The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2015)
- Volume: 69, Issue: 1
- ISSN: 0365-1029
Access Full Article
top
Abstract
topHow to cite
topMariusz Plaszczyk. "The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 69.1 (2015): null. <http://eudml.org/doc/289836>.
@article{MariuszPlaszczyk2015,
abstract = {If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrT*M between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.},
author = {Mariusz Plaszczyk},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Riemannian manifold; higher order prolongation of a vector bundle; natural tensor; natural operator},
language = {eng},
number = {1},
pages = {null},
title = {The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds},
url = {http://eudml.org/doc/289836},
volume = {69},
year = {2015},
}
TY - JOUR
AU - Mariusz Plaszczyk
TI - The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2015
VL - 69
IS - 1
SP - null
AB - If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrT*M between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.
LA - eng
KW - Riemannian manifold; higher order prolongation of a vector bundle; natural tensor; natural operator
UR - http://eudml.org/doc/289836
ER -
References
top- Epstein, D. B. A., Natural tensors on Riemannian manifolds, J. Differential Geom. 10 (1975), 631–645.
- Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, Vol. I, J. Wiley- Interscience, New York–London, 1963.
- Kolar, I., Connections on higher order frame bundles and their gauge analogies, Variations, Geometry and Physics, Nova Sci. Publ., New York, 2009, 167–188.
- Kolar, I., Michor, P. W., Slovak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.
- Kurek, J., Mikulski, W. M., The natural transformations between r-tangent and rcotangent bundles over Riemannian manifolds, Ann. Univ. Mariae Curie-Skłodowska Sect. A 68 (2) (2014), 59–64.
- Kurek, J., Mikulski, W. M., The natural operators lifting connections to tensor powers of the cotangent bundle, Miskolc Mathematical Notes 14, No. 2 (2013), 517–524.
- Mikulski, W. M., Lifting connections to the r-jet prolongation of the cotangent bundle, Math. Appl. (Brno) 3 (2014), 115–124.
NotesEmbed ?
topYou 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.