Results Concerning Almost Complex Structures on the Six-Sphere

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.