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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear connections and almost complex structures


Author: Jean-Marc Terrier
Journal: Proc. Amer. Math. Soc. 49 (1975), 59-65
MSC: Primary 53C15
MathSciNet review: 0380664
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Abstract: An almost complex structure is defined on $ P$, the principal bundle of linear frames over an arbitrary even-dimensional smooth manifold $ M$ with a given linear connection. Complexifying connections are those which induce a complex structure on $ P$. For two-dimensional manifolds, every linear connection is of this kind.

In the special case where $ M$ itself is an almost complex manifold, a relationship between the two almost complex structures is found and provides a very simple proof of the fact that the existence of an almost complex connection without torsion implies the integrability of the given almost complex structure. As a second application, we give a geometrical interpretation of an identity between the torsion of an almost complex structure on $ M$ and the torsion of an almost complex connection over $ M$.


References [Enhancements On Off] (What's this?)

  • [1] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974 (27 #2945)
  • [2] -, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969. MR 38 #6501.
  • [3] Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 0355886 (50 #8360)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0380664-7
PII: S 0002-9939(1975)0380664-7
Keywords: Almost complex structures, complex manifolds, linear connections, complexifying connections, almost complex connections
Article copyright: © Copyright 1975 American Mathematical Society