About Generalized Complex Structures on $$\mathbb {S}^6$$

Abstract We study the existence of generalized complex structures on the six-dimensional sphere $$\mathbb {S}^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:math> . We work with the generalized tangent bundle $$\mathbb {T}\mathbb {S}^6\rightarrow \mathbb {S}^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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 $$\mathbb {S}^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:math> from its usual nearly Kähler structure inherited from the octonions product.