The Frölicher-Nijenhuis bracket on some functional spaces

Ivan Kolář; Marco Modungo

Annales Polonici Mathematici (1998)

  • Volume: 68, Issue: 2, page 97-106
  • ISSN: 0066-2216

Abstract

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Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as C ∞ ( E ₁ ₓ , E ₂ ₓ ) , for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated covariant differential and curvature.

How to cite

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Ivan Kolář, and Marco Modungo. "The Frölicher-Nijenhuis bracket on some functional spaces." Annales Polonici Mathematici 68.2 (1998): 97-106. <http://eudml.org/doc/270521>.

@article{IvanKolář1998,
abstract = {Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as $C^\{∞\}(E₁ₓ,E₂ₓ)$, for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated covariant differential and curvature.},
author = {Ivan Kolář, Marco Modungo},
journal = {Annales Polonici Mathematici},
keywords = {bundle of smooth maps; connection on a functional bundle; Frölicher-Nijenhuis bracket; Schrödinger connection; fiber bundles; tangent prolongation; differential operators; connection; covariant differential; curvature},
language = {eng},
number = {2},
pages = {97-106},
title = {The Frölicher-Nijenhuis bracket on some functional spaces},
url = {http://eudml.org/doc/270521},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Ivan Kolář
AU - Marco Modungo
TI - The Frölicher-Nijenhuis bracket on some functional spaces
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 2
SP - 97
EP - 106
AB - Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as $C^{∞}(E₁ₓ,E₂ₓ)$, for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated covariant differential and curvature.
LA - eng
KW - bundle of smooth maps; connection on a functional bundle; Frölicher-Nijenhuis bracket; Schrödinger connection; fiber bundles; tangent prolongation; differential operators; connection; covariant differential; curvature
UR - http://eudml.org/doc/270521
ER -

References

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  1. [1] A. Cabras and I. Kolář, Connections on some functional bundles, to appear. 
  2. [2] A. Frölicher, Smooth structures, in: Category Theory 1981, Lecture Notes in Math. 962, Springer, 1982, 69-81. 
  3. [3] A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms, I, Indag. Math. 18 (1956), 338-359. 
  4. [4] A. Jadczyk and M. Modugno, An outline of a new geometrical approach to Galilei general relativistic quantum mechanics, to appear. 
  5. [5] A. Jadczyk and M. Modugno, Galilei general relativistic quantum mechanics, preprint 1993, 1-220. 
  6. [6] I. Kolář, On the second tangent bundle and generalized Lie derivatives, Tensor (N.S.) 38 (1982), 98-102. Zbl0512.58002
  7. [7] I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer, 1993. Zbl0782.53013
  8. [8] Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibred manifolds, in: Lecture Notes in Math. 792, Springer, 1982, 307-355. 
  9. [9] L. Mangiarotti and M. Modugno, Graded Lie algebras and connections on fibred spaces, J. Math. Pures Appl. 83 (1984), 111-120. Zbl0494.53033
  10. [10] P. Michor, Gauge Theory for Fiber Bundles, Bibliopolis, Napoli, 1991. Zbl0953.53001
  11. [11] A. Vanžurová, On geometry of the third order tangent bundle, Acta Univ. Palack. Olomuc. Math. 24 (1985), 81-96. Zbl0619.53013

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