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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References

  1. Blanchard A., Recherche de structures analytiques complexes sur certaines variétés, C. R. Acad. Sci. Paris 236 (1953), 657-659.
  2. Bor G., Hernández-Lamoneda L., The canonical bundle of a Hermitian manifold, Bol. Soc. Mat. Mexicana 5 (1999), 187-198.
  3. Borel A., Serre J.P., Détermination des $p$-puissances réduites de Steenrod dans la cohomologie des groupes classiques. Applications, C. R. Acad. Sci. Paris 233 (1951), 680-682.
  4. Bryant R., S.-S. Chern's study of almost-complex structures on the six-sphere, arXiv:1405.3405.
  5. Hopf H., Zur Topologie der komplexen Mannigfaltigkeiten, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, 167-185.
  6. Karoubi M., Leruste C., Algebraic topology via differential geometry, London Mathematical Society Lecture Note Series, Vol. 99, Cambridge University Press, Cambridge, 1987.
  7. LeBrun C., Orthogonal complex structures on $S^6$, Proc. Amer. Math. Soc. 101 (1987), 136-138.
  8. McDuff D., Salamon D., $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, Vol. 52, 2nd ed., Amer. Math. Soc., Providence, RI, 2012.
  9. Milnor J.W., Stasheff J.D., Characteristic classes, Annals of Mathematics Studies, Vol. 76, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1974.
  10. Newlander A., Nirenberg L., Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391-404.
  11. Salamon S.M., Hermitian geometry, in Invitations to Geometry and Topology, Oxford Graduate Texts in Mathematics, Vol. 7, Oxford University Press, Oxford, 2002, 233-291.
  12. Tang Z., Curvature and integrability of an almost Hermitian structure, Internat. J. Math. 17 (2006), 97-105, .

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