Geodesic
On the completeness of total spaces of horizontally conformal submersions
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 493-504.

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In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of $g$\HH natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.
Classification : 53C07, 53C24, 53C25
Keywords: Vector bundle; spherically symmetric metric; complete Riemannian metric; complete metric space; Hopf-Rinow theorem. 
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     title = {On the completeness of total spaces of horizontally conformal submersions},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a12/}
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Abbassi, Mohamed Tahar Kadaoui; Lakrini, Ibrahim. On the completeness of total spaces of horizontally conformal submersions. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 493-504. http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a12/