• 제목/요약/키워드: characteristic Jacobi operator

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THE UNIT TANGENT SPHERE BUNDLE WHOSE CHARACTERISTIC JACOBI OPERATOR IS PSEUDO-PARALLEL

  • Cho, Jong Taek;Chun, Sun Hyang
    • 대한수학회보
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    • 제53권6호
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    • pp.1715-1723
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    • 2016
  • We study the characteristic Jacobi operator ${\ell}={\bar{R}({\cdot},{\xi}){\xi}$ (along the Reeb flow ${\xi}$) on the unit tangent sphere bundle $T_1M$ over a Riemannian manifold ($M^n$, g). We prove that if ${\ell}$ is pseudo-parallel, i.e., ${\bar{R}{\cdot}{\ell}=L{\mathcal{Q}}({\bar{g}},{\ell})$, by a non-positive function L, then M is locally flat. Moreover, when L is a constant and $n{\neq}16$, M is of constant curvature 0 or 1.

REEB FLOW INVARIANT UNIT TANGENT SPHERE BUNDLES

  • Cho, Jong Taek;Chun, Sun Hyang
    • 호남수학학술지
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    • 제36권4호
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    • pp.805-812
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    • 2014
  • For unit tangent sphere bundles $T_1M$ with the standard contact metric structure (${\eta},\bar{g},{\phi},{\xi}$), we have two fundamental operators that is, $h=\frac{1}{2}{\pounds}_{\xi}{\phi}$ and ${\ell}=\bar{R}({\cdot},{\xi}){\xi}$, where ${\pounds}_{\xi}$ denotes Lie differentiation for the Reeb vector field ${\xi}$ and $\bar{R}$ denotes the Riemmannian curvature tensor of $T_1M$. In this paper, we study the Reeb ow invariancy of the corresponding (0, 2)-tensor fields H and L of h and ${\ell}$, respectively.