Results Concerning Almost Complex Structures on the Six-Sphere
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209538 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Results Concerning Almost Complex Structures on the Six-Sphere / S.O. Wilson // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862715253786148864 |
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| author | Wilson, S.O. |
| author_facet | Wilson, S.O. |
| citation_txt | Results Concerning Almost Complex Structures on the Six-Sphere / S.O. Wilson // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T17:56:54Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209538 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:56:54Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wilson, S.O. 2025-11-24T10:46:47Z 2018 Results Concerning Almost Complex Structures on the Six-Sphere / S.O. Wilson // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C15; 32Q60; 53A07 arXiv: 1610.09620 https://nasplib.isofts.kiev.ua/handle/123456789/209538 https://doi.org/10.3842/SIGMA.2018.034 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. I gratefully acknowledge Queens College's sabbatical/fellowship leave program, which provided me with time to conduct some of this research. I thank Arthur Parzygnat for comments on a preliminary version of this paper, and also thank the referees for their suggestions, which have improved this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Results Concerning Almost Complex Structures on the Six-Sphere Article published earlier |
| spellingShingle | Results Concerning Almost Complex Structures on the Six-Sphere Wilson, S.O. |
| title | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_full | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_fullStr | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_full_unstemmed | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_short | Results Concerning Almost Complex Structures on the Six-Sphere |
| title_sort | results concerning almost complex structures on the six-sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209538 |
| work_keys_str_mv | AT wilsonso resultsconcerningalmostcomplexstructuresonthesixsphere |