Results Concerning Almost Complex Structures on the Six-Sphere

TL;DR

This paper proves the non-existence of certain almost complex structures on the six-sphere under specific differential conditions, using novel techniques like an almost-complex Gauss map.

Contribution

It introduces new non-existence results for almost complex structures on the six-sphere, employing refined differential inequalities and an almost-complex Gauss map.

Findings

No almost complex structure J satisfies the commuting condition with its covariant derivative.

No integrable J satisfies the condition $ abla_{JX} J = J abla_X J$ on the six-sphere.

Refined differential inequalities characterize the non-existence results.

Abstract

For the standard metric on the six-dimensional sphere, with Levi-Civita connection $\nabla$, we show there is no almost complex structure $J$ such that $\nabla_X J$ and $\nabla_{JX} J$ commute for every $X$, nor is there any integrable $J$ such that $\nabla_{JX} J = J \nabla_X J$ for every $X$. The latter statement generalizes a previously known result on the non-existence of integrable orthogonal almost complex structures on the six-sphere. Both statements have refined versions, expressed as intrinsic first order differential inequalities depending only on $J$ and the metric. The new techniques employed include an almost-complex analogue of the Gauss map, defined for any almost complex manifold in Euclidean space.