# The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek; Włodzimierz M. Mikulski

Annales UMCS, Mathematica (2015)

- Volume: 68, Issue: 2, page 59-64
- ISSN: 2083-7402

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topJan Kurek, and Włodzimierz M. Mikulski. "The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds." Annales UMCS, Mathematica 68.2 (2015): 59-64. <http://eudml.org/doc/270006>.

@article{JanKurek2015,

abstract = {If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) between the tangent TM and the cotangent T∗ M bundles of M. In the present note, we generalize this isomorphism to the one T(r)M ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.},

author = {Jan Kurek, Włodzimierz M. Mikulski},

journal = {Annales UMCS, Mathematica},

keywords = {Riemannian manifold; higher order vector tangent bundle; higher order cotangent bundle; natural tensor; natural operator},

language = {eng},

number = {2},

pages = {59-64},

title = {The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds},

url = {http://eudml.org/doc/270006},

volume = {68},

year = {2015},

}

TY - JOUR

AU - Jan Kurek

AU - Włodzimierz M. Mikulski

TI - The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

JO - Annales UMCS, Mathematica

PY - 2015

VL - 68

IS - 2

SP - 59

EP - 64

AB - If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) between the tangent TM and the cotangent T∗ M bundles of M. In the present note, we generalize this isomorphism to the one T(r)M ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.

LA - eng

KW - Riemannian manifold; higher order vector tangent bundle; higher order cotangent bundle; natural tensor; natural operator

UR - http://eudml.org/doc/270006

ER -

## References

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- [6] Mikulski, W. M., Some natural operators on vector fields, Rend Math. Appl (7) 12, no. 3 (1992), 783-803. Zbl0766.58005
- [7] Nijenhuis, A., Natural bundles and their general properties Diff. Geom. in Honor of K. Yano, Kinokuniya, Tokyo (1972), 317-334.
- [8] Paluszny, M., Zajtz, A., Foundation of the Geometry of Natural Bundles, Lect. Notes Univ. Caracas, 1984.

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