### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 034, 21 pages      arXiv:1610.09620      https://doi.org/10.3842/SIGMA.2018.034

### Results Concerning Almost Complex Structures on the Six-Sphere

Scott O. Wilson
Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd., Queens, NY 11367, USA

Received November 20, 2017, in final form April 09, 2018; Published online April 17, 2018

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.

Key words: six-sphere; almost complex; integrable.

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