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http://dx.doi.org/10.4134/BKMS.b150897

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

Cho, Jong Taek (Department of Mathematics Chonnam National University)
Chun, Sun Hyang (Department of Mathematics Chosun University)
Publication Information
Bulletin of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1715-1723 More about this Journal
Abstract
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.
Keywords
unit tangent sphere bundle; contact metric structure; characteristic Jacobi operator;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 Y. Nikolayevsky, On Osserman manifolds of dimension 16, Contemporary geometry and related topics, 379-398, Univ. Belgrade Fac. Math., Belgrade, 2006.
2 L. Vanhecke and T. J. Willmore, Interactions of tubes and spheres, Math. Anal. 21 (1983), no. 1, 31-42.
3 K. Yano and S. Ishihara, Tangent and Cotangent Bundles, M. Dekker Inc., 1973.
4 D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Second edition, Progr. Math. 203, Birkhauser Boston, Inc., Boston, MA, 2010.
5 E. Boeckx, J. T. Cho, and S. H. Chun, Flow-invariant structures on unit tangent bundles, Publ. Math. Debrecen 70 (2007), no. 1-2, 167-178.
6 E. Boeckx and L. Vanhecke, Characteristic reflections on unit tangent sphere bundles, Houston J. Math. 23 (1997), no. 3, 427-448.
7 J. T. Cho and S. H. Chun, On the classification of contact Riemannian manifolds satisfying the condition (C), Glasg. Math. J. 45 (2003), no. 3, 475-492.   DOI
8 J. T. Cho and J.-I. Inoguchi, Pseudo-symmetric contact 3-manifolds, J. Korean Math. Soc. 42 (2005), no. 5, 913-932.   DOI
9 P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73-88.
10 P. Gilkey, A. Swann, and L. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 183, 299-320.   DOI
11 A. Gray, Classification des varietes approximativement kahleriennes de courbure sectionelle holomorphe constante, J. Reine Angew. Math. 279 (1974), 797-800.
12 O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math. 250 (1971), 124-129.
13 Y. Nikolayevsky, Osserman manifolds of dimension 8, Manuscr. Math. 115 (2004), no. 1, 31-53.   DOI
14 Y. Nikolayevsky, Osserman conjecture in dimension n ${\neq}$ 8, 16, Math. Ann. 331 (2005), no. 3, 505-522.   DOI