# The Frölicher-Nijenhuis bracket on some functional spaces

Annales Polonici Mathematici (1998)

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

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topIvan 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

top- [1] A. Cabras and I. Kolář, Connections on some functional bundles, to appear.
- [2] A. Frölicher, Smooth structures, in: Category Theory 1981, Lecture Notes in Math. 962, Springer, 1982, 69-81.
- [3] A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms, I, Indag. Math. 18 (1956), 338-359.
- [4] A. Jadczyk and M. Modugno, An outline of a new geometrical approach to Galilei general relativistic quantum mechanics, to appear.
- [5] A. Jadczyk and M. Modugno, Galilei general relativistic quantum mechanics, preprint 1993, 1-220.
- [6] I. Kolář, On the second tangent bundle and generalized Lie derivatives, Tensor (N.S.) 38 (1982), 98-102. Zbl0512.58002
- [7] I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer, 1993. Zbl0782.53013
- [8] Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibred manifolds, in: Lecture Notes in Math. 792, Springer, 1982, 307-355.
- [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] P. Michor, Gauge Theory for Fiber Bundles, Bibliopolis, Napoli, 1991. Zbl0953.53001
- [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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