The complex structure on the six dimensional sphere

TL;DR

This paper proves the existence of a complex structure on the six-dimensional sphere and explicitly computes its integrable almost complex tensor using octonion automorphisms, revealing a complex and perplexing tensor field.

Contribution

It provides a new proof of the complex structure on the six-sphere and explicitly computes its tensor using octonion automorphisms, which is a novel and detailed approach.

Findings

Existence of a complex structure on the six-sphere confirmed.

Explicit computation of the integrable almost complex tensor achieved.

The resulting tensor field is complex and intricate.

Abstract

Proof of existence of a complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tensor field on the six-sphere is perplexing.