On the Kolář connection

Włodzimierz M. Mikulski

Archivum Mathematicum (2013)

  • Volume: 049, Issue: 4, page 223-240
  • ISSN: 0044-8753

Abstract

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Let Y M be a fibred manifold with m -dimensional base and n -dimensional fibres and E M be a vector bundle with the same base M and with n -dimensional fibres (the same n ). If m 2 and n 3 , we classify all canonical constructions of a classical linear connection A ( Γ , Λ , Φ , Δ ) on Y from a system ( Γ , Λ , Φ , Δ ) consisting of a general connection Γ on Y M , a torsion free classical linear connection Λ on M , a vertical parallelism Φ : Y × M E V Y on Y and a linear connection Δ on E M . An example of such A ( Γ , Λ , Φ , Δ ) is the connection ( Γ , Λ , Φ , Δ ) by I. Kolář.

How to cite

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Mikulski, Włodzimierz M.. "On the Kolář connection." Archivum Mathematicum 049.4 (2013): 223-240. <http://eudml.org/doc/260831>.

@article{Mikulski2013,
abstract = {Let $Y\rightarrow M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\rightarrow M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma $ on $Y\rightarrow M$, a torsion free classical linear connection $\Lambda $ on $M$, a vertical parallelism $\Phi \colon Y\times _ME\rightarrow VY$ on $Y$ and a linear connection $\Delta $ on $E\rightarrow M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.},
author = {Mikulski, Włodzimierz M.},
journal = {Archivum Mathematicum},
keywords = {general connection; linear connection; classical linear connection; vertical parallelism; natural operators; general connection; linear connection; classical linear connection; vertical parallelism; natural operators},
language = {eng},
number = {4},
pages = {223-240},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the Kolář connection},
url = {http://eudml.org/doc/260831},
volume = {049},
year = {2013},
}

TY - JOUR
AU - Mikulski, Włodzimierz M.
TI - On the Kolář connection
JO - Archivum Mathematicum
PY - 2013
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 049
IS - 4
SP - 223
EP - 240
AB - Let $Y\rightarrow M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\rightarrow M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma $ on $Y\rightarrow M$, a torsion free classical linear connection $\Lambda $ on $M$, a vertical parallelism $\Phi \colon Y\times _ME\rightarrow VY$ on $Y$ and a linear connection $\Delta $ on $E\rightarrow M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.
LA - eng
KW - general connection; linear connection; classical linear connection; vertical parallelism; natural operators; general connection; linear connection; classical linear connection; vertical parallelism; natural operators
UR - http://eudml.org/doc/260831
ER -

References

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  1. Doupovec, M., Mikulski, W. M., 10.1007/s10114-010-7333-2, Acta Math. Sinica 26 (1) (2010), 169–184. (2010) Zbl1186.53036MR2584996DOI10.1007/s10114-010-7333-2
  2. Gancarzewicz, J., Horizontal lifts of linear connections to the natural vector bundles, Differential geometry (Santiago de Compostela, 1984), vol. 131, Pitman, Boston, MA, 1985, pp. 318–341. (1985) 
  3. Janyška, J., Vondra, J., 10.1016/S0034-4877(10)00002-9, Rep. Math. Phys. 64 (3) (2009), 395–415. (2009) Zbl1195.53040MR2602937DOI10.1016/S0034-4877(10)00002-9
  4. Kolář, I., 10.1142/S021988781000452X, Int. J. Geom. Methods Mod. Phys. (2010), 705–711. (2010) MR2669064DOI10.1142/S021988781000452X
  5. Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer Verlag, 1993. (1993) 

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