• Title, Summary, Keyword: bundle of linear frames

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HORIZONTAL SUBSPACES IN THE BUNDLE OF LINEAR FRAMES

  • Park, Joon-Sik
    • Honam Mathematical Journal
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    • v.34 no.4
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    • pp.513-517
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    • 2012
  • 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.

GENERALIZED AFFINE DEVELOPMENTS

  • Park, Joon-Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.1
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    • pp.65-72
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    • 2015
  • The (affine) development of a smooth curve in a smooth manifold M with respect to an arbitrarily given affine connection in the bundle of affine frames over M is well known (cf. S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, Vol.1). In this paper, we get the generalized affine development of a smooth curve $x_t$ ($t{\in}[0,1]$) in M into the affine tangent space at $x_0$ (${\in}M$) with respect to a given generalized affine connection in the bundle of affine frames over M.

TORSION TENSOR FORMS ON INDUCED BUNDLES

  • Kim, Hyun Woong;Park, Joon-Sik;Pyo, Yong-Soo
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.4
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    • pp.793-798
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    • 2013
  • Let ${\phi}$ be a map of a manifold M into another manifold N, L(N) the bundle of all linear frames over N, and ${\phi}^{-1}$(L(N)) the bundle over M which is induced from ${\phi}$ and L(N). Then, we construct a structure equation for the torsion form in ${\phi}^{-1}$(L(N)) which is induced from a torsion form in L(N).