The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

Mariusz Plaszczyk

Annales UMCS, Mathematica (2015)

  • Volume: 69, Issue: 1, page 91-108
  • ISSN: 2083-7402

Abstract

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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 → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM 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

How to cite

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Mariusz Plaszczyk. "The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds." Annales UMCS, Mathematica 69.1 (2015): 91-108. <http://eudml.org/doc/270870>.

@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 → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM 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 UMCS, Mathematica},
keywords = {and phrases Riemannian manifold; higher order prolongation of a vector bundle; natural tensor; natural operator; Riemannian manifold},
language = {eng},
number = {1},
pages = {91-108},
title = {The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds},
url = {http://eudml.org/doc/270870},
volume = {69},
year = {2015},
}

TY - JOUR
AU - Mariusz Plaszczyk
TI - The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds
JO - Annales UMCS, Mathematica
PY - 2015
VL - 69
IS - 1
SP - 91
EP - 108
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 → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM 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 - and phrases Riemannian manifold; higher order prolongation of a vector bundle; natural tensor; natural operator; Riemannian manifold
UR - http://eudml.org/doc/270870
ER -

References

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  1. [1] Epstein, D. B. A., Natural tensors on Riemannian manifolds, J. Differential Geom. 10 (1975), 631-645. Zbl0321.53039
  2. [2] Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, Vol. I, J. Wiley- Interscience, New York-London, 1963. Zbl0119.37502
  3. [3] Kolář, I., Connections on higher order frame bundles and their gauge analogies, Variations, Geometry and Physics, Nova Sci. Publ., New York, 2009, 167-188. Zbl1208.58003
  4. [4] Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. Zbl0782.53013
  5. [5] 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. Zbl1312.58003
  6. [6] 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. Zbl1299.53070
  7. [7] Mikulski, W. M., Lifting connections to the 

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