scholarly journals Mathematical Modeling in Problems of Homogeneous (Pseudo)Riemaimian Geometry

2020 ◽  
pp. 95-98
Author(s):  
P.N. Klepikov
Keyword(s):  
Mathematical Modeling ◽  
Differential Geometry ◽  
Weyl Tensor ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Conformally Flat ◽  
Computer Math ◽  
Symbolic Calculations ◽  
The Moment ◽  
Modern Mathematics

Currently, mathematical and computer modeling, as well as systems of symbolic calculations, are actively used in many areas of mathematics. Popular computer math systems as Maple, Mathematica, MathCad, MatLab allow not only to perform calculations using symbolic expressions but also solve algebraic and differential equations (numerically and analytically) and visualize the results. Differential geometry, like other areas of modern mathematics, uses new computer technologies to solve its own problems. The applying is not limited only to numerical calculations; more and more often, computer mathematics systems are used for analytical calculations. At the moment, there are many examples that prove the effectiveness of systems of analytical calculations in the proof of theorems of differential geometry.This paper demonstrates how symbolic computation packages can be used to classify neither conformally flat nor Ricci parallel four-dimensional Lie groups with leftinvariant (pseudo)Riemannian metric of the algebraic Ricci soliton with the zero Schouten-Weyl tensor.

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

2016 ◽  
Vol 13 (02) ◽  
pp. 1650015 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Symmetric Space ◽  
Weyl Tensor ◽  
Killing Vector ◽  
Complete Classification ◽  
Space Time ◽  
Conformal Killing ◽  
Conformally Flat ◽  
Conformal Curvature ◽  
Wave Space ◽  
Divergence Free

In this paper we present some new results about [Formula: see text]-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for [Formula: see text] and a pp-wave space-time in [Formula: see text]. In all cases an algebraic classification for the Weyl tensor is provided for [Formula: see text] and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat [Formula: see text], [Formula: see text], space-time is conformal to the Robertson–Walker space-time.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals η-Ricci Solitons on Sasakian 3-Manifolds

2017 ◽  
Vol 55 (2) ◽  
pp. 143-156
Author(s):  
Pradip Majhi ◽  
Uday Chand De ◽  
Debabrata Kar
Keyword(s):  
Ricci Tensor ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Conformally Flat ◽  
Curvature Condition ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Parallel Ricci Tensor

AbstractIn this paper we studyη-Ricci solitons on Sasakian 3-manifolds. Among others we prove that anη-Ricci soliton on a Sasakian 3-manifold is anη-Einstien manifold. Moreover we considerη-Ricci solitons on Sasakian 3-manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Beside these we study conformally flat andφ-Ricci symmetricη-Ricci soliton on Sasakian 3-manifolds. Alsoη-Ricci soliton on Sasakian 3-manifolds with the curvature conditionQ.R= 0 have been considered. Finally, we construct an example to prove the non-existence of properη-Ricci solitons on Sasakian 3-manifolds and verify some results.

Influence of the Shaper Design of Pneumohydraulic Impact Device on the Form and Duration of the Impact Impulse

2015 ◽  
Vol 756 ◽  
pp. 394-401 ◽  
Author(s):  
Oyuna O. Angatkina ◽  
Petr Y. Krauinsch ◽  
Valentina N. Deryusheva
Keyword(s):  
Mathematical Modeling ◽  
Work Environment ◽  
Mechanical Engineering ◽  
Specific Area ◽  
Impact Device ◽  
Impact Impulse ◽  
The Moment ◽  
The Impact

Impact devices are used in various areas of mechanical engineering and deal with a variety of materials. At the moment, it is determined that an increase in the duration of the impact impulse ensures the constant contact of a headgear (tool) with the work environment. This increases the efficiency of impact devices. We have proposed a shaper that allows to control the shape and duration of the impact impulse. In this paper, results of mathematical modeling of the pneumohydraulic impact device with elastic and viscoelastic shaper is considered. Several designs of impact devices are presented. Parameters which affect the impact impulse are determined. These results can be used in constructing pneumohydraulic impact devices for a specific area of implication.

scholarly journals A NOTE ON THE QUANTUM-MECHANICAL RICCI FLOW

2009 ◽  
Vol 24 (27) ◽  
pp. 4999-5006
Author(s):  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER ◽  
P. FERNÁNDEZ DE CÓRDOBA
Keyword(s):  
Quantum Mechanics ◽  
Configuration Space ◽  
Ricci Flow ◽  
Riemannian Metric ◽  
Quantum Mechanical ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Flow Equations ◽  
Emergent Quantum Mechanics

We obtain Schrödinger quantum mechanics from Perelman's functional and from the Ricci-flow equations of a conformally flat Riemannian metric on a closed two-dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

scholarly journals Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

2020 ◽  
pp. 96-101
Author(s):  
M.V. Kurkina ◽  
S.P. Semenov ◽  
V.V. Slavsky ◽  
O.V. Samarina ◽  
O.A. Petuhova ◽  
...  
Keyword(s):  
Convex Functions ◽  
Riemannian Geometry ◽  
Duality Theory ◽  
Digital Processing ◽  
Riemannian Metric ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Legendre Transformation ◽  
Conformally Flat ◽  
Tropical Mathematics

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

scholarly journals Generalized potential functions in differential geometry and information geometry

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940002 ◽  
Author(s):  
F. M. Ciaglia ◽  
G. Marmo ◽  
J. M. Pérez-Pardo
Keyword(s):  
Differential Geometry ◽  
Inverse Problem ◽  
Riemannian Manifold ◽  
Information Geometry ◽  
Riemannian Metric ◽  
Negative Answer ◽  
Potential Functions ◽  
Point Of View ◽  
Geometric Structures ◽  
Generalized Potential

Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view.

ENERGY AND HEAT FLUX

1995 ◽  
Vol 04 (03) ◽  
pp. 357-365
Author(s):  
J. GARIEL ◽  
N.O. SANTOS ◽  
G. LE DENMAT
Keyword(s):  
Heat Flux ◽  
General Solution ◽  
Total Energy ◽  
Weyl Tensor ◽  
Spherically Symmetric ◽  
Conformally Flat ◽  
Flat Spacetime ◽  
Spherically Symmetric Metric

The total energy in a sphere containing an isotropic shear-free conducting heat fluid is studied in the frame of a spherically symmetric metric. Firstly, we examine the role played by the heat flux. Secondly, we point out the contribution to the energy by the Weyl tensor. We obtain different formulas for the total energy, and those formulas are shown to be equivalent. We derive the general solution for a conformally flat spacetime, and give an example for a nonconformally flat spacetime.

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