Reduction theorems for principal and classical connections

Doupovec, Miroslav; Mikulski, Włodzimierz M.

doi:10.1007/s10114-010-7333-2

Cited by 9 publications

(11 citation statements)

References 9 publications

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“…Previous approach to connections is suitable for conceptual considerations and operations with connections, such as prolongations of connections, natural operators, and some classifications. For us the following theorem is quite useful; see [8] for the proof.…”

Section: Jet Prolongation Of a Fibered Manifold And Higher Order Connmentioning

confidence: 99%

Semiholonomic Second Order Connections Associated with Material Bodies

Hrdina

Vašík

2013

Journal of Applied Mathematics

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The thermomechanical behavior of a material is expressed mathematically by means of one or more constitutive equations representing the response of the body to the history of its deformation and temperature. These settings induce a set of connections which can express local properties. We replace two of them by a second order connection and prove that the holonomity of this connection classifies our materials.

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Section: Jet Prolongation Of a Fibered Manifold And Higher Order Connmentioning

confidence: 99%

Semiholonomic Second Order Connections Associated with Material Bodies

Hrdina

Vašík

2013

Journal of Applied Mathematics

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“…The concept of jet prolongations of principal and associated bundles is a fundamental tool in higher order differential geometry, the theory of differential invariants, and in applications (see, e.g., Brajerčík [2], Doupovec and Mikulski [3], Janyška [4], Kolář, Michor and Slovák [7], Kowalski and Sekizawa [9], Krupka [11], Kureš [13], Paták and Krupka [15]). This paper is a contribution to the structure theory of the prolongations.…”

Section: Introductionmentioning

confidence: 99%

Principal bundle structure on jet prolongations of frame bundles

Brajerčík

Demko

Krupka

2014

Mathematica Slovaca

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ABSTRACT. In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r F X of the frame bundle F X over a manifold X. Our construction reduces the well-known principal prolongation W r F X of F X with structure group G r n . For a structure group of J r F X we find a suitable subgroup of G r n . We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r F X corresponds with the standard actions of differential groups on tensor spaces.

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“…"We decompose Z ∈ T y Y into the horizontal part h(Z) = Γ(y, Z o ), Z o ∈ T x M , x = p(y) and the vertical part vZ = Φ(y, Z 1 ), Z 1 ∈ E x . We take a vector field X on M such that j 1 x X = Λ(Z o ) and construct its Γ-lift ΓX : Y → T Y . Further, we consider a section s of E such that j 1 x s = ∆(Z 1 ).…”

Section: Introductionmentioning

confidence: 99%

“…We take a vector field X on M such that j 1 x X = Λ(Z o ) and construct its Γ-lift ΓX : Y → T Y . Further, we consider a section s of E such that j 1 x s = ∆(Z 1 ). For every Z ∈ T y Y we define ψ(Z) = j 1 y ΓX + ϕ(s) .…”

Section: Introductionmentioning

confidence: 99%

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On the Kolář connection

Mikulski¹

2013

Arch. Math. (Brno)

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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, ∆) on Y from a system (Γ, Λ, Φ, ∆) 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 ∆ on E → M. An example of such A(Γ, Λ, Φ, ∆) is the connection (Γ, Λ, Φ, ∆) by I. Kolář.

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Reduction theorems for principal and classical connections

Cited by 9 publications

References 9 publications

Semiholonomic Second Order Connections Associated with Material Bodies

Semiholonomic Second Order Connections Associated with Material Bodies

Principal bundle structure on jet prolongations of frame bundles

On the Kolář connection

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