scholarly journals Connections from trivializations

2016 ◽  
Vol 70 (2) ◽  
pp. 83
Author(s):  
Jan Kurek ◽  
Włodzimierz Mikulski
Keyword(s):  
Fiber Bundle ◽  
Structural Group ◽  
Natural Operator ◽  
Principal Fiber Bundle ◽  
Linear Connections ◽  
Torsion Free ◽  
Natural Operators

Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.

scholarly journals The constructions of general connections on second jet prolongation

2014 ◽  
Vol 68 (1) ◽  
Author(s):  
Mariusz Plaszczyk
Keyword(s):  
Second Order ◽  
Linear Connections ◽  
Jet Prolongation ◽  
Torsion Free ◽  
Natural Operators

AbstractWe determine all natural operators D transforming general connections Γ on fibred manifolds Y → M and torsion free classical linear connections ∇ on M into general connections D(Γ,∇) on the second order jet prolongation J2Y → M of Y → M

scholarly journals The constructions of general connections on the fibred product of q copies of the first jet prolongation

2018 ◽  
Vol 72 (1) ◽  
pp. 77
Author(s):  
Mariusz Plaszczyk
Keyword(s):  
Linear Connections ◽  
Jet Prolongation ◽  
Torsion Free ◽  
Natural Operators ◽  
Fibred Product

We describe all natural operators \(A\) transforming general connections \(\Gamma\) on fibred manifolds \(Y \rightarrow M\) and torsion-free classical linear connections \(\Lambda\) on \(M\) into general connections \(A(\Gamma,\Lambda)\) on the fibred product \(J^{&lt;q&gt;}Y \rightarrow M\) of \(q\) copies of the first jet prolongation \(J^{1}Y \rightarrow M\).<br /><br />

scholarly journals Reduction for natural operators on projectable connections

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
W. M. Mikulski ◽  
J. Tomáš
Keyword(s):  
Simple Proof ◽  
Linear Connections ◽  
Torsion Free ◽  
Natural Operators ◽  
General Reduction

AbstractWe present a very simple proof of a general reduction for natural operators on torsion free projectable classical linear connections.

Topological invariance of the Hall conductance and quantization

2015 ◽  
Vol 29 (24) ◽  
pp. 1550135
Author(s):  
Paul Bracken
Keyword(s):  
Chern Class ◽  
Fiber Bundle ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Hall Conductance ◽  
Integer Multiple ◽  
Principal Fiber Bundle ◽  
Topological Invariance ◽  
First Chern Class

It is shown that the Kubo equation for the Hall conductance can be expressed as an integral which implies quantization of the Hall conductance. The integral can be interpreted as the first Chern class of a [Formula: see text] principal fiber bundle on a two-dimensional torus. This accounts for the conductance given as an integer multiple of [Formula: see text]. The formalism can be extended to deduce the fractional conductivity as well.

scholarly journals The induced connections on total spaces of fibred manifolds

2015 ◽  
Vol 97 (111) ◽  
pp. 149-160
Author(s):  
Włodzimierz Mikulski
Keyword(s):  
Linear Connection ◽  
Linear Connections ◽  
Fibred Manifold ◽  
Torsion Free ◽  
General Connection

Let Y ? M be a fibred manifold with m-dimensional base and n-dimensional fibres. If m ? 2 and n ? 3, we classify all linear connections A(?, ?, ?) : TY ? J1(TY ? Y) in TY ? Y (i.e., classical linear connections on Y) depending canonically on a system (?, ?, ?) consisting of a general connection ? : Y ? J1Y in Y ? M, a torsion free classical linear connection ? : TM ? J1(TM ? M) on M and a linear connection ? : V Y ? J1(VY ? Y ) in the vertical bundle VY ? Y.

scholarly journals The natural operators similar to the twisted courant bracket on couples of vector fields and p-forms

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4071-4078
Author(s):  
Włodzimierz Mikulski
Keyword(s):  
Vector Fields ◽  
Jacobi Identity ◽  
Natural Numbers ◽  
Normalization Condition ◽  
Courant Bracket ◽  
Natural Operator ◽  
Bilinear Operators ◽  
Natural Operators

Given natural numbers m and p with m ? p + 2 ? 3, all Mfm-natural operators A sending closed (p+2)-forms H on m-manifolds M into R-bilinear operators AH transforming pairs of couples of vector fields and p-forms on M into couples of vector fields and p-forms on M are found. If m ? p + 2 ? 3, all Mfm-natural operators A (as above) such that AH satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket [-,-]H is the unique Mfm-natural operator AH (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced.

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