  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.
