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 On naturality of some construction of connections

2020 ◽  
Vol 74 (1) ◽  
pp. 57
Author(s):  
Jan Kurek ◽  
Włodzimierz Mikulski
Keyword(s):  
Linear Connection ◽  
Bundle Functor ◽  
Fibred Manifold ◽  
Natural Transformations ◽  
General Connection

Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the  well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.

2-ρ-DERIVATIONS ON A ρ-ALGEBRA AND APPLICATIONS TO THE QUATERNIONIC ALGEBRA

2007 ◽  
Vol 04 (03) ◽  
pp. 457-469 ◽  
Author(s):  
CĂTĂLIN CIUPALĂ
Keyword(s):  
Linear Connection ◽  
Linear Connections

In this paper, we introduce 2-ρ-derivations on a ρ-algebra A, and define 2-linear connections on a ρ-bimodule M over A using these 2-derivations. Then we introduce and study the curvature of a linear connection. Our results are applied to the particular case of the quaternionic algebra ℍ.

scholarly journals Prolongations of linear connections to the frame bundle

1983 ◽  
Vol 28 (3) ◽  
pp. 367-381
Author(s):  
Luis A. Cordero ◽  
Manuel de Leon
Keyword(s):  
Linear Connection ◽  
Frame Bundle ◽  
Linear Connections ◽  
Complete Lift

In this paper we construct the prolongation of a linear connection Γ on a manifold Μ to the bundle space of its frame bundle, and show that such prolongated connection coincides with the so-called complete lift of Γ to .

scholarly journals On the Holonomy Groups of Linear Connections

1956 ◽  
Vol 10 ◽  
pp. 97-100 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Hideki Ozeki
Keyword(s):  
Linear Group ◽  
Affine Space ◽  
General Linear Group ◽  
Affine Connection ◽  
Holonomy Group ◽  
Linear Connection ◽  
General Linear ◽  
Linear Connections ◽  
Holonomy Groups ◽  
Lie Subgroup

In this note we show in § 1, as the main result, that any connected Lie subgroup of the general linear group GL(n, R) can be realized as the holonomy group of a linear connection, i.e. the homogeneous holonomy group of the associeted affine connection, defined on an affine space of dimension n (n ≧ 2).

INDUCED CONNECTIONS ON TOTAL SPACES OF FIBER BUNDLES

2010 ◽  
Vol 07 (04) ◽  
pp. 705-711 ◽  
Author(s):  
IVAN KOLÁŘ
Keyword(s):  
Linear Connection ◽  
Total Space ◽  
Fiber Bundles ◽  
Natural Operators ◽  
General Connection

We present a construction transforming a general connection Γ on a fibered manifold Y → M and a classical connection Λ on its base M into a classical connection on the total space Y by means of a vertical parallelism Φ and an auxiliary linear connection Δ. The relations to the theory of gauge-natural operators are discussed.

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 Ricci and Bianchi identities forh-normalΓ-linear connections onJ1(T,M)

2003 ◽  
Vol 2003 (34) ◽  
pp. 2177-2191 ◽  
Author(s):  
Mircea Neagu
Keyword(s):  
Fibre Bundle ◽  
Linear Connection ◽  
Local Features ◽  
Bianchi Identities ◽  
First Order ◽  
Linear Connections

The aim of this paper is to describe the local Ricci and Bianchi identities of anh-normalΓ-linear connection on the first-order jet fibre bundleJ1(T,M). We present the physical and geometrical motives that determined our study and introduce theh-normalΓ-linear connections onJ1(T,M), emphasizing their particular local features. We describe the expressions of the local components of torsion and curvatured-tensors produced by anh-normalΓ-linear connection∇Γ, and analyze the local Ricci identities induced by∇Γ, together with their derived local deflectiond-tensors identities. Finally, we expose the local expressions of Bianchi identities which geometrically connect the local torsion and curvatured-tensors of connection∇Γ.

Curvature of Cr Manifolds

2013 ◽  
Vol 59 (1) ◽  
pp. 43-72
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Existence And Uniqueness ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Differential Operators ◽  
Linear Connection ◽  
Horizontal Distribution ◽  
Einstein Equations ◽  
Space Forms ◽  
Torsion Free

Abstract We prove the existence and uniqueness of a torsion-free and h-metric linear connection ▽(CR connection) on the horizontal distribution of a CR manifold M. Then we define the CR sectional curvature of M and obtain a characterization of the CR space forms. Also, by using the CR Ricci tensor and the CR scalar curvature we define the CR Einstein gravitational tensor field on M. Thus, we can write down Einstein equations on the horizontal distribution of the 5-dimensional CR manifold involved in the Penrose correspondence. Finally, some CR differential operators are defined on M and two examples are given to illustrate the theory developed in the paper. Most of the results are obtained for CR manifolds that do not satisfy the integrability conditions

scholarly journals Existence of Linear Connections with Respect to Which Given Tensor Fields are Parallel or Recurrent

1964 ◽  
Vol 24 ◽  
pp. 67-108 ◽  
Author(s):  
Yung-Chow Wong
Keyword(s):  
Tensor Product ◽  
Covariant Derivative ◽  
Linear Connection ◽  
Tensor Fields ◽  
Linear Connections

Let M be an n-dimensional (n≥2) connected C∞-manifold with a linear connection. For simplicity, tensor fields on M will simply be called tensors on M. A tensor S on M is said to be parallel if its covariant derivative is everywhere zero in M, i.e., if ▽S = 0. S is said to be recurrent if its covariant derivative is equal to the tensor product of a covector and S itself, i.e., if ▽S = W⊗S, where W is called the recurrence covector.

XXII.—Tensor Fields and Connections on Cross-Sections in the tangent bundle of a differentiable manifold

1967 ◽  
Vol 67 (4) ◽  
pp. 277-288
Author(s):  
Kentaro Yano
Keyword(s):  
Vector Field ◽  
Cross Section ◽  
Tangent Bundle ◽  
Complex Structure ◽  
Linear Connection ◽  
Differentiable Manifold ◽  
Tensor Fields ◽  
Linear Connections ◽  
Almost Complex ◽  
The Cross

SynopsisTensor fields and linear connections in an n-dimensional differentiable manifold M can be extended, in a natural way, to the tangent bundle T(M) of M to give tensor fields of the same type and linear connections in T(M) respectively. We call such extensions complete lifts to T(M) of tensor fields and linear connections in M.On the other hand, when a vector field V is given in M, V determines a cross-section which is an n-dimensional submanifold in the 2n-dimensional tangent bundle T(M).We study first the behaviour of complete lifts of tensor fields on such a cross-section. The complete lift of an almost complex structure being again an almost complex structure, we study especially properties of the cross-section as a submanifold in an almost complex manifold.We also study properties of cross-sections with respect to the linear connection which is the complete lift of a linear connection in M and with respect to the linear connection induced by the latter on the cross-section. To quote a typical result: A necessary and sufficient condition for a cross-section to be totally geodesic is that the vector field V in M defining the cross-section in T(M) be an affine Killing vector field in M.

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