Linear almost complex structures on \(S^ 6\) (Q920414)

As an approach to the unsolved problem whether the sphere \(S^ 6\) admits a complex structure, the author utilizes the following two notions: the twistor space \(Z(S^ 6)\) defined on \(S^ 6\) by the fiber bundle \(P(S^ 6,SO(6))/U(n)\), and a normed 8-dimensional R-algebra (with multiplicative unit and equipped with an inner product such as it is available in the Cayley algebra). Constructing the standard almost complex structure on \(S^ 6\), the author finds a one-to-one correspondence between \(R^ 8\)- algebra and linear sections of \(Z(S^ 6)\), so that these sections are all equivalent by Hurwitz's theorem. This equivalence is proved here to be completely described by the action of spin(7) over \(S^ 7\).