On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields I

Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE′3 can transform into a curved three-dimensionalRiemannian metric space IM′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE3 (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM3, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE′3ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE′3ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE′3. The projective ∅IE′3ab is imperceptibly embedded in ∅IE′3 and IE′3ab in IE′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM′3 only is reached.

The Relation between General Relativity’s Metrics and Special Relativity’s Gravitational Scalar Generalized Potentials and Case Studies on the Schwarzschild Metric, Teleparallel Gravity, and Newtonian Potential

This paper shows that gravitational results of general relativity (GR) can be reached by using special relativity (SR) via a SR Lagrangian that derives from the corresponding GR time dilation and vice versa. It also presents a new SR gravitational central scalar generalized potential V=V(r,r.,ϕ.), where r is the distance from the center of gravity and r.,ϕ. are the radial and angular velocity, respectively. This is associated with the Schwarzschild GR time dilation from where a SR scalar generalized potential is obtained, which is exactly equivalent to the Schwarzschild metric. Thus, the Precession of Mercury’s Perihelion, the Gravitational Deflection of Light, the Shapiro time delay, the Gravitational Red Shift, etc., are explained with the use of SR only. The techniques used in this paper can be applied to any GR spacetime metric, Teleparallel Gravity, etc., in order to obtain the corresponding SR gravitational scalar generalized potential and vice versa. Thus, the case study of Newtonian Gravitational Potential according to SR leads to the corresponding non-Riemannian metric of GR. Finally, it is shown that the mainstream consideration of the Gravitational Red Shift contains two approximations, which are valid in weak gravitational fields only.

Boundary rigidity for Randers metrics

If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

A structure of the solenoidal 2D vector and 2-tensor fields given in a domain with the conformal Riemannian metric

The Helmholtz decomposition of a vector field on potential and solenoidal parts is much more natural from physical and geometric points of view then representations through the components of the vector in the Cartesian coordinate system of Euclidean space. The structure, representation through potentials and detailed decomposition for 2D symmetric m-tensor fields in a case of the Euclidean metric is known. For the Riemannian metrics similar results are known for vector fields. We investigate the properties of the solenoidal vector and 2-tensor two-dimensional fields given in the Riemannian domain with the conformal metric and establish the connections between the fields and metrics.

On the induced metric of a photon qubit for different points of action in Bloch sphere

Travelling with harmonical actions about the referential line, the moment the photon qubit emits form the atom as a packet, not only embedded by the envelopes of quadratic nature overlap but also by the non-quadratic overlap effect as well in concurrent. This paper followed on by the subsistence of both ways of overlapping, focuses mainly on the induced metric of photon qubit, which concluded to be in 3-sphere by the local parameterization so called as Hopf coordinates. The protruded photon qubit in the induced Riemannian metric, for different points of the Bloch sphere, are in a manner that the stereographic projection of volume enveloped in 3-sphere determines the circular geometry in 2-sphere with the cross-sectional function in perpendicular to the continuum line of travel.

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

Variational formulas for submanifolds of fixed degree

We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler–Lagrange equations. The resulting mean curvature operator can be of third order.