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

TL;DR

This paper explores the relationship between almost complex structures on the six sphere and the induced algebraic structures on the seven sphere, questioning whether integrability implies Lie group properties.

Contribution

It investigates the link between integrability of almost complex structures on S^6 and the algebraic properties of the associated parallelism on S^7, proposing new fundamental questions.

Findings

Connects almost complex structures on S^6 with parallelism on S^7.

Raises questions about whether integrability implies Lie group structure.

Suggests that positive answers would imply S^6 is not a complex manifold.

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.