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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209538 |
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| Zitieren: | 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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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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Results Concerning Almost Complex Structures on the Six-Sphere |
| spellingShingle |
Results Concerning Almost Complex Structures on the Six-Sphere Wilson, S.O. |
| title_short |
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_sort |
results concerning almost complex structures on the six-sphere |
| author |
Wilson, S.O. |
| author_facet |
Wilson, S.O. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209538 |
| 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. |
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AT wilsonso resultsconcerningalmostcomplexstructuresonthesixsphere |
| first_indexed |
2025-12-07T17:56:54Z |
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2025-12-07T17:56:54Z |
| _version_ |
1850886083317858304 |