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  • $\begingroup$ I ask you to write this answer in terms of the tangent bundle $T_M$, its vector bundle of endomorphisms $End(T_M)$, possibly using the Serre-Swan theorem and the module of smooth sections of $T_M$. $\endgroup$
    – user122276
    Commented Apr 15, 2021 at 13:10
  • 8
    $\begingroup$ From your comments, my impression is that my description of an example is not to your taste. I have never seen an almost complex structure described using the Serre-Swan theorem or in terms of vectors fields as forming a module, so I will need some idea of how you prefer to write almost complex structures if I am to provide a clearer description. Perhaps you can give an example of almost complex structures from your preferred perspective. $\endgroup$
    – Ben McKay
    Commented Apr 15, 2021 at 15:40
  • $\begingroup$ as a differential geometer you are used to using local coordinates, with background in algebra/geometry I use sheaves of functions and sheaves of sections of vector bundles. Hence an almost complex structure in my language is an endomorphism $J$ of the tangent bundle (which is a projective $R$-module of rank $2$) with square $=-Id$. If in a local trivialization $J$ is defined using "rational functions", $J$ is algebraic. If $J$ is locally defined using smooth functions it is smooth. $\endgroup$
    – user122276
    Commented Apr 15, 2021 at 17:27
  • $\begingroup$ The real Lie group $SU(3)$ is 8-dimensional and its real tangent bundle is trivial of rank 8: $T_{SU(3)}\cong SU(3) \times \mathbb{R}^8$. It seems to me that $\Omega_{\epsilon}$ can not be inverted as you claim - it is a map from a real Lie group to a 6-dimensional real vector space. Is there a mistake? $\endgroup$
    – user122276
    Commented Apr 16, 2021 at 9:47
  • $\begingroup$ Right: it is 8-dimensional. Fixed $\endgroup$
    – Ben McKay
    Commented Apr 16, 2021 at 9:50