Horizontal Lifts Research Articles

Horizontal lift problems in a pull-back bundle of tensor bundles defined by projection of the cotangent bundles

Using projection (submersion) of the cotangent bundle T*M over a manifold M, we define a semi-tensor (pull-back) bundle tM of type (p,q). The purpose of this paper is to investigate horizontal lifts of vector fields in a special class of semi-tensor (pull-back) bundle tM of type (p,q).

Some notes on lifts of the F((υ+1),λ²(υ-1))-structure on cotangent and tangent bundle

The F^{v+1}-λ²F^{v-1}=0 structure (v≥3) have been studied by Kim J. B. K75 . Later, Srivastava S.K studied on the complete lifts of (1,1) tensor field F satisfying structure F^{v+1}-λ²F^{v-1}=0 and extended in Mⁿ to cotangent bundle. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete and horizontal lifts of F^{v+1}-λ²F^{v-1}=0. Later, we get the results of Tachibana operators applied to vector and covector fields according to the complete and horizontal lifts of F((v+1),λ²(v-1)) -structure and the conditions of almost holomorfic vector fields in cotangent bundle T^{∗}(Mⁿ). Finally, we have studied the purity conditions of Sasakian metric with respect to the lifts of F^{v+1}-λ²F^{v-1}=0-structure. In the second part, all results obtained in the first section were investigated according to the complete and horizontal lifts of the F^{v+1}-λ²F^{v-1}=0 structure in tangent bundle T(Mⁿ).

The transformation of the involute curves using by lifts on $R^{3}$ to tangent space $TR^{3}$

"How we can speak about the features of involute curve on space $TR^{3}$ by looking at the characteristics of the first curve $\alpha $?" In this paper, we investigate the answer of this question using by lifts. In this direction firstly, we define the involute curve of any curve with respect to the vertical, complete and horizontal lifts on space $R^{3}$ to its tangent space $TR^{3}=R^{6}.$ Secondly, we examine the Frenet-Serret aparatus $\left\{ T^{\ast }(s),N^{\ast }(s),B^{\ast }(s),\kappa ^{\ast },\tau ^{\ast }\right\} $ and the unit Darboux vector $\tilde{D}^{\ast }$ of the involute curve $\alpha ^{\ast }$ according to the vertical, complete and horizontal lifts on $TR^{3}$ depending on the lifting of Frenet-Serret aparatus $\left\{ T(s),N(s),B(s),\kappa ,\tau \right\} $ of the first curve $\alpha $ on space $R^{3}.$ In addition, we include all special cases the curvature $\kappa ^{\ast }(s)$ and torsion $\tau ^{\ast }(s)$ of the Frenet-Serret aparatus of the involute curve $\alpha ^{\ast }$ with respect to lifts on space $R^{3}$ to its tangent space $TR^{3}.$ As a result of this transformation on space $R^{3}$ to its tangent space $TR^{3}$, we could have some information about the features of involute curve of any curve on space $TR^{3}$ by looking at the characteristics of the first curve $\alpha$. Moreover, we get the transformation of the involute curves using by lifts on $R^{3}$ to tangent space $TR^{3}$. Finally, some examples are given for each curve transformation to validated our theorical claims.

Lifts of connections to the bundle of (1,1) type tensor frames

In this paper we consider the bundle of (1,1) type tensor frames over a smooth manifold, define the horizontal and complete lifts of symmetric linear connection into this bundle. Also we study the properties of the geodesic lines corresponding to the complete lift of the linear connection and investigate the relations between Sasaki metric and lifted connections on the bundle of (1,1) type tensor frames.

Соціальні ліфти як механізм реалізації державної молодіжної політики

The purpose of this article is to characterize the functioning of social lifts as one of the effective mechanisms for implementing state youth policy. It is determined that in the transition to the postindustrial phase of civilization development the main goal of the state youth policy in Ukraine should be: creating conditions for self-realization of every young person and development of youth initiatives on the principles of openness, system and comprehensiveness by restoring and improving social lifts. At the same time, vertical (traditional) social lifts connected with the hierarchical structure of society and horizontal social lifts created with the use of information and communication networks and virtual environments should be preserved. It is demonstrated that the state youth policy should be focused on the creation and improvement of legal, socio-economic, research, organizational conditions for successful socialization and self-realization of Ukrainian youth, the use of their creative potential in the interests of society. It is revealed that in the state youth policy it is necessary to take into account the following main stratifying factors: economic well-being (income, wealth, property), which is based on the indicators of the average monetary income in the system “rich – poor” the marginalization of a large part of the population, caused by a significant declining mobility in the context of the general crisis and radical reforms, is mainly forced under the influence of external factors related to socio-economic and socio-cultural transformation of society as a whole; falling importance of education as a factor of upward mobility, due to inequality in obtaining educational opportunities for low-income groups.

Metrics and Connections on the Bundle of Affinor Frames

In this paper the authors consider the bundle of affinor frames over a smooth manifold, define the Sasaki metric on this bundle, and investigate the Levi-Civita connection of Sasaki metric. Also the authors determine the horizontal lifts of symmetric linear connection from a manifold to the bundle of affinor frames and study the geodesic curves corresponding to the horizontal lift of the linear connection.

Lifts of Derivations in the Coframe Bundle

In this paper, using a decomposition formula for any derivations in the coframe bundle, a complete and horizontal lifts of derivations from manifold to its coframe bundle are determined and their properties are studied.

On the Integrability Conditions and Operators of the F((K + 1),(K − 1))− Structure Satisfying F K+1 + F K−1 = 0, (F 6= 0, K 1 2) on Cotangent Bundle and Tangent Bundle

This paper consists of two main sections. In the first part, we find the integrability conditions of the horizontal lifts of $F((K+1),(K-1))-$ structure satisfying $F^{K+1}+F^{K-1}=0,$ $(F\neq 0,$ $K\eqslantgtr 2)$. Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of $F((K+1),(K-1))-$structure in cotangent bundle $T^{\ast }(M^{n})$. Finally, we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of the structure. In the second part, all results obtained in the first section were obtained according to the complete and horizontal lifts of the structure in tangent bundle $T(M^{n})$.

Model and solution approaches for retrieval operations in a multi-tier shuttle warehouse system

In the Multi-tier Shuttle Warehousing System (MSWS), shuttles are responsible for horizontal movements and lifts are responsible for vertical movements. In this paper, a retrieval task scheduling problem for a tier-captive MSWS under single command cycle is considered. This paper aims to determine the sequence of retrieval tasks to minimize the makespan. A mixed integer programming model is developed for the retrieval task scheduling problem. When the number of tasks is less than 10, the mixed integer programming model can be solved by the GUROBI solver to find the exact solutions within a short period. Moreover, a tabu search algorithm, an ant colony algorithm and a simulated annealing algorithm are designed and applied to solve large size problems with up to 200 tasks. By analyzing experimental results, the ant colony algorithm shows stable and superior performance in solving large size problems.

A Lichnerowicz estimate for the spectral gap of a sub-Laplacian

For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.

On the Lifts of F(2K + S; S)-Structure Satisfying F^(2K+S) + F^S = 0; (F≠ 0; K≥ 0 ,S≥1 ) on Cotangent and Tangent Bundle

This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the horizontal lifts of F(2K + S, S)−structure Satisfying F 2K+S + F S = 0. Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of F(2K + S, S)−structure in cotangent bundle T ∗ (Mn). Finally, we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of the structure. In the second part, all results obtained in the first section were obtained according to the complete and horizontal lifts of F(2K + S, S)−structure in tangent bundle T(Mn).

On integrability conditions, operators, and the purity conditions of the Sasakian metric with respect to lifts of F l (7 ; 1) -structure on the cotangent bundle

There are many structures in the cotangent bundle. These include the complete and horizontal lifts of the $F_{\lambda }(7,1)$-structure. The $F_{\lambda }(7,1)$-structure was first extended in $M^{n}$ to $T^{\ast }(M^{n})$ by Das, Nivas, and Pathak. Later, the horizontal and complete lift of the $F_{a}(K,1)$-structure in the tangent bundle was given by Prasad and Chauhan. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete lifts of the $F_{\lambda }(7,1)$-structure. Later, we get the results of the Tachibana operators applied to vector and covector fields according to the complete lifts of the $% F_{\lambda }(7,1)$-structure in the cotangent bundle $T^{\ast }(M_{n})$. Finally, we study the purity conditions of the Sasakian metric with respect to the complete lifts of the $F_{\lambda }(7,1)$-structure. In the second part, all results obtained in the first section are obtained according to the horizontal lifts of the $F_{\lambda }(7,1)$-structure in cotangent bundle $T^{\ast }(M_{n})$.

Coframe bundle and problems of lifts on its cross-sections

The main purpose of this paper is to study the complete and horizontal lifts of vector and tensor fields of type (1,1) on cross-sections in the coframe bundle. Explicit formulas of these lifts are obtained. Finally, complete lifts of almost complex structures restricted to almost analytic cross-sections are investigated.

Kaluza–Klein type Ricci solitons on unit tangent sphere bundles

We consider g-natural pseudo-Riemannian metrics of Kaluza–Klein type on the unit tangent sphere bundle of a Riemannian manifold of constant sectional curvature and give necessary and sufficient conditions for these metrics to give rise to a Ricci soliton.On the one hand, we obtain a rigidity result in dimension three, showing that there are no nontrivial Ricci solitons among g-natural metrics of Kaluza–Klein type on the unit tangent sphere bundle of any Riemannian surface. On the other hand, while Ricci solitons determined by tangential lifts remain trivial in arbitrary dimension, horizontal lifts of vector fields related to the geometry of the base manifold (namely, homothetic vector fields) produce nontrivial Ricci solitons metrics of Kaluza–Klein type. Gradient Ricci solitons of Kaluza–Klein type are also completely characterized.

Shuttle-based operating policies for multiple-lift compact automated parking systems based on queuing networks

We study a new multiple-lift compact automated parking system, which is a new technology with a circular structure using carousel equipped with the shuttles for the horizontal movement and multiple lifts equipped with the platforms for the vertical movement. This paper investigates dedicated and random shuttles operating policies. We divided the carousel into areas based on the number of lifts. We propose a dedicated shuttle operating policy under which each area shuttles can only serve one dedicated lift. A random shuttle operating policy under which shuttles can serve any lift. We get expected throughput time models for the two shuttle operating policies under the same system physical configurations based on queuing network models. We compare the analytical models with simulation models, and the results show that the proposed models provide accurate estimates. A numerical experiment is conducted to illustrate the trade-offs between dedicated shuttle and random shuttle operating policies. The results show that the retrieval time in random shuttle operating policy is larger than the retrieval time in dedicated shuttle operating policy. We also get the optimal system structure, including the number of the lifts.

English

We study harmonic metrics with respect to the class of invariant metrics on non-reductive homogeneous four dimensional manifolds. In particular, we consider harmonic lifted metrics with respect to the Sasaki lifts, horizontal lifts and complete lifts of the metrics under study.

On Tachibana and Vishnevskii Operators Associated with Certain Structures in the Tangent Bundle

The aim of the present work is to study the complete, vertical and horizontal lifts using Tachibana and Visknnevskii operators along generalized almost r-contact structure in tangent bundle. We also prove certain theorems on Tachibana and Visknnevskii operators with Lie derivative and lifts.

Efficient Parallel Transport in the Group of Diffeomorphisms via Reduction to the Lie Algebra.

This paper presents an efficient, numerically stable algorithm for parallel transport of tangent vectors in the group of diffeomorphisms. Previous approaches to parallel transport in large deformation diffeomorphic metric mapping (LDDMM) of images represent a momenta field, the dual of a tangent vector to the diffeomorphism group, as a scalar field times the image gradient. This "scalar momenta" constraint couples tangent vectors with the images being deformed and leads to computationally costly horizontal lifts in parallel transport. This paper uses the vector momenta formulation of LDDMM, which decouples the diffeomorphisms from the structures being transformed, e.g., images, point sets, etc. This decoupling leads to parallel transport expressed as a linear ODE in the Lie algebra. Solving this ODE directly is numerically stable and significantly faster than other LDDMM parallel transport methods. Results on 2D synthetic data and 3D brain MRI demonstrate that our algorithm is fast and conserves the inner products of the transported tangent vectors.

Vertical and horizontal lifts of multivector fields and applications

Let Q be a smoothmanifold of dimension n ≥ 1. In this paper,we define the vertical lift of multivector fields from Q to T Q and we give some applications in the Poisson geometry. In particular we describe the structure of singular foliation induced by the vertical lift of Poisson structures defined below. On the other hand, given a second order vector field on Q, we defined the horizontal lift of multivector fields from Q to TQ and we study some properties.

Non-abelian higher gauge theory and categorical bundle

A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is an attempt to generalize the bundle structure in order to describe horizontal lifts of surfaces. A such attempt is particularly difficult for the non-abelian case. Some structures have been proposed to realize this goal (twisted bundle, gerbes with connection, bundle gerbe, 2-bundle). Each of them uses a category in place of the total space manifold of the usual principal bundle structure. Some of them replace also the structure group by a category (more precisely a Lie crossed module viewed as a category). But the base space remains still a simple manifold (possibly viewed as a trivial category with only identity arrows). We propose a new principal categorical bundle structure, with a Lie crossed module as structure groupoid, but with a base space belonging to a bigger class of categories (which includes non-trivial categories), that we call affine 2-spaces. We study the geometric structure of the categorical bundles built on these categories (which is a more complicated structure than the 2-bundles) and the connective structures on these bundles. Finally we treat an example interesting for quantum dynamics which is associated with the Bloch wave operator theory.