The six-dimensional sphere S6\mathbb{S}^6 is one of only two spheres admitting almost complex structures (Borel–Serre), carrying a canonical nearly Kähler one induced by octonion multiplication; yet whether it admits an integrable complex structure — the classical Hopf problem — remains open, and H2(S6,R)=0H^2(\mathbb{S}^6,\mathbb{R})=0 rules out symplectic structures. Etayo, Gómez-Nicolás and Santamaría posed the Hopf problem in generalized geometry: does S6\mathbb{S}^6 admit any integrable generalized complex structure on its generalized tangent bundle TS6=TS6⊕T∗S6\mathbb{T}\mathbb{S}^6=T\mathbb{S}^6\oplus T^*\mathbb{S}^6? They proved that no such structure can be built as a pointwise spherical combination of the structures induced by the octonionic nearly Kähler data, and an unrefereed 2026 preprint by Ricciarini excludes constant types 1 and 2 via an Euler-class obstruction (constant type 0 being the classical H2=0H^2=0 obstruction to symplectic structures), together with gluing constructions from locally flat models. What survives untouched is the full existence question: the constant type-3 (complex-type) case is equivalent to the classical Hopf problem, and type-changing structures — the genuinely new possibility opened by generalized geometry — are subject to no known obstruction or construction on S6\mathbb{S}^6. The problem asks for either an explicit integrable structure of any admissible type configuration, or a proof that none exists.

Background

An almost complex structure on an even-dimensional manifold MM is a bundle endomorphism J ⁣:TM→TMJ\colon TM\to TM with J2=−IdJ^2=-\mathrm{Id}; it is integrable when it arises from holomorphic coordinates, which by the Newlander–Nirenberg theorem is equivalent to the vanishing of a certain tensor built from JJ, the Nijenhuis tensor. A question going back to Hopf in the 1940s (see the historical account of Agricola et al.) asks whether S6\mathbb{S}^6 admits an integrable almost complex structure — the classical Hopf problem. Borel and Serre proved in 1953 that, among all spheres, only S2\mathbb{S}^2 and S6\mathbb{S}^6 admit almost complex structures at all. The sphere S6\mathbb{S}^6 carries a canonical one, induced by multiplication of imaginary octonions (the octonions are the eight-dimensional normed division algebra, S6\mathbb{S}^6 sits inside its seven-dimensional imaginary part, and the exceptional Lie group G2G_2 is the automorphism group of the octonions); this structure makes S6\mathbb{S}^6 nearly Kähler — an almost Hermitian structure (J,g)(J,g) whose fundamental 2-form ω(⋅,⋅)=g(J⋅,⋅)\omega(\cdot,\cdot)=g(J\cdot,\cdot) has totally skew covariant derivative — but its Nijenhuis tensor is famously nonzero, and Blanchard, in a proof popularized by LeBrun, showed that no complex structure on S6\mathbb{S}^6 can be orthogonal with respect to the round metric. Since H2(S6,R)=0H^2(\mathbb{S}^6,\mathbb{R})=0, the sphere also admits no symplectic structure (a closed nondegenerate 2-form), so neither of the two classical integrable geometries is available on S6\mathbb{S}^6.

Generalized geometry, introduced by Hitchin and developed by Gualtieri, replaces TMTM by the generalized tangent bundle TM=TM⊕T∗M\mathbb{T}M=TM\oplus T^*M, endowed with the canonical indefinite pairing G0(X+ξ,Y+η)=12(ξ(Y)+η(X))\mathcal{G}_0(X+\xi,Y+\eta)=\frac{1}{2}(\xi(Y)+\eta(X)), which is symmetric of signature (n,n)(n,n) when dim⁡M=2n\dim M=2n, and with the Dorfman bracket ⟦X+ξ,Y+η⟧=[X,Y]+LXη−iYdξ\llbracket X+\xi,Y+\eta\rrbracket=[X,Y]+\mathcal{L}_X\eta-i_Yd\xi, a natural bilinear bracket on sections of TM\mathbb{T}M extending the Lie bracket of vector fields (LX\mathcal{L}_X denotes the Lie derivative and iYi_Y the contraction with YY). A generalized almost complex structure is an endomorphism J ⁣:TM→TM\mathcal{J}\colon\mathbb{T}M\to\mathbb{T}M with J2=−Id\mathcal{J}^2=-\mathrm{Id}; following the source authors, it is called strong when it is skew-adjoint for the pairing, G0(Ju,v)=−G0(u,Jv)\mathcal{G}_0(\mathcal{J}u,v)=-\mathcal{G}_0(u,\mathcal{J}v) for all sections u,vu,v (equivalently, since J2=−Id\mathcal{J}^2=-\mathrm{Id}, G0(Ju,Jv)=G0(u,v)\mathcal{G}_0(\mathcal{J}u,\mathcal{J}v)=\mathcal{G}_0(u,v)), and weak otherwise. It is integrable when its +i+i-eigenbundle LJ1,0={u−iJu ⁣:u∈Γ(TM)}\mathbb{L}^{1,0}_{\mathcal{J}}=\{u-i\mathcal{J}u\colon u\in\Gamma(\mathbb{T}M)\} is closed under the Dorfman bracket, equivalently when the associated generalized Nijenhuis map vanishes identically; for isotropic subbundles, closure under the Dorfman bracket is equivalent to closure under its skew-symmetrization (the Courant bracket), so the two integrability notions agree for strong structures. Integrable generalized complex structures simultaneously generalize complex and symplectic structures: at each point the structure has a type kk, with 0≤k≤n=dim⁡M/20\le k\le n=\dim M/2, given by the dimension of the intersection of the +i+i-eigenbundle with the complexified cotangent subspace, and on the regular locus the structure is locally the product of a complex kk-dimensional and a symplectic 2(n−k)2(n-k)-dimensional model. The local normal-form theory of Bailey and Gualtieri shows the type may jump along loci, but only by even amounts, giving type-changing integrable structures that are genuinely neither complex nor symplectic.

From an almost Hermitian triple (J,g,ω)(J,g,\omega), Gualtieri's constructions induce generalized almost complex structures Jλ,J(X+ξ)=JX+λJ∗ξ\mathcal{J}_{\lambda,J}(X+\xi)=JX+\lambda J^*\xi for λ∈{1,−1}\lambda\in\{1,-1\} (where J∗J^* is the dual action on covectors), the metric structure Jg\mathcal{J}_g, and the 2-form structure Jω\mathcal{J}_\omega (the latter two built from gg and ω\omega through the musical isomorphisms); Gualtieri proved that J−1,J\mathcal{J}_{-1,J} is integrable exactly when JJ is integrable, and Jω\mathcal{J}_\omega exactly when ω\omega is symplectic. Etayo, Gómez-Nicolás and Santamaría posed the Hopf problem in generalized geometry and proved that S6\mathbb{S}^6 admits no weak generalized complex structure expressible as a pointwise spherical combination aJ1,J+bJg+cJωa\mathcal{J}_{1,J}+b\mathcal{J}_g+c\mathcal{J}_\omega with smooth functions a,b,ca,b,c satisfying a2+b2+c2=1a^2+b^2+c^2=1. Ricciarini subsequently excluded, in an unrefereed preprint, constant types 1 and 2 by an Euler-class argument, and structures glued from locally flat models over the stereographic cover, while observing that the constant type-3 case is exactly the classical Hopf problem. The existence question in its full generality — the subject of this problem — remains open.

Problem Statement

Does the six-dimensional sphere S6\mathbb{S}^6, with its standard smooth structure, admit a generalized complex structure — that is, a strong (pairing-compatible) generalized almost complex endomorphism J\mathcal{J} of TS6=TS6⊕T∗S6\mathbb{T}\mathbb{S}^6=T\mathbb{S}^6\oplus T^*\mathbb{S}^6, satisfying J2=−Id\mathcal{J}^2=-\mathrm{Id} and G0(Ju,v)=−G0(u,Jv)\mathcal{G}_0(\mathcal{J}u,v)=-\mathcal{G}_0(u,\mathcal{J}v) for all sections u,vu,v (equivalently G0(Ju,Jv)=G0(u,v)\mathcal{G}_0(\mathcal{J}u,\mathcal{J}v)=\mathcal{G}_0(u,v)), whose +i+i-eigenbundle LJ1,0={u−iJu}\mathbb{L}^{1,0}_{\mathcal{J}}=\{u-i\mathcal{J}u\} is closed under the Dorfman bracket (equivalently, whose generalized Nijenhuis map vanishes identically)?

(1) Existence: exhibit such a structure explicitly, of any type configuration, and prove its integrability on all of S6\mathbb{S}^6. Two subcases are qualitatively distinct: a constant type-3 (complex-type) structure, which amounts to a positive solution of the classical Hopf problem since such a structure is a BB-field transform of an integrable complex structure; or a type-changing structure, which would be the first integrable structure on S6\mathbb{S}^6 that is neither complex nor symplectic.

(2) Non-existence: prove that no integrable strong generalized complex structure of any type configuration — constant type 3 or type-changing — exists on S6\mathbb{S}^6. Such a proof must in particular resolve the classical Hopf problem negatively, because a constant type-3 integrable structure on S6\mathbb{S}^6 is equivalent to an integrable complex structure on S6\mathbb{S}^6.

For scope: constant type 0 is excluded classically, since H2(S6,R)=0H^2(\mathbb{S}^6,\mathbb{R})=0 precludes symplectic structures, and an unrefereed 2026 preprint excludes constant types 1 and 2 by an Euler-class argument; neither exclusion touches the two branches above. Since a strong structure is in particular a weak generalized complex structure in the source authors' convention (dropping the G0\mathcal{G}_0-compatibility requirement), any exclusion covering the weak class would also cover the strong class.

References

  1. Fernando Etayo, Pablo Gómez-Nicolás, Rafael Santamaría, "About generalized complex structures on S6\mathbb{S}^6", Mediterranean Journal of Mathematics 22 (2025), no. 5, article 138. DOI: 10.1007/s00009-025-02916-0; arXiv:2405.05681. https://arxiv.org/abs/2405.05681
  2. Andrea Ricciarini, "On the integrability of generalized almost complex structures on S6\mathbb{S}^6", preprint (2026, unrefereed). arXiv:2606.03379. https://arxiv.org/abs/2606.03379
  3. Marco Gualtieri, "Generalized complex geometry", Annals of Mathematics (2) 174 (2011), 75–123. DOI: 10.4007/annals.2011.174.1.3. https://doi.org/10.4007/annals.2011.174.1.3
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