Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T06:54:11.132Z Has data issue: false hasContentIssue false
  • English
  • Français

Prolongations of G-Structures to Tangent Bundles

Published online by Cambridge University Press:  22 January 2016

Akihiko Morimoto
Akihiko Morimoto*
Affiliation:
Mathematical Institute, Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of the present paper is to study the prolongations of G-structures on a manifold M to its tangent bundle T(M), G being a Lie subgroup of GL(n,R) with n = dim M. Recently, K. Yano and S. Kobayashi [9] studied the prolongations of tensor fields on M to T(M) and they proposed the following question: Is it possible to associate with each G-structure on M a naturally induced G-structure on T(M), where G′ is a certain subgroup of GL(2n,R)? In this paper we give an answer to this question and we shall show that the prolongations of some special tensor fields by Yano-Kobayashi — for instance, the prolongations of almost complex structures — are derived naturally by our prolongations of the classical G-structures. On the other hand, S. Sasaki [5] studied a prolongation of Riemannian metrics on M to a Riemannian metric on T(M), while the prolongation of a (positive definite) Riemannian metric due to Yano-Kobayashi is always pseudo-Riemannian on T(M) but never Riemannian. We shall clarify the circumstances for this difference and give the reason why the one is positive definite Riemannian and the other is not.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 67 - 108
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Bernard, D.: Sur la géométrie différentielle des G-structures, Ann. Inst. Fourier, 10 (1960), 151–270.CrossRefGoogle Scholar
[2] Chern, S.S.: The geometry of G-structures, Bull. Amer. Math. Soc., 72 (1966), 167–219.Google Scholar
[3] Gray, J.W.: Some global properties of contact structures, Ann. of Math., 69 (1959), 421–450.Google Scholar
[4] Kobayashi, S.: Theory of connections, Ann. Mat. Pura Appl., 43 (1957), 119–194.Google Scholar
[5] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J., 10 (1958), 338–354.Google Scholar
[6] Hatakeyama, S. Sasaki-Y.: On differentiate manifolds with contact structure, J. Math. Soc. Japan, 14 (1962), 249–271.Google Scholar
[7] Steenrod, N.: The topology of fibre bundles, Princeton, 1951.CrossRefGoogle Scholar
[8] Sternberg, S.: Lectures on differential geometry, Englewood Cliffs, 1964.Google Scholar
[9] Kobayashi, K. Yano-S.: Prolongations of tensor fields and connections to tangent bundles I, J. Math. Soc. Japan, 18 (1966), 194–210.Google Scholar