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Tensor Fields and Their Parallelism

Published online by Cambridge University Press:  22 January 2016

Minoru Kurita
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Much has been studied about an almost complex structure these ten years. One of the problems about the structure is to find an affine connection which makes a given almost complex tensor field parallel. A Riemannian connection is a one without torsion for which the fundamental tensor field of a Riemannian manifold is parallel. Affine connections on the group manifold were investigated fully by E. Cartan in [1]. In this paper we treat in general some tensor fields and affine connections which make the fields parallel. Moreover some studies about certain tensor fields are given.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 133 - 151
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

References

[1] Cartan, E.: La géométrie des groupes de transformation. Jour. Math, pures et appliquées t. 6 (1927), pp. 1–119.Google Scholar
[2] Chern, S. S.: On integral geometry in Klein spaces. Ann. of Math. vol. 43, (1942), pp. 178–189.CrossRefGoogle Scholar
[3] Chern, S. S.: Pseudogroupes continus infinis. Colloq. de la géométrie différentielles, Strasbourg (1953), pp. 119–136.Google Scholar
[4] Frölicher, A.: Zur Differentialgeometrie der komplexen Strukturen, Math. Ann. Bd. 129 (1955), s. 50–95.CrossRefGoogle Scholar
[5] Frölicher, A. and Nijenhuis, A.: Theory of vector-valued differential forms, Part 1. Indag. Math. vol. 18 (1956), pp. 338–359.CrossRefGoogle Scholar
[6] Willmore, T. J.: Connexions for systems of parallel distributions Quart. Jour, of Math. Oxford., vol. 7 (1955), pp. 269–276.CrossRefGoogle Scholar
[7] Kurita, M.: On the vector in homogeneous spaces. Nagoya Math. Jour. vol. 5 (1953), pp. 1–33.CrossRefGoogle Scholar
[8] Kurita, M.: On the volume in homogeneous spaces. Nagoya Math. Jour. vol. 15 (1959), pp. 201–217.CrossRefGoogle Scholar
[9] Newlander, A. and Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. of Math., vol. 65 (1957), pp. 391–404.CrossRefGoogle Scholar
[10] Nijenhuis, A.: Xu-1 -forming sets of eigenvectors. Indag. Math., vol. 13 (1951), pp. 200–212.CrossRefGoogle Scholar
[11] Nomizu, K.: Invariant affine connections in homogeneous spaces. Amer. Math. Jour., vol. 76 (1954), pp. 33–56.CrossRefGoogle Scholar
[12] Schouten, J. A.: Ricci·Calculus. Springer (1954).Google Scholar
[13] Walker, A. G.: Connexions for parallel distributions in the large. Quart. Jour, of Math. Oxford, vol. 6 (1955), pp. 301–308.CrossRefGoogle Scholar
[14] Yano, K.: Quelques remarques sur les variétés à structure presque complexe. Bull. Soc. Math. France, t. 83 (1955), pp. 57–80.Google Scholar
[15] Yano, K.: On Walker differentiation in almost product or almost complex spaces. Indag. Math., vol. 20 (1958), pp. 573–580.CrossRefGoogle Scholar
[16] Yano, K.: Affine connexions in an almost product space. Kȯdal Math. Reports, vol. 11 (1959) pp. 1–24.Google Scholar