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HORIZONTAL SUBSPACES IN THE BUNDLE OF LINEAR FRAMES

  • Park, Joon-Sik
  • Received : 2012.08.15
  • Accepted : 2012.09.07
  • Published : 2012.12.25

Abstract

Let L(M) be the bundle of all linear frames over a smooth manifold M, $u$ an arbitrarily given point of L(M), and ${\nabla}:\mathfrak{X}(M){\times}\mathfrak{X}(M){\rightarrow}\mathfrak{X}(M)$ a linear connection on M. Then the following result is well known: the horizontal subspace at the point $u$ may be written in terms of local coordinates of $u{\in}L(M)$ and Christoel's symbols defined by ${\nabla}$. This result is very fundamental on the study of the theory of connections. In this paper we show that the local expression of the horizontal subspace at the point u does not depend on the choice of a local coordinate system around the point $u{\in}L(M)$, which is rarely seen.

Keywords

bundle of linear frames;connection;horizontal subspace

References

  1. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
  2. S. Kobayashi and K. Nomizu, Foundation of Differential Geometry, Vol.I, Wiley-Interscience, New York, 1963.
  3. I. Mogi and M. Itoh, Differential Geometry and Gauge Theory (in Japanese), Kyoritsu Publ., 1986.
  4. Walter A. Poor, Differential Geometric Structures, McGraw-Hill, Inc., 1081.

Acknowledgement

Supported by : Busan University of Foreign Studies