About generalized complex structures on ${\mathbb{S}}^6$

Abstract

We study the existence of generalized complex structures on the six-dimensional sphere ${\mathbb{S}}^6$. We work with the generalized tangent bundle $\mathbb{T}{\mathbb{S}}^6\to {\mathbb{S}}^6$ and define the integrability of generalized geometric structures in terms of the Dorfman bracket. Specifically, we prove that there is not a direct way to induce a generalized complex structure on ${\mathbb{S}}^6$ from its usual nearly Kähler structure inherited from the octonions product.