<p>We study the existence of generalized complex structures on the six-dimensional sphere <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2916_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {S}^6\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation>. We work with the generalized tangent bundle <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2916_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {T}\mathbb {S}^6\rightarrow \mathbb {S}^6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>6</mn> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>6</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2916_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {S}^6\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation> from its usual nearly Kähler structure inherited from the octonions product.</p>

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

  • Fernando Etayo,
  • Pablo Gómez-Nicolás,
  • Rafael Santamaría

摘要

We study the existence of generalized complex structures on the six-dimensional sphere \(\mathbb {S}^6\) S 6 . We work with the generalized tangent bundle \(\mathbb {T}\mathbb {S}^6\rightarrow \mathbb {S}^6\) T S 6 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 \(\mathbb {S}^6\) S 6 from its usual nearly Kähler structure inherited from the octonions product.