A note on Kirchhoff's theorem for almost complex spheres I

Abstract

By a theorem of Kirchhoff if the six sphere admits an almost complex structure then the seven sphere is parallelizable, more crucial, he exhibited an explicit global frame constructed out of the given almost complex structure. This result implicitly equips the seven sphere with a definite H-space multiplication. We propose to address the existence problem of complex structures on the six sphere studying the associated parallelism-multiplications on the seven sphere. We ask to what extent the integrability condition of the almost complex structure amounts to the constancy of the structure functions of the global frame defining the parallelism, i.e, if this parallelism comes from a Lie group structure. At a more fundamental level we inquire if the integrability condition of the almost complex structure entails the homotopy associativity of the induced multiplication. A positive answer to these questions would rule out the six sphere of being a complex manifold since the seven sphere is not a Lie group, not even a homotopy associative H-space.