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

Mariusz Plaszczyk


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.


Riemannian manifold; higher order prolongation of a vector bundle; natural tensor; natural operator

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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.

Data publikacji: 2015-11-30 09:21:12
Data złożenia artykułu: 2015-09-03 12:56:40


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