Open AccessThe natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds
Author(s) -
Mariusz Plaszczyk
Publication year - 2015
Publication title -
annales universitatis mariae curie-skłodowska. sectio a, mathematica
Language(s) - English
Resource type - Journals
eISSN - 2083-7402
pISSN - 0365-1029
DOI - 10.17951/a.2015.69.1.91
Subject(s) - tangent bundle , order (exchange) , cotangent bundle , riemannian manifold , base (topology) , mathematics , isomorphism (crystallography) , combinatorics , physics , trigonometric functions , pure mathematics , mathematical analysis , tangent space , geometry , crystallography , chemistry , crystal structure , finance , economics
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 J r TM → J r T*M between the r-th order prolongation J r TM of tangent TM and the r-th order prolongation J r T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D M (g) : J r TM → J r T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.