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

Mohamed Tahar Kadaoui Abbassi

Commentationes Mathematicae Universitatis Carolinae (2004)

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

Abstract

top
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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.