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 , the principal bundle of linear frames over an arbitrary even-dimensional smooth manifold with a given linear connection. Complexifying connections are those which induce a complex structure on . For two-dimensional manifolds, every linear connection is of this kind.
In the special case where 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 and the torsion of an almost complex connection over .
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- -, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969. MR 38 #6501.
- S. Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, Berlin and New York, 1972. MR 0355886 (50:8360)
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Keywords: Almost complex structures, complex manifolds, linear connections, complexifying connections, almost complex connections
Article copyright: © Copyright 1975 American Mathematical Society