standard Euclidean inner-product. Denote by S n and S n+1 the unit spheres in R n+1 and R n+2 respectively. Lemma 8. At each y ∈ S n , the almost complex structure J y can be extended to a map J y : R n+2 → R n+2 such that J 2 y = -I. Moreover, if J y is compatible with the inner product on S n , then J y ∈ SO(n + 2).

Proof. Given y ∈ S n denote by V y the n-dimensional vector subspace of R n+2 parallel to the tangent space T y (S n ) in R n+2 and J y the linear endomorphism of V y corresponding to the linear endomorphism of T y (S n ) given by J. Define a linear transformation J y : R n+2 → R n+2 by J y (e) = y, J y (y) = -e and J y (z) = J y (z) for z ∈ V y . It follows from J 2 = -I that J 2 y = -I.

Let x ∈ R n+2 , then it can be written uniquely as follows:

x = αe + βy, α, β ∈ R, β ≥ 0, and y ∈ S n . (6)

We will refer to y as the equatorial projection of x. Define the linear transformation:

σ x : R n+2 → R n+2 , σ x := αI + β J y . (7)
Lemma 9. The map σ x is an automorphism of R n+2 . Moreover, if x ∈ S n+1 and J y is compatible with the inner product on S n , then σ x ∈ SO(n + 2).

Proof. It is straightforward to check that 1 α 2 +β 2 σ t x = 1 α 2 +β 2 (αI -β J y ) = σ -1
x , which yields the claim when α 2 + β 2 = 1.

When x ∈ S n+1 , note also that σ x (e) = x, and denote each restriction of σ x to R n+1 by

σ x := σ x | R n+1 , x ∈ S n+1 .
Lemma 10. When x ∈ S n+1 , each linear frame σ x =: R n+1 → T x (S n+1 ) is an isomorphism.

Proof. Recall that R n+1 is spanned by y (the equatorial projection of x) and V y . Clearly σ ±e = ±I : R n+1 → T ±e S n+1 , respectively. For a point x ∈ S n+1 × := S n+1 \ {±e} and a vector v ∈ R n+1 = Ry ⊕ V y , using the standard inner-product of R n+2 and recalling that y, e = 0, we obtain

σ x (v) = αv -β v, y e + β Jx (v ⊥ ), where v ⊥ ∈ V y . Note that σ x (y) = αy -βe and σ x (v) = αv + β Jx (v) when v ∈ V y . Observe that as v ⊥ ∈ V y then Jx (v ⊥ ) ∈ V y , so Jx (v ⊥
) is orthogonal to both y and e, hence to x = αe + βy.

It is then straightforward to verify that σ x indeed maps into T x S n+1 :

σ x (v), x = αv -β v, y e + β Jx (v ⊥ ), x = α v, x -β v, y e, x + β Jx (v ⊥ ), x = αβ v, y -αβ v, y = 0.
Finally, σ x is a linear map and, as J2

x = -I n+1 , σ x is an isomorphism Jx (σ x (v)) = α Jx (v) + β J2 x (v) = α Jx (v) -βv so v = ασ x (v) + β Jx (σ x (v))
. and σ x is an isomorphism from R n+1 to T x (S n+1 ).

Remark 11. Kirchhoff's theorem does not assume any property of the almost complex structure J.

Remark 12. If moreover J is an almost Hermitian structure, i.e., compatible with some Riemannian metric g on S n (this is always possible), then the theorem above can be found in [Ste1951, Theorem 41.19]. Steenrod noted that in this case the global frame σ is in fact orthogonal, that is, σ x ( σ x ) t = I, for x ∈ S n+1 . More generally, we have σ x ( σ x ) t = x 2 I, for x ∈ R n+2 . This will be used in Section 4.2. In what follows, x will denote the Euclidean norm of x.

Remark 13. The vector fields {X i (x) := σ x (e i )} i=1,••• ,n+1 defining the parallelism in Theorem 2 are given explicitly by X i (x) = x 0 e i -x i e 0 + β(x)J y (e i -y, e i y) ,

where {e i } i=0,••• ,n+1 is the canonical basis of R n+2 .
Remark 14. The linear frame σ is smooth at all points of S n+1 except at e and -e, where it is merely Lipschitz continuous. The following approximation theorem, originally proved in somewhat outdated language by Steenrod [Ste1951, Theorem 6.7], guarantees that every continuous section of a smooth fibre bundle can be approximated arbitrarily closely (in a suitable topology) by a smooth section:

Theorem 3. [Osb1982, Chap. III, Prop. 5.11 & 5.12] Let E → X be a smooth vector bundle over a smooth manifold X. Denote by F the C 0 (X)-module of (continuous) sections X → E, and let . : F → C 0 (X) be any Euclidean norm. For any strictly positive function ε ∈ C 0 (X) and continuous section σ : X → E, there is then a smooth section σ : X → E such that σ -σ < ε. In particular, if there is a continuous nowhere-vanishing section σ : X → E, there is also a smooth nowhere-vanishing section σ : X → E.

Therefore, unless otherwise mentioned, we may assume σ is a smooth linear frame that coincides on S n+1 × := S n+1 \ {e, -e} with the one constructed in the above theorem, except at a small neighbourhood of the poles; we will refer to it in this case as the smooth Kirchhoff frame.

Remark 15. We can use Proposition 2 to conclude that S n+1 is parallelisable, by Kirchhoff's theorem, since the map ν : R n+1 × R n+1 → R n+1 defined by ν(v, z) := σ z v satisfies the hypotheses of Proposition 2.

Example 16. Let us illustrate the construction in the familiar n = 2 case, when the standard vector crossproduct on R 3 defined by quaternion multiplication induces an (almost) complex structure J on S 2 . At the poles we have σ ±e0 = ±I; at any other x = ±e 0 , we have β = 1 -x 2 0 = 0 and the equatorial projection is y = 1 β (x -x 0 e 0 ) ∈ S 2 , so

y = 1 1 -x 2 0     0 x 1 x 2 x 3     , [J y ] = 1 1 -x 2 0     0 0 0 0 0 0 -x 3 x 2 0 x 3 0 -x 1 0 -x 2 x 1 0     .
The induced Kirchhoff frame on S 3 is obtained explicitly as follows:

X 1 (x) = x 0 e 1 -x 1 e 0 + (1 -x 2 0 )J y (e 1 ) =     -x 1 x 0 0 0     +     0 0 0 0 0 0 -x 3 x 2 0 x 3 0 -x 1 0 -x 2 x 1 0         0 1 0 0     =     -x 1 x 0 x 3 -x 2     X 2 (x) = x 0 e 2 -x 2 e 0 + (1 -x 2 0 )J y (e 2 ) =     -x 2 0 x 0 0     +     0 0 0 0 0 0 -x 3 x 2 0 x 3 0 -x 1 0 -x 2 x 1 0         0 0 1 0     =     -x 2 -x 3 x 0 x 1     X 3 (x) = x 0 e 3 -x 3 e 0 + (1 -x 2 0 )J y (e 3 ) =     -x 3 0 0 x 0     +     0 0 0 0 0 0 -x 3 x 2 0 x 3 0 -x 1 0 -x 2 x 1 0         0 0 0 1     =     -x 3 x 2 -x 1 x 0     ie., σ x =     x 0 -x 1 -x 2 -x 3 x 1 x 0 -x 3 x 2 x 2 x 3 x 0 -x 1 x 3 -x 2 x 1 x 0     ∈ SO(4).
These ideas lead to the following question, regarding the Hopf problem. Assume, for a moment, that S 6 admits an integrable almost complex structure J. By Kirchhoff's Theorem 2, this induces a parallelism on the sphere S 7 , and we have the explicit form of the corresponding global frame in terms of J.

Question 1. To what extent does the integrability condition on J on S 6 amount to the constancy of the structure functions of the smooth Kirchhoff frame defining the parallelism on S 7 ?

We will return to this Question in §4.1. Notice finally that, even if in general the structure functions of the Kirchhoff frame are not constant, we can certainly aim at expressing the full torsion tensor for this {e}-structure on the punctured sphere S 7 × in terms of the almost complex structure J on S 6 .

Example: Kirchhoff frames on S 7

We will now specialise the language of Kirchhoff's proof to the context of the unit sphere S 7 ⊂ R 8 and its (equatorial) sphere S 6 := {x ∈ S 7 | x 8 = 0} and fill in some of the details of the previous section. We still denote the North pole by the vector e := (1, 0, . . . , 0) ∈ S 7 . We assume that S 6 is equipped with an almost complex structure J, that is for any y ∈ S 6 , J y : T y S 6 → T y S 6 with J 2 y = -I. Given x ∈ S 7 , we write x = cos φ e + sin φ y, for the zenith angle φ ∈ [0, π] and the equatorial projection y ∈ S 6 . We denote by V y the tangent space T y S 6 of S 6 at the point y, so that T y S 7 = Ry ⊕ V y .

At each point x ∈ S 7 ⊂ R 8 , we extend the almost complex structure J on S 6 to J : R 8 → R 8 by Jx (e) = y, Jx (y) = -e, Jx (z) = J y (z), ∀z ∈ V y .

Note that J2

x = -I 8 . This definition could easily be extended into an almost complex structure on R 8 . Setting α = cos φ and β = sin φ, the Kirchhoff frame is defined at each point x ∈ S 7 by

σ x : R 7 → T x S 7 v → σ x (v) = αv + β Jx (v).
The Kirchhoff frame assumes a particularly simple form in spherical coordinates. At any given y ∈ S 6 , let {f i } i=1,...,6 be a basis of V y , so that {y, f i } i=1,...,6 spans R 7 ∈ R 8 . Now, introducing the zenith coordinate vector ∂ φ = -βe + αy, we obtain a basis of T x S 7 × as {∂ φ , f i } i=1,...,6 , in which the Kirchhoff frame field is expressed as the (7 × 7)-matrix:

     -1 0 . . . 0 0 . . . J y 0      (8)
Remark 17. While the matrix expression (8) of the Kirchhoff frame field is simple enough, it is written with respect to a moving frame -which will change from point to point. An alternative choice is to consider now the canonical basis (up to order) E = {e 0 = e, e 1 , . . . , e 7 } of R 8 . For a point x = αe + βy ∈ S 7 , with equatorial projection y = (0, y 1 , . . . , y 7 ) ∈ S 6 ⊂ R 7 , we know that Jx (e 0 ) = Jx (e) = y and Jx (e i ) = Jx e i , y y + (e i -e i , y y) = y i Jx (y) + Jx (e i -e i , y y) = y i Jx (y) + J y (e i -e i , y y) , for i = 1, . . . , 7. since e i -e i , y y ∈ V y . Therefore, in the basis E, the operator Jx is given by the (8 × 8)-matrix:

[ Jx ] E =      0 -y 1 • • • -y 7 y 1 . . . [J y • γ y ] y 7     
where [J y • γ y ] is the matrix of the operator J y • γ y : R 7 → R 7 in the basis {e 1 , . . . , e 7 }, and

γ y =      1 -y 2 1 -y 1 y 2 . . . -y 1 y 7 -y 1 y 2 1 -y 2 2 . . . -y 2 y 7 . . . . . . . . . . . . -y 1 y 7 -y 2 y 7 . . . 1 -y 2 7      : v → v -v, y y ∈ V y ⊂ R 7
is an orthogonal projection. Then the Kirchhoff frame field reads

[σ x ] E = α I 8 + β[ Jx ] E .

A metric on S 7 adapted to generalised Kirchhoff parallelisms

All the classical discussion so far considers S 6 ⊂ S 7 as round spheres in the Euclidean metric of R 8 . Let us now try and address the same construction from the more general perspective of spherical metrics g induced on a smooth 7-sphere by an arbitrary Hermitian metric g on its equatorial 6-sphere, compatible with a given almost complex structure J. Suppose, without loss of generality up to homotopy, that the embedding of (S 6 , g) in R 7 is star-shaped with respect to the origin. Then g has a natural polar extension metric over R 8 , obtained in the following way. Let S7 := {x = cos φ e + sin φ y | y ∈ S 6 , φ ∈ [0, π]}.

be the new unit 7-sphere, which is homotopic in R 8 to the standard Euclidean S 7 . Then, for any point p ∈ R 8 , we define r = |p| g by p =: rx, where x ∈ S7 is unique by assumption, and let g := dr 2 + r 2 dφ 2 + r 2 (sin 2 φ)g.

Notice in particular that g(e, y) = 0, for any y ∈ S 6 . We can then define the group of isometries of (R 8 , g):

SO(8) = {A ∈ End(T R 8 ) : A t gA = g}.
Please observe that this notation is merely suggestive, since the total space, in this case, is not in itself a matrix Lie group. The compatibility condition between g and an almost complex structure J on R 8 is

g(u, v) = g( Ju, Jv) = u t Jt g Jv,
which is equivalent to Jt g J = g.

Lemma 18. If g is compatible with an almost complex structure J on S 6 , then its polar extension g is compatible with J on R 8 , i.e.,

g( Ju, Jv) = g(u, v), ∀u, v ∈ T p R 8 , p ∈ R 8 .
Proof. Let x = p |p|g = cos φe + sin φy ∈ S7 , with equatorial projection y ∈ S 6 . If both u, v ∈ T y S 6 , then the conclusion is immediate, because g| S 6 = g. If u ∈ (T y S 6 ) ⊥, then it is contained in the 2-plane Π := span{x, ∂ φ } = span{e, y}, and by definition J : Π → Π is a rotation by 90 degrees. The conclusion follows.

Adopting slightly more convenient notation, we redefine the (basepoint-dependent) transpose by A t = g-1 A t g, and the corresponding orthonormal frame bundle over S7 is characterised by pointwise projection onto the first column:

SO(8) = {A ∈ End(T R 8 ) : A t = A -1 } π -→ ( S7 , g).
The associated bundle of Lie algebras is so(8) := {B ∈ End(T R 8 | S7 )) : B t = -B} = g-1 so(8).

Example 19. The Kirchhoff almost complex structure J on R 8 is compatible with g so Jt g J = g therefore g J + Jt g = 0, so that J is in so(8, g). Then the Kirchhoff frame constructed from J is σ = αI + β J, with α 2 + β 2 = 1 takes values in SO(8):

σ t gσ = αI + β Jt g αI + β J = α 2 g + αβ Jt g + g J + β 2 Jt g J = g.

Torsion of a parallelism

Our procedure to compute the torsion of parallelisms on S7 will follow closely that which was carried out for the Euclidean 3-sphere in [LSE2019, §3.1], and we refer the reader there for additional details.

In particular, we avoid repeating here the abstract definition of the torsion of a geometric structure as the vertical differential of a homogeneous section. The tangent space to the orthonormal frame bundle at z ∈ SO(8) is given by:

T z SO(8) = 1 2 (M -zM tz) : M ∈ GL(8, R) = z so(8). (9)
The vertical distribution at the identity of SO(8) is

V I := ker p * = M ∈ so(8) : M = 0 0 0 M , M ∈ so(7)
and, at z ∈ SO(8), V z = zV I . Here we define so(7) := {C ∈ GL(7, R) :

C t = -C}, under the identification R 7 = {p ∈ R 8 : p, e = 0} = {p ∈ R 8 : φ = π 2 }. A g-orthonormal parallelism is a section σ : S7 → SO(8) x → σ(x) = (x, σ 1 (x), . . . , σ 7 (x))
of the orthonormal frame bundle SO(8) π -→ ( S7 , g). At each x ∈ S7 , its differential is given by

dσ x : T x S7 → T σ(x) SO(8) X → dσ x (X) =: Proj T σ(x) SO(8) (M )
with M = (X, dσ 1 (X), . . . , dσ 7 (X)). Therefore

dσ x (X) = 1 2 M -σ(x)M tσ(x) = σ(x). 1 2 σ(x) tM -M tσ(x) = σ(x) 0 -v t v M ,
with M ∈ so(7) and v ∈ R 7 , so the torsion of the parallelism is the vertical component

T (X x ) = d V σ x (X) = σ(x) 0 0 0 M . Denoting gx by •, • , we have σ(x) tM =      x, X x, dσ 1 (X) . . . x, dσ 7 (X) σ 1 , X σ 1 , dσ 1 (X) . . . σ 1 , dσ 7 (X) . . . . . . . . . σ 7 , X σ 7 , dσ 1 (X) . . . σ 7 , dσ 7 (X)      ,
and, since dσ i (X) = X H + ( ∇X σ i ) V , several entries simplify as follows:

x, X = σ i , dσ i (X) = 0, x, dσ i (X) = -σ i , X σ i , dσ j (X) = σ i , ∇X σ j , for i, j = 1, ..., 7.

We obtain, cf. (9):

dσ x (X) = σ(x)     0 -σ 1 , X -σ 2 , X -σ 3 , X σ 1 , X 0 σ 1 , ∇X σ 2 σ 1 , ∇X σ 3 σ 2 , X σ 2 , ∇X σ 1 0 σ 2 , ∇X σ 3 σ 3 , X σ 3 , ∇X σ 1 σ 3 , ∇X σ 2 0     ∈ T σ(x) SO(8) ⇒ d V σ x (X) = σ(x)         0 0 0 0 0 0 0 σ 1 , ∇X σ 2 . . . σ 1 , ∇X σ 7 0 σ 2 , ∇X σ 1 0 . . . 0 . . . . . . σ 6 , ∇X σ 7 0 σ 7 , ∇X σ 1 . . . σ 7 , ∇X σ 6 0        
In words, the typical (off-diagonal) elements of d V σ are the 1-forms:

(d V σ) ij = g( ∇σ j , σ i ), 1 ≤ i = j ≤ 7.
Indeed, those are the Christoffel symbols of ∇ in the frame field σ, if we care to raise an index etc, which are precisely the 'structure functions' introduced in §2.1. By Koszul's formula, we find:

T k ij := (d V i σ) k j = g( ∇i σ j , σ k ) = sign(ijk)g([σ i , σ j ], σ k ), 1 ≤ i, j ≤ 7. (10)
In §4.1, we will specialise the above discussion to the particular case of Kirchhoff frames. Remark 20. An alternative approach to computing the torsion of a parallelism is to consider it a section of the SO(7)-principal bundle of orthonormal frames of (S 7 , g), ie. the Euclidean unit 7-sphere endowed with the unusual Riemannian metric from (R 8 , g). While its total space can no longer be seen as a matrix Lie group, its fibre of frames on the point x ∈ S 7 is equal to SO(7) = {A ∈ GL(7, R) : A -1 = g-1 x A t gx }. As we work with the vertical part of the differential of σ, the computations are the same.

Reflections on the Hopf problem 4.1 Integrability on S 6 and torsion of smooth Kirchhoff parallelisms

Let us now address the issue raised in Question 1 and reason, for argument's sake, under the outrageous assumption that J is actually an integrable complex structure on S 6 . Around a given point y ∈ S 6 , one may choose a local system of coordinates (U, {θ 1 , . . . , θ 6 }) such that J∂ θj = ∂ θj+1 , for j = 1, 3, 5, where we write ∂ θj for ∂ ∂θj . Then, in a neighbourhood of x = αe + βy ∈ S7 , we have a local system of coordinates (φ, θ 1 , . . . , θ 6 ). The Kirchhoff frame field is then given by the image of the local frame {∂ φ , ∂ θ1 , . . . , ∂ θ6 } by the matrix (8):

σ 0 = σ (φ,θ1,...,θ6) (y) = ∂ φ σ j = σ (φ,θ1,...,θ6) (∂ θj ) = J y (∂ θj ), for j = 1, . . . , 6.
Proposition 21. The existence of a complex structure on S 6 would induce a smooth torsion-free parallelism on the punctured sphere S7 × .

Proof. It becomes easy to compute the brackets of the vector fields of the Kirchhoff frame field at any point where it is smooth, ie. away from the poles, since our assumption that J is integrable implies that, for any j = 1, . . . , 6, J y (∂ θj ) = ±∂ θ k , for some 1 ≤ k ≤ 6:

[σ 0 , σ j ] = [∂ φ , J y (∂ θj )] = ±[∂ φ , ∂ θ k ] = 0, [σ j1 , σ j2 ] = [J y (∂ θj 1 ), J y (∂ θj 2 )] = ±[∂ θ k 1 , ∂ θ k 2 ] = 0, for 1 ≤ j 1 , j 2 ≤ 6.
Here we are simply using the fact that local coordinate vector fields commute. Replacing the above vanishings in Koszul's formula (10), we find

T k ij = 0, ∀1 ≤ i, j ≤ 7,
i.e. the generalised Kirchhoff parallelism induced on S7 by an integrable almost complex structure J on S 6 is torsion-free wherever smooth.

In the light of Remarks 11-15, applied to S 6 ⊂ S7 , Proposition 21 motivates the following observations:

Remark 22. If the associated connection of σ can be shown to be complete, this would trigger from Wolf's Proposition 1 the conclusion that the 7-sphere punctured at its poles, away from which the Kirchhoff parallelism is indeed smooth, is a 'Lie group' (possibly quotiented by a discrete subgroup). This rather unimpressive conclusion opens however the following analytical programme. Remember that we only need that the torsion functions of a given parallelism be constant, not necessarily vanishing, in order to guarantee a Lie group structure on a simply-connected manifold. If this could be achieved from the Kirchhoff parallelism on S7 , under the initial assumption of an integrable J on S 6 , then by contradiction Hopf's problem would be answered in the negative. Thus, first, one might try to extend this torsion-free parallelism, along with its complete connection, over the poles with nonconstant torsion concentrated around the polar caps. Then, one may hope to deform it homotopically, eg. by a suitable geometric flow, towards a smooth limit with constant torsion everywhere. As a teaser, let us mention that Shi-type estimates for a general flow of {e}-structures have a particularly amenable form, cf. [FLMSE2022, Remark 1.35]. In any event, this would still be, of course, a very challenging problem.

Remark 23. Due to the nature of H-spaces, as we will discuss below, it suffices to achieve through this process a continuous associative multiplication on S 7 , in order to obtain a contradiction.

H-space structures on spheres

We can rephrase Kirchhoff's theorem as follows: if S n admits an almost complex structure J, then S n+1 is a an H-space, ie. it admits a continuous multiplication with a two-sided identity element. This is trivial at glance because a parallelisable sphere is well-known to be an H-space [Ada1960]. The point is that the induced multiplication on S n+1 is written explicitly in terms of J.

In what follows we keep the notation of Theorem 2. Let J be an almost complex structure on S 6 , and let σ be the global frame given by Kirchhoff's theorem. Define the map: m : S 7 × S 7 -→ S 7 , m(x, y) := σ x (y) / σ x (y) .

It follows from Kirchhoff's theorem this map is a well defined multiplication on S 7 with e as two-sided identity. Thus every almost complex structure J on S 6 defines an H-space structure on S 7 . Moreover, recall that we can always assume the given almost complex structure J is compatible with some Riemannian metric g on S 7 . Then, using Remark 12, we can define a multiplication on R 8 : m : R 8 × R 8 -→ R 8 , m(x, y) := σ x (y) , which satisfies the norm product rule m(x, y) 2 = x 2 y 2 , and has a two-sided identity e. The multiplication m restricts to the H-space multiplication m on S 7 . By a celebrated theorem of Adams [Ada1960], the only spheres that admit an H-space structure are S 0 , S 1 , S 3 and S 7 . However, the next theorem by Wallace shows that S 7 does not admit an associative multiplication: Theorem 4. [Wal1953, Corollary 2] If a compact manifold M admits a continuous, associative multiplication with identity, then it is a topological group.

In fact, combining the above theorem with von Neumann's solution of Hilbert's Fifth problem for compact groups [vN1933] would imply that such M has actually a Lie group structure. A stronger result in this direction was proved by James:

Theorem 5. [Jam1957, Theorem 1.4] There exists no homotopy-associative multiplication on S n unless n = 1 or 3.

As we have already seen in (5), the non-associativity of the octonions causes the non-integrability of the almost complex structure induced on S 6 by them. We would like to relate the (likely) non-existence of a complex structure on S 6 with the lack of associative -or more generally homotopy-associativemultiplications on S 7 . Question 2. Does the integrability of the almost complex structure J imply the (homotopy-)associativity of the multiplications m and m?

Working with the multiplication m instead of m has the advantage of involving the additive structure of R 8 . As observed in Remark 7, we employed the alternativity of the octonions in establishing the connection between the Nijenhuis tensor and the associator. It would be reasonable to expect the answer to Question 2 to involve a similar requirement. In this regard, it is interesting to mention the following result, by Norman: Theorem 6. [Nor1963, Corollary 9.3] Any multiplication on a sphere satisfies the Moufang law, up to homotopy:

(x • y) (z • x) = (x (y • z)) x.
Remark 24. In case J be the almost complex structure of S 6 induced by the octonions, Question 1 is related to Question 2. This happens because J comes a priori from an ambient multiplication of R 8 , in other words, if we interpret J as a map J : S 6 -→ SO(8, R), x -→ J x it extends as a linear map between R 8 and SO(8, R).

Remark 25. However, in the general case of an almost complex structure on S 6 , Questions 1 and 2 stand at different levels and are not immediately linked. Question 1 asks whether it is possible to integrate the parallelism, i.e., if there exists a multiplication the differential of which essentially induces the global frame. Question 2 asks directly if the multiplication induced by the global frame is (homotopy-)associative. This opens the possibility to combine analytic-homotopic tools, such as geometric flows, to study the problem in the same vein of Remark 22.

Acknowledgements:

The authors are grateful to the members of the MathAmSud collaboration Geometric Structures and Moduli Spaces (21-Math-06), for several stimulating discussions during the Workshop at Universidad Nacional de Córdoba in August-September 2022. We also thank Eduardo Toffolo for the picture in §3.

All three authors were supported by a MathAmSud grant (2021-2023), from the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) -Finance Code 001 [88881.520221/2020-01]. They are also members of the Fapesp-ANR [2021/04065-6] BRIDGES binational collaboration (2022-2026). HSE has been partially funded by the Brazilian National Council for Scientific and Technological Development (CNPq) Productivity Grant 1D [307217/2017-5].