| Title:
|
On natural metrics on tangent bundles of Riemannian manifolds (English) |
| Author:
|
Abbassi, Mohamed Tahar Kadaoui |
| Author:
|
Sarih, Maâti |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
41 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
71-92 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Riemannian manifold |
| Keyword:
|
tangent bundle |
| Keyword:
|
natural operation |
| Keyword:
|
$g$-natural metric |
| Keyword:
|
Geodesic flow |
| Keyword:
|
incompressibility |
| MSC:
|
53A55 |
| MSC:
|
53B20 |
| MSC:
|
53C07 |
| MSC:
|
53C20 |
| MSC:
|
53D25 |
| idZBL:
|
Zbl 1114.53015 |
| idMR:
|
MR2142144 |
| . |
| Date available:
|
2008-06-06T22:45:10Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107936 |
| . |
| Reference:
|
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