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Harmonic morphisms and Riemannian geometry of tangent bundles

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Abstract

Let (TM, G) and \({(T_1 M,\tilde G)}\) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and \({\pi_1:(T_1 M,\tilde G) \rightarrow (M,g)}\), we give necessary and sufficient conditions for π and π 1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection \({\Phi:(TM-\{0\},G)\to (T_1 M,\tilde G)}\) with respect to Riemannian g-natural metrics \({G,\tilde G}\) of Kaluza–Klein type.

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Authors and Affiliations

  1. Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Lecce, Italy

    Giovanni Calvaruso & Domenico Perrone

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  1. Giovanni Calvaruso
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  2. Domenico Perrone
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Correspondence to Giovanni Calvaruso.

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Calvaruso, G., Perrone, D. Harmonic morphisms and Riemannian geometry of tangent bundles. Ann Glob Anal Geom 39, 187–213 (2011). https://doi.org/10.1007/s10455-010-9230-4

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  • DOI: https://doi.org/10.1007/s10455-010-9230-4

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