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

Abstract

top
There is a class of metrics on the tangent bundle T M 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 T M . 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 T M from some quadratic forms on O M × ℝ m to find metrics (not necessary Riemannian) on T M , we prove that all g -natural metrics on T M can be obtained by Musso-Tricerri’s generalized scheme. We calculate also the Levi-Civita connection of Riemannian g -natural metrics on T M . As application, we sort out all Riemannian g -natural metrics with the following properties, respectively: 1) The fibers of T M are totally geodesic. 2) The geodesic flow on T M is incompressible. We shall limit ourselves to the non-oriented situation.

How to cite

top

Abbassi, 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 -

References

top
  1. Abbassi K. M. T., Note on the classification Theorems of g -natural metrics on the tangent bundle of a Riemannian manifold ( M , g ) , Comment. Math. Univ. Carolin. 45 4 (2004), 591–596. Zbl1097.53013MR2103077
  2. Abbassi K. M. T., Sarih M., On the differential geometry of the tangent and the tangent sphere bundles with Cheeger-Gromoll metric, preprint. 
  3. Abbassi K. M. T., Sarih M., Killing vector fields on tangent bundles with Cheeger-Gromoll metric, Tsukuba J. Math. 27 (2) (2003), 295–306. Zbl1060.53019MR2025729
  4. Abbassi K. M. T., Sarih M., The Levi-Civita connection of Riemannian natural metrics on the tangent bundle of an oriented Riemannian manifold, preprint. 
  5. Abbassi K. M. T., Sarih M., On Riemannian g -natural metrics of the form a · g s + b · g h + c · g v on the tangent bundle of a Riemannian manifold ( M , g ) , to appear in Mediter. J. Math. 
  6. Besse A. L., Manifolds all of whose geodesics are closed, Ergeb. Math. (93), Springer-Verlag, Berlin, Heidelberg, New York 1978. (1978) Zbl0387.53010MR0496885
  7. Borisenko A. A., Yampol’skii A. L., Riemannian geometry of fiber bundles, Russian Math. Surveys 46 (6) (1991), 55–106. (1991) MR1164201
  8. Cheeger J., Gromoll D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. (1972) Zbl0246.53049MR0309010
  9. Dombrowski P., On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73–82. (1962) Zbl0105.16002MR0141050
  10. Epstein D. B. A., Natural tensors on Riemannian manifolds, J. Differential Geom. 10 (1975), 631–645. (1975) Zbl0321.53039MR0415531
  11. Epstein D. B. A., Thurston W. P., Transformation groups and natural bundles, Proc. London Math. Soc. 38 (1979), 219–236. (1979) Zbl0409.58001MR0531161
  12. Kobayashi S., Nomizu K., Foundations of differential geometry, Intersci. Pub. New York (I, 1963 and II, 1967). (1963) Zbl0119.37502MR0152974
  13. Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin 1993. (1993) Zbl0782.53013MR1202431
  14. Kowalski O., Curvature of the induced Riemannian metric of the tangent bundle of Riemannian manifold, J. Reine Angew. Math. 250 (1971), 124–129. (1971) MR0286028
  15. 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
  16. Kowalski O., Sekizawa M., On tangent sphere bundles with small or large constant radius, Ann. Global Anal. Geom. 18 (2000), 207–219. Zbl1011.53025MR1795094
  17. Krupka D., Janyška J., Lectures on Differential Invariants, University J. E. Purkyně, Brno 1990. (1990) MR1108622
  18. Musso E., Tricerri F., Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. (4) 150 (1988), 1–20. (1988) Zbl0658.53045MR0946027
  19. Nijenhuis A., Natural bundles and their general properties, in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972, 317–334. (1972) Zbl0246.53018MR0380862
  20. Palais R. S., Terng C. L., Natural bundles have finite order, Topology 16 (1977), 271–277. (1977) Zbl0359.58004MR0467787
  21. 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). (1958) Zbl0086.15003MR0112152
  22. Sekizawa M., Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14 (2) (1991), 407–417. (1991) Zbl0768.53020MR1138176
  23. Slovák J., On natural connections on Riemannian manifolds, Comment. Math. Univ. Carolin. 30 (1989), 389–393. (1989) Zbl0679.53025MR1014139
  24. Stredder P., Natural differential operators on Riemannian manifolds and representations of the orthogonal and the special orthogonal groups, J. Differential Geom. 10 (1975), 647–660. (1975) MR0415692
  25. Terng C. L., Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978), 775–828. (1978) Zbl0422.58001MR0509074
  26. Willmore T. J., An introduction to differential geometry, Oxford Univ. Press 1959. (1959) Zbl0086.14401MR0159265
  27. Yano K., Ishihara S., Tangent and cotangent bundles, Differential Geometry, Marcel Dekker Inc. New York 1973. (1973) Zbl0262.53024MR0350650
  28. Yano K., Kobayashi S., Prolongations of tensor fields and connections to tangent bundles, J. Math. Soc. Japan (I, II, 18, (2–3) (1966), III, 19 (1967)). (1966) Zbl0147.21501

Citations in EuDML Documents

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

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.