| Title:
|
Flow prolongation of some tangent valued forms (English) |
| Author:
|
Cabras, Antonella |
| Author:
|
Kolář, Ivan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
493-504 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the prolongation of semibasic projectable tangent valued $k$-forms on fibered manifolds with respect to a bundle functor $F$ on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear $r$-th order connection on the base manifold, where $r$ is the base order of $F$. We find a general condition under which the Frölicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued $k$-forms in the case $F$ is a fiber product preserving bundle functor on the category of fibered manifolds with $m$-dimensional bases and local diffeomorphisms as base maps. (English) |
| Keyword:
|
semibasic tangent valued $k$-form |
| Keyword:
|
Frölicher-Nijenhuis bracket |
| Keyword:
|
bundle functor |
| Keyword:
|
flow prolongation of vector fields |
| Keyword:
|
connection |
| Keyword:
|
curvature |
| MSC:
|
53C05 |
| MSC:
|
58A20 |
| idZBL:
|
Zbl 1174.53312 |
| idMR:
|
MR2411105 |
| . |
| Date available:
|
2009-09-24T11:56:28Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128273 |
| . |
| Reference:
|
[1] A. Cabras and I. Kolář: Prolongation of tangent valued forms to Weil bundles.Archivum Math. (Brno) 31 (1995), 139–145. MR 1357981 |
| Reference:
|
[2] M. Doupovec and I. Kolář: Iteration of fiber product preserving bundle functors.Monatsh Math. 134 (2001), 39–50. MR 1872045, 10.1007/s006050170010 |
| Reference:
|
[3] J. Gancarzewicz, W. Mikulski and Z. Pogoda: Lifts of some tensor fields and connections to product preserving bundles.Nagoya Math. J. 135 (1994), 1–41. MR 1295815, 10.1017/S0027763000004931 |
| Reference:
|
[4] I. Kolář: On the geometry of fiber product preserving bundle functors.Proceedings of 8th ICDGA, Silesian University, Opava (2002), 85–92. MR 1978765 |
| Reference:
|
[5] I. Kolář: On the flow prolongation of vector fields.Proceedings of Geometric Seminar, Kazan 24 (2003), 69–80. |
| Reference:
|
[6] I. Kolář, P. W. Michor and J. Slovák: Natural Operations in Differential Geometry.Springer-Verlag, 1993. MR 1202431 |
| Reference:
|
[7] I. Kolář and W. M. Mikulski: On the fiber product preserving bundle functors.Differential Geometry and Its Applications 11 (1999), 105–115. MR 1712139, 10.1016/S0926-2245(99)00022-4 |
| Reference:
|
[8] I. Kolář and W. M. Mikulski: Natural lifting of connections to vertical bundles.Suppl. Rendiconti Circolo Mat. Palermo, Serie II 63 (2000), 97–102. MR 1758084 |
| Reference:
|
[9] I. Kolář and M. Modugno: Natural maps on the iterated jet prolongation of a fibered manifold.Annali do Mat. pura ed applicata CLVIII (1991), 151–165. MR 1131848 |
| Reference:
|
[10] I. Kolář and M. Modugno: The Frölicher-Nijenhuis bracket on some functional spaces.Annales Polonici Mat. LXVII (1998), 97–106. MR 1610540 |
| Reference:
|
[11] L. Mangiarotti and M. Modugno: New operators on jet spaces.Ann. Fac. Sci. Toulouse 2 (1983), 171–198. MR 0735661 |
| Reference:
|
[12] M. Modugno: Differential calculus on fibered manifolds.Manuscript. |
| Reference:
|
[13] A. Morimoto: Prolongations of connections to bundles of infinitely near points.J. Diff. Geometry 11 (1976), 479–498. MR 0445422, 10.4310/jdg/1214433720 |
| . |
