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1 – 10 of 30Bhumeshwar Patle, Shyh-Leh Chen, Brijesh Patel, Sunil Kumar Kashyap and Sudarshan Sanap
With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a…
Abstract
Purpose
With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively.
Design/methodology/approach
Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively.
Findings
This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement.
Originality/value
In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.
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Sreerag C., Gokul R., Vinaykumar J. and Rajyalakshmi G.
In any machining process, the surface profile of the workpiece is continuously changing with respect to time and input parameters. In a conventional machining process, input…
Abstract
Purpose
In any machining process, the surface profile of the workpiece is continuously changing with respect to time and input parameters. In a conventional machining process, input parameters are feed and depth of cut whilst other parameters are considered to be constant throughout the process.
Design/methodology/approach
The direct and indirect participation of this instantaneous curvature can be used to optimize the strategy of cutting operation in terms of different parameters like heat generation-induced stresses, etc. The concepts of the metric tensor and Riemannian curvature tensor are made use in this study as a representation of curvature itself. The objective of this study is to create a mathematical methodology that can be implemented on a highly flexible machining process to find an optimum cutting strategy for a particular output parameter.
Findings
The study also includes different case studies for the validation of this newly introduced mathematical methodology.
Originality/value
The study will also find its position in other mechanical processes like forging and casting where instantaneous curvature affects various mechanical properties.
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Lakehal Belarbi and Hichem Elhendi
Let (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki…
Abstract
Purpose
Let (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.
Design/methodology/approach
In this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.
Findings
The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of
Originality/value
The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of
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Sudhakar Kumar Chaubey and Uday Chand De
The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the…
Abstract
Purpose
The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly
Design/methodology/approach
The authors have used the tensorial approach to achieve the goal.
Findings
A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.
Originality/value
The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.
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H. Aruna Kumara, V. Venkatesha and Devaraja Mallesha Naik
Besse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance…
Abstract
Purpose
Besse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f-cosymplectic manifolds.
Design/methodology/approach
The paper opted the tensor calculus on manifolds to find the solution of the CPE.
Findings
In this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\tilde{f} is Einstein. Next, the authors find that a three dimensional almost f-cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti‐commuting.
Originality/value
The paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.
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Ghodratallah Fasihi-Ramandi and Shahroud Azami
In this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.
Abstract
Purpose
In this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.
Design/methodology/approach
In the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.
Findings
In the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.
Originality/value
In this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.
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Cotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric…
Abstract
Purpose
Cotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds.
Design/methodology/approach
The authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations.
Findings
First the authors consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next the authors assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to.
Originality/value
The results of this paper are new and interesting. Also, the Proposition 3.2 will be helpful in further study of this space.
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Aykut Akgün and Mehmet Gülbahar
Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.
Abstract
Purpose
Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.
Design/methodology/approach
Some properties of Di-geodesic and Di-umbilical bi-slant submanifolds are examined.
Findings
The Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.
Originality/value
The topic is original, and the manuscript has not been submitted to any other journal.
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H.M. Manjunatha, S.K. Narasimhamurthy and Zohreh Nekouee
The purpose of this paper is to study the Bertotti–Kasner space-time and its geometric properties.
Abstract
Purpose
The purpose of this paper is to study the Bertotti–Kasner space-time and its geometric properties.
Design/methodology/approach
This paper is based on the features of λ-tensor and the technique of six-dimensional formalism introduced by Pirani and followed by W. Borgiel, Z. Ahsan et al. and H.M. Manjunatha et al. This technique helps to describe both the geometric properties and the nature of the gravitational field of the space-times in the Segre characteristic.
Findings
The Gaussian curvature quantities specify the curvature of Bertotti–Kasner space-time. They are expressed in terms of invariants of the curvature tensor. The Petrov canonical form and the Weyl invariants have also been obtained.
Originality/value
The findings are revealed to be both physically and geometrically interesting for the description of the gravitational field of the cylindrical universe of Bertotti–Kasner type as far as the literature is concerned. Given the technique of six-dimensional formalism, the authors have defined the Weyl conformal
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Xinrong Hu, Shuangshuang Li, Tao Peng, Shi Dong, Junjie Zhang, Changnian Chen, Zlli Zhang, Shuqin Cui, Ruhan He, Min Li and Junping Liu
Fabric has complicated anisotropic mechanical behavior because of the woven pattern and complex physical properties. However, most current fabric simulation models are not…
Abstract
Purpose
Fabric has complicated anisotropic mechanical behavior because of the woven pattern and complex physical properties. However, most current fabric simulation models are not satisfied because the models are usually geometrical models with stiffness parameters.
Design/methodology/approach
In this paper, the authors present a modeling technique to simulate fabric with Riemann manifold. The proposed nonlinear model is formed with ridge wave-curved surface based on the Riemann zero curvature, and the authors develop a solution to conserve the surface area. It decomposes the m × n matrix constituting the fabric into several batches and processes the fabric dots in batches. In this model, the distance between any two adjacent particles of the fabric's is assumed to be equal, and the area of the curved surface is always constant, and the inclination and decay of the ridge wave-curved surface are also considered.
Findings
As the result, the simulated shape is lifelike. In time cost performance, the model improves the efficiency of the fabric styling and meets the requirements of real-time simulation.
Originality/value
The proposed nonlinear model is formed with ridge wave-curved surface based on the Riemann zero curvature, and the authors develop a solution to conserve the surface area.
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