On jet like bundles of vector bundles. (English) Zbl 1538.58002
Summary: We describe completely the so called jet like functors of a vector bundle \(E\) over an \(m\)-dimensional manifold \(M\), i.e. bundles \(FE\) over \(M\) canonically depending on \(E\) such that \(F(E_1\times_M E_2)=FE_1\times_MFE_2\) for any vector bundles \(E_1\) and \(E_2\) over \(M\). Then we study how a linear vector field on \(E\) can induce canonically a vector field on \(FE\).
MSC:
| 58A05 | Differentiable manifolds, foundations |
| 58A20 | Jets in global analysis |
| 58A32 | Natural bundles |
Keywords:
bundle functor; gauge bundle functor; natural transformation; (gauge) natural operator; vector bundle; module bundle; jetReferences:
| [1] | M. Doupovec, I. Kol´aˇr:Iteration of fiber product preserving bundle functors, Monatsh. Math. 134(2001), 39-50. · Zbl 0999.58001 |
| [2] | M. Doupovec, J. Kurek, W. M. Mikulski:Vertical functors on fibered manifolds, Differ. Geom. Appl.54(2017), 100-110. · Zbl 1375.58001 |
| [3] | C. Ehresmann:Extension du calcul des jets non holonomes, C. R. A. S. Paris239(1954), 1762-1764. · Zbl 0057.15603 |
| [4] | I. Kol´aˇr, P. W. Michor, J. Slov´ak:Natural Operations in Differential Geometry, Springer Verlag, 1993. · Zbl 0782.53013 |
| [5] | I. Kol´aˇr, W. M. Mikulski:On the fiber product preserving bundle functors, Differ. Geom. Appl. 11(1999), 105-115. · Zbl 0935.58001 |
| [6] | D. Krupka:Lagrange theory in fibered manifolds, Rep. Math. Phys.2(1971), 121-133. · Zbl 0219.49022 |
| [7] | J. Kurek, W. M. Mikulski:Lifting of linear vector fields to fiber product preserving vertical gauge bundles, Colloq. Math.103(2005), 33-40. · Zbl 1086.58002 |
| [8] | P. Libermann:Introduction to the theory of semi-holonomic jets, Arch. Math. (Brno)33 (1997), 173-189. · Zbl 0915.58004 |
| [9] | W. M. Mikulski:On the fiber product preserving gauge bundle functors on vector bundles, Ann. Polon. Math.82(2003), 251-264. · Zbl 1126.58300 |
| [10] | W. M. Mikulski:The natural operators lifting linear vector fields from a vector bundle into itsrth prolongations, Ann. Polon. Math.82(2003), 155-170. · Zbl 1083.58003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
