# Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$

Commentationes Mathematicae Universitatis Carolinae (2004)

- Volume: 45, Issue: 4, page 591-596
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topAbbassi, Mohamed Tahar Kadaoui. "Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$." Commentationes Mathematicae Universitatis Carolinae 45.4 (2004): 591-596. <http://eudml.org/doc/249337>.

@article{Abbassi2004,

abstract = {In [7], it is proved that all $g$-natural metrics on tangent bundles of $m$-dimensional 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.},

author = {Abbassi, Mohamed Tahar Kadaoui},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; curvatures; Riemannian manifold; tangent bundle; natural operation; -natural metric; curvatures},

language = {eng},

number = {4},

pages = {591-596},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$},

url = {http://eudml.org/doc/249337},

volume = {45},

year = {2004},

}

TY - JOUR

AU - Abbassi, Mohamed Tahar Kadaoui

TI - Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2004

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 45

IS - 4

SP - 591

EP - 596

AB - In [7], it is proved that all $g$-natural metrics on tangent bundles of $m$-dimensional 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.

LA - eng

KW - Riemannian manifold; tangent bundle; natural operation; $g$-natural metric; curvatures; Riemannian manifold; tangent bundle; natural operation; -natural metric; curvatures

UR - http://eudml.org/doc/249337

ER -

## References

top- Abbassi K.M.T., Sarih M., On natural metrics on tangent bundles of Riemannian manifolds, to appear in Arch. Math. (Brno). Zbl1114.53015MR2142144
- Abbassi K.M.T., Sarih M., The Levi-Civita connection of Riemannian natural metrics on the tangent bundle of an oriented Riemannian manifold, preprint.
- 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). Zbl1068.53016MR2106375
- 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. (1998) MR1629795
- Dombrowski P., On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73-82. (1962) Zbl0105.16002MR0141050
- Kobayashi S., Nomizu K., Foundations of Differential Geometry, Interscience Publishers, New York (I, 1963 and II, 1967). Zbl0526.53001MR0152974
- Kolář I., Michor P.W., Slovák J., Natural Operations in Differential Geometry, Springer, Berlin, 1993. MR1202431
- 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. (1988) Zbl0656.53021MR0974641
- Krupka D., Janyška J., Lectures on Differential Invariants, University J.E. Purkyně, Brno, 1990. MR1108622
- Nijenhuis A., Natural bundles and their general properties, in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp.317-334. Zbl0246.53018MR0380862
- 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. (1962) Zbl0109.40505MR0112152
- Willmore T.J., An Introduction to Differential Geometry, Oxford Univ. Press, Oxford, 1959. Zbl0086.14401MR0159265
- Yano K., Ishihara S., Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker Inc., New York, 1973. Zbl0262.53024MR0350650

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.