Linear connections and almost complex structures
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- by Jean-Marc Terrier PDF
- Proc. Amer. Math. Soc. 49 (1975), 59-65 Request permission
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
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- Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 59-65
- MSC: Primary 53C15
- DOI: https://doi.org/10.1090/S0002-9939-1975-0380664-7
- MathSciNet review: 0380664