# 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

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topAbbassi, 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 -

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## Citations in EuDML Documents

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

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