• Title, Summary, Keyword: vector bundle

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Nontrivial Complex Equivariant Vector Bundles over $S^1$ (원 위에서의 Nontrivial Complex Equivariant Vector Bundle)

  • Kim, Sung-Sook
    • The Journal of Natural Sciences
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    • v.10 no.1
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    • pp.13-16
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    • 1998
  • Every complex vector bundle over $S^1$ splits sum of line bundle and the first Chern class classify complex line bundle. This implies every complex vector bundle over $S^1$ is trivial. In this paper, we show the existence of some nontrivial complex vector bundle over $S^1$ in the equivariant case.

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Equivariant vector bundle structures on real line bundles

  • Shu, Dong-Youp
    • Communications of the Korean Mathematical Society
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    • v.11 no.1
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    • pp.259-263
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    • 1996
  • Let G be a topological group and X a G space. For a given nonequivariant vector bundle over X there does not always exist a G equivariant vector bundle structure. In this paper we find some sufficient conditions for nonequivariant real line bundles to have G equivariant vector bundle structures.

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ANTI-LINEAR INVOLUTIONS ON A G-VECTOR BUNDLE

  • Kim, Sung-Sook;Shin, Joon-Kook
    • Communications of the Korean Mathematical Society
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    • v.14 no.1
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    • pp.211-216
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    • 1999
  • We study the anti-linear involutions on a real algebraic vector bundle with bundle with a compact real algebraic group action.

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DEFINABLE Cr FIBER BUNDLES AND DEFINABLE CrG VECTOR BUNDLES

  • Kawakami, Tomohiro
    • Communications of the Korean Mathematical Society
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    • v.23 no.2
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    • pp.257-268
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    • 2008
  • Let G and K be compact subgroups of orthogonal groups and $0{\leq}r<x<{\infty}$. We prove that every topological fiber bundle over a definable $C^r$ manifold whose structure group is K admits a unique strongly definable $C^r$ fiber bundle structure up to definable $C^r$ fiber bundle isomorphism. We prove that every G vector bundle over an affine definable $C^rG$ manifold admits a unique strongly definable $C^rG$ vector bundle structure up to definable $C^rG$ vector bundle isomorphism.

SECOND ORDER TANGENT VECTORS IN RIEMANNIAN GEOMETRY

  • Kwon, Soon-Hak
    • Journal of the Korean Mathematical Society
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    • v.36 no.5
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    • pp.959-1008
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    • 1999
  • This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

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UNIFORMITY OF HOLOMORPHIC VECTOR BUNDLES ON INFINITE-DIMENSIONAL FLAG MANIFOLDS

  • Ballico, E.
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.85-89
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    • 2003
  • Let V be a localizing infinite-dimensional complex Banach space. Let X be a flag manifold of finite flags either of finite codimensional closed linear subspaces of V or of finite dimensional linear subspaces of V. Let E be a holomorphic vector bundle on X with finite rank. Here we prove that E is uniform, i.e. that for any two lines $D_1$ R in the same system of lines on X the vector bundles E$\mid$D and E$\mid$R have the same splitting type.

LIMITS OF TRIVIAL BUNDLES ON CURVES

  • Ballico, Edoardo
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.43-61
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    • 2020
  • We extend the work of A. Beauville on rank 2 vector bundles on a smooth curve in several directions. We give families of examples with large dimension, add new existence and non-existence results and prove the existence of indecomposable limits with arbitrary rank. To construct the large dimensional families we use the examples of limits of rank 2 trivial bundles on ℙ2 and ℙ3 due to C. Banica. We also consider a more flexible notion: limits of trivial bundles on nearby curves.

ON CONFORMALLY FLAT UNIT VECTOR BUNDLES

  • Bang, Keumseong
    • Korean Journal of Mathematics
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    • v.6 no.2
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    • pp.303-311
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    • 1998
  • We study the conformally flat unit vector bundle $E_1$ of constant scalar curvature for the bundle ${\pi}:E^{n+2}{\rightarrow}M^n$ over an Einstein manifold M.

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