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On the Vector in Homogeneous Spaces

Published online by Cambridge University Press:  22 January 2016

Minoru Kurita
Minoru Kurita*
Affiliation:
Mathematical Institute, Nagoya University
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The main purpose of this paper is to investigate the parallelism of vectors in homogeneous spaces. The definition of a vector and the condition for spaces under which a covariant differential of a vector is also a vector were given by E. Cartan [4] in a very intuitive way. Here I formulate this in a stricter way by his method of moving frame. Even if a homogeneous space has the property that the covariant differential of a vector is of the same kind, another definition of covariant differential may also have the required property. I will give a necessary and sufficient condition under which the definition of covariant differential is unique. Once the covariant differential has been defined it is easy to introduce a parallelism of vectors in the space. But the parallelism depends in general on the path along which we translate a vector. The condition for the spaces with an absolute parallelism can be obtained.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 5 , February 1953 , pp. 1 - 33
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

[1] Blaschke, W.. Integralgeometrie 1 (Actualités scientifique et industrielles. 1935.)Google Scholar
[2] Cartan, E.. Sur les invariants intégraux de certains espaces homogènes clos et les propriétés topologique de ces espaces. (Selecta 1937.)Google Scholar
[3] Cartan, E.. La théorie des groupes finis et continus et la géométrie differential par la méthode du repères mobiles. (Gauthier Villar 1937.)Google Scholar
[4] Cartan, E.. L’extension du calcul tensoriel aux géométrie non affines. (Annals of Mathematics Vol. 38, 1937.)Google Scholar
[5] Cartan, E.. Leçon sur la géométrie des espaces de Riemann. (Gauthier Villar 1946.)Google Scholar