On the completeness of total spaces of horizontally conformal submersions

Mohamed Tahar Kadaoui Abbassi; Ibrahim Lakrini

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 493-504
  • ISSN: 1804-1388

Abstract

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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 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.

How to cite

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Abbassi, Mohamed Tahar Kadaoui, and Lakrini, Ibrahim. "On the completeness of total spaces of horizontally conformal submersions." Communications in Mathematics 29.3 (2021): 493-504. <http://eudml.org/doc/297760>.

@article{Abbassi2021,
abstract = {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 $g$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.},
author = {Abbassi, Mohamed Tahar Kadaoui, Lakrini, Ibrahim},
journal = {Communications in Mathematics},
keywords = {Vector bundle; spherically symmetric metric; complete Riemannian metric; complete metric space; Hopf-Rinow theorem},
language = {eng},
number = {3},
pages = {493-504},
publisher = {University of Ostrava},
title = {On the completeness of total spaces of horizontally conformal submersions},
url = {http://eudml.org/doc/297760},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Abbassi, Mohamed Tahar Kadaoui
AU - Lakrini, Ibrahim
TI - On the completeness of total spaces of horizontally conformal submersions
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 493
EP - 504
AB - 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 $g$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.
LA - eng
KW - Vector bundle; spherically symmetric metric; complete Riemannian metric; complete metric space; Hopf-Rinow theorem
UR - http://eudml.org/doc/297760
ER -

References

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