Results Concerning Almost Complex Structures on the Six-Sphere

Abstract

For the standard metric on the six-dimensional sphere, with Levi-Civita connection ∇, we show there is no almost complex structure J such that ∇XJ and ∇JXJ commute for every X, nor is there any integrable J such that ∇JXJ = J∇XJ 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.