| Title:
|
Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$ (English) |
| Author:
|
Abbassi, Mohamed Tahar Kadaoui |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
4 |
| Year:
|
2004 |
| Pages:
|
591-596 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In [7], it is proved that all $g$-natural metrics on tangent bundles of $m$-dimen\-sional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on $m$ and on the assumption that the base manifold is oriented, or non-oriented, respectively. The result was originally stated in [8] for the oriented case, but the smoothness was assumed and not explicitly proved. In this note, we shall prove that, both in the oriented and non-oriented cases, the functions generating the $g$-natural metrics are, in fact, smooth on the set of all nonnegative real numbers. (English) |
| Keyword:
|
Riemannian manifold |
| Keyword:
|
tangent bundle |
| Keyword:
|
natural operation |
| Keyword:
|
$g$-natural metric |
| Keyword:
|
curvatures |
| MSC:
|
53A55 |
| MSC:
|
53C07 |
| MSC:
|
58A32 |
| idZBL:
|
Zbl 1097.53013 |
| idMR:
|
MR2103077 |
| . |
| Date available:
|
2009-05-05T16:47:47Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119487 |
| . |
| Reference:
|
[1] Abbassi K.M.T., Sarih M.: On natural metrics on tangent bundles of Riemannian manifolds.to appear in Arch. Math. (Brno). Zbl 1114.53015, MR 2142144 |
| Reference:
|
[2] Abbassi K.M.T., Sarih M.: The Levi-Civita connection of Riemannian natural metrics on the tangent bundle of an oriented Riemannian manifold.preprint. |
| Reference:
|
[3] Abbassi K.M.T., Sarih M.: On some hereditary properties of Riemannian $g$-natural metrics on tangent bundles of Riemannian manifolds.to appear in Differential Geom. Appl. (2004). Zbl 1068.53016, MR 2106375 |
| Reference:
|
[4] Calvo M. del C., Keilhauer G.G.R: Tensor fields of type $(0,2)$ on the tangent bundle of a Riemannian manifold.Geom. Dedicata 71 (2) (1998), 209-219. MR 1629795 |
| Reference:
|
[5] Dombrowski P.: On the geometry of the tangent bundle.J. Reine Angew. Math. 210 (1962), 73-82. Zbl 0105.16002, MR 0141050 |
| Reference:
|
[6] Kobayashi S., Nomizu K.: Foundations of Differential Geometry.Interscience Publishers, New York (I, 1963 and II, 1967). Zbl 0526.53001, MR 0152974 |
| Reference:
|
[7] Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry.Springer, Berlin, 1993. MR 1202431 |
| Reference:
|
[8] Kowalski O., Sekizawa M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles - a classification.Bull. Tokyo Gakugei Univ. (4) 40 (1988), 1-29. Zbl 0656.53021, MR 0974641 |
| Reference:
|
[9] Krupka D., Janyška J.: Lectures on Differential Invariants.University J.E. Purkyně, Brno, 1990. MR 1108622 |
| Reference:
|
[10] Nijenhuis A.: Natural bundles and their general properties.in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp.317-334. Zbl 0246.53018, MR 0380862 |
| Reference:
|
[11] Sasaki S.: On the differential geometry of tangent bundles of Riemannian manifolds.Tohôku Math. J., I, 10 (1958), 338-354 II, 14 (1962), 146-155. Zbl 0109.40505, MR 0112152 |
| Reference:
|
[12] Willmore T.J.: An Introduction to Differential Geometry.Oxford Univ. Press, Oxford, 1959. Zbl 0086.14401, MR 0159265 |
| Reference:
|
[13] Yano K., Ishihara S.: Tangent and Cotangent Bundles: Differential Geometry.Marcel Dekker Inc., New York, 1973. Zbl 0262.53024, MR 0350650 |
| . |
