On the integrability of generalized almost complex structures on $\mathbb{S}^6$
We study integrability of generalized almost complex structures on the six-dimensional sphere $\mathbb{S}^6$. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection $[,]_\nabla$ (in particular the Levi-Civita connection), and the Courant integrability for strong generalized almost complex structures. After recalling the necessary background on the generalized tangent bundle and on spherical combinations of the canonical generalized structures determined by an almost Hermitian triple $(J,g,ω)$, we derive local coordinate criteria for $[,]_\nabla$-integrability of weak generalized structures. Applying these formulae to the nearly Kähler structure on S^6 induced by the octonionic product, we prove that no nontrivial spherical combinations $J=aJ_{1,J} + bJ_g + cJ_ω$ with smooth coefficients such that $a^2+b^2+c^2=1$ (except $J_g$) is integrable with respect to $[,]_{\nabla^{LC}}$. We then turn to Courant integrability: we give sufficient local conditions for Courant integrability of strong generalized almost complex structures, prove a gluing result for local Courant algebroids and b-field transforms, and use it to exhibit obstruction results characterizing the impossibility of constructing, via certain gluing procedures, a Courant integrable strong generalized almost complex structures on $\mathbb{S}^6$.