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The complex structure on the six dimensional sphere

Gabor Etesi

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.

The complex structure on the six dimensional sphere

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.

Paper Structure

This paper contains 4 sections, 5 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Proof of existence
  3. Explicit construction
  4. Appendix: inner automorphisms and $\pi_6({\rm G}_2)\cong{\mathbb Z}_3$

Key Result

Theorem 2.1

Take the family ${\mathfrak s}_u\subset{\mathfrak g}_2^{\mathbb C}$ of Samelson subalgebras with $u\in P({\mathfrak h}^{\mathbb C} )\setminus P({\mathfrak h})$ and consider the induced compact complex $7$-manifolds $Y_u$ which are all diffeomorphic to ${\rm G}_2\subset{\rm G}_2^{\mathbb C}$. Then fo

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof