On the integrability of generalized almost complex structures on 𝕊⁶

@misc{ricciarini2026integrability,
  title = {{On the integrability of generalized almost complex structures on $\mathbbS^6$}},
  author = {Andrea Ricciarini},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.03379},
  eprint = {2606.03379},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.03379},
}

TY  - UNPB
TI  - On the integrability of generalized almost complex structures on 𝕊^6
AU  - Andrea Ricciarini
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2606.03379
UR  - https://arxiv.org/abs/2606.03379
AB  - We study integrability of generalized almost complex structures on the six-dimensional sphere 𝕊^6. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection [· ,· ]_∇ (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 [· ,· ]_∇ -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) are integrable with respect to [· ,· ]_∇ ^LC. We then turn to Courant integrability: we give local characterizations 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 obstructions to constructing, via certain gluing procedures, smooth strong generalized complex structures on 𝕊^6.
ER  -