Generalized Cheeger-Gromoll metrics and the Hopf map |
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Authors: | Michele Benyounes Seiki Nishikawa |
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Affiliation: | a Département de Mathématiques, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France b Mathematical Institute, Tohoku University, Sendai 980-8578, Japan |
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Abstract: | We show that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle, equipped with the induced metric, onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented. |
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Keywords: | primary, 53C07 secondary, 55R25 |
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