Linear connections and almost complex structures
Proc. Amer. Math. Soc. 49 (1975), 59-65
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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 .
Kobayashi and Katsumi
Nomizu, Foundations of differential geometry. Vol I,
Interscience Publishers, a division of John Wiley & Sons, New
York-London, 1963. MR 0152974
-, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969. MR 38 #6501.
Kobayashi, Transformation groups in differential geometry,
Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und
ihrer Grenzgebiete, Band 70. MR 0355886
- S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. MR 27 #2945. MR 0152974 (27:2945)
- -, 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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