On natural metrics on tangent bundles of Riemannian manifolds

Mohamed Tahar Kadaoui Abbassi; Maâti Sarih

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 1, page 71-92
  • ISSN: 0044-8753

Abstract

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There is a class of metrics on the tangent bundle T M of a Riemannian manifold ( M , g ) (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric g [Kow-Sek1]. We call them “ g -natural metrics" on T M . To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on T M from some quadratic forms on O M × m to find metrics (not necessary Riemannian) on T M , we prove that all g -natural metrics on T M can be obtained by Musso-Tricerri’s generalized scheme. We calculate also the Levi-Civita connection of Riemannian g -natural metrics on T M . As application, we sort out all Riemannian g -natural metrics with the following properties, respectively: 1) The fibers of T M are totally geodesic. 2) The geodesic flow on T M is incompressible. We shall limit ourselves to the non-oriented situation.

How to cite

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Abbassi, Mohamed Tahar Kadaoui, and Sarih, Maâti. "On natural metrics on tangent bundles of Riemannian manifolds." Archivum Mathematicum 041.1 (2005): 71-92. <http://eudml.org/doc/249513>.

@article{Abbassi2005,
abstract = {There is a class of metrics on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric $g$ [Kow-Sek1]. We call them “$g$-natural metrics" on $TM$. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on $TM$ from some quadratic forms on $OM \times \mathbb \{R\}^m$ to find metrics (not necessary Riemannian) on $TM$, we prove that all $g$-natural metrics on $TM$ can be obtained by Musso-Tricerri’s generalized scheme. We calculate also the Levi-Civita connection of Riemannian $g$-natural metrics on $TM$. As application, we sort out all Riemannian $g$-natural metrics with the following properties, respectively: 1) The fibers of $TM$ are totally geodesic. 2) The geodesic flow on $TM$ is incompressible. We shall limit ourselves to the non-oriented situation.},
author = {Abbassi, Mohamed Tahar Kadaoui, Sarih, Maâti},
journal = {Archivum Mathematicum},
keywords = {Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; Geodesic flow; incompressibility; natural operation; -natural metric; geodesic flow},
language = {eng},
number = {1},
pages = {71-92},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On natural metrics on tangent bundles of Riemannian manifolds},
url = {http://eudml.org/doc/249513},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Abbassi, Mohamed Tahar Kadaoui
AU - Sarih, Maâti
TI - On natural metrics on tangent bundles of Riemannian manifolds
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 1
SP - 71
EP - 92
AB - There is a class of metrics on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric $g$ [Kow-Sek1]. We call them “$g$-natural metrics" on $TM$. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on $TM$ from some quadratic forms on $OM \times \mathbb {R}^m$ to find metrics (not necessary Riemannian) on $TM$, we prove that all $g$-natural metrics on $TM$ can be obtained by Musso-Tricerri’s generalized scheme. We calculate also the Levi-Civita connection of Riemannian $g$-natural metrics on $TM$. As application, we sort out all Riemannian $g$-natural metrics with the following properties, respectively: 1) The fibers of $TM$ are totally geodesic. 2) The geodesic flow on $TM$ is incompressible. We shall limit ourselves to the non-oriented situation.
LA - eng
KW - Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; Geodesic flow; incompressibility; natural operation; -natural metric; geodesic flow
UR - http://eudml.org/doc/249513
ER -

References

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Citations in EuDML Documents

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  1. Mohamed Tahar Kadaoui Abbassi, Giovanni Calvaruso, Domenico Perrone, Some examples of harmonic maps for g -natural metrics
  2. M. T. K. Abbassi, Giovanni Calvaruso, g -natural metrics of constant curvature on unit tangent sphere bundles
  3. Mohamed Tahar Kadaoui Abbassi, Note on the classification theorems of g -natural metrics on the tangent bundle of a Riemannian manifold ( M , g )
  4. Abderrahim Zagane, Mustapha Djaa, Geometry of Mus-Sasaki metric
  5. Kamil Niedziałomski, On the geometry of frame bundles

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