scholarly journals On an R-Randers mth-Root Space

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
P. N. Pandey ◽  
Shivalika Saxena
Keyword(s):  
Finsler Space ◽  
Riemannian Metric ◽  
Root Space ◽  
Metric Tensor ◽  
Nonlinear Connection

We consider an n-dimensional Finsler space Fn(n>2) with the metric L(x,y)=F(x,y)+α(x,y), where F is an mth-root metric and α is a Riemannian metric. We call such space as an R-Randers mth-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers mth-root space. Some other properties of such space have also been discussed.

scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

2010 ◽  
Vol 07 (03) ◽  
pp. 485-503 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Inner Product ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Separable States ◽  
Pure States ◽  
Complex Projective

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

scholarly journals Lagrange Spaces with (γ,β)-Metric

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Suresh K. Shukla ◽  
P. N. Pandey
Keyword(s):  
Lagrange Equations ◽  
Metric Tensor ◽  
Nonlinear Connection

We study Lagrange spaces with (γ,β)-metric, where γ is a cubic metric and β is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a (γ,β)-metric. Several other properties of such space are also discussed.

scholarly journals On the ℒ-duality of a Finsler space with exponential metric αeβ/α

2018 ◽  
Vol 10 (1) ◽  
pp. 167-177
Author(s):  
Ramdayal Singh Kushwaha ◽  
Gauree Shanker
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Finsler Metrics ◽  
Exponential Metric

Abstract The (α, β)-metrics are the most studied Finsler metrics in Finsler geometry with Randers, Kropina and Matsumoto metrics being the most explored metrics in modern Finsler geometry. The ℒ-dual of Randers, Kropina and Matsumoto space have been introduced in [3, 4, 5], also in recent the ℒ-dual of a Finsler space with special (α, β)-metric and generalized Matsumoto spaces have been introduced in [16, 17]. In this paper, we find the ℒ-dual of a Finsler space with an exponential metric αeβ/α, where α is Riemannian metric and β is a non-zero one form.

scholarly journals Hypersurfaces Of a Finsler Space

1956 ◽  
Vol 8 ◽  
pp. 487-503 ◽  
Author(s):  
Hanno Rund
Keyword(s):  
Classical Theory ◽  
Covariant Derivative ◽  
Present Author ◽  
Finsler Space ◽  
Metric Tensor

Introduction. Certain aspects of the theory of subspaces of a Finsler space had been treated by the present author in earlier papers (7). These developments were based on an approach essentially different from the classical theory of Cartan (2) and subsequent writers, whose use of the element of support enables one to introduce the so-called “euclidean connection,” which effects the vanishing of the covariant derivative of the metric tensor.

ON THE PROJECTIVE ALGEBRA OF SOME (α, β)-METRICS OF ISOTROPIC S-CURVATURE

2013 ◽  
Vol 10 (10) ◽  
pp. 1350048 ◽  
Author(s):  
BAHMAN REZAEI ◽  
MEHDI RAFIE-RAD
Keyword(s):  
Sectional Curvature ◽  
Lie Algebra ◽  
Finsler Space ◽  
Riemannian Metric ◽  
Constant Sectional Curvature ◽  
Lie Bracket ◽  
Finite Dimensional ◽  
Projective Algebra

In this paper, we study projective algebra, p(M, F), of special (α, β)-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We characterize p(M, F) of Matsumoto and square metrics of isotropic S-curvature of dimension n ≥ 3 as a certain Lie sub-algebra of the Killing algebra k(M, α). We also show that F has a maximum projective symmetry if and only if F either is a Riemannian metric of constant sectional curvature or locally Minkowskian.

scholarly journals Deformation of complex Finsler metrics

2018 ◽  
Vol 26 (3) ◽  
pp. 229-244
Author(s):  
Annamária Szász-Friedl
Keyword(s):  
Finsler Space ◽  
Linear Connection ◽  
Finsler Metrics ◽  
Infinitesimal Deformation ◽  
Metric Tensor ◽  
First Order ◽  
Special Part ◽  
Non Linear ◽  
Order Deformation ◽  
Complex Finsler Metrics

AbstractThe aim of this paper is to describe the infinitesimal deformation (M, V) of a complex Finsler space family {(M, Lt)}t∈ℝ and to study some of its geometrical objects (metric tensor, non-linear connection, etc). In this circumstances the induced non-linear connection on (M, V) is defined. Moreover we have elaborate the inverse problem, the problem of the first order deformation of the metric. A special part is devoted to the study of particular cases of the perturbed metric.

Relativistic Dynamics

2019 ◽  
pp. 385-425
Author(s):  
David D. Nolte
Keyword(s):  
General Theory ◽  
Particle Physics ◽  
Special Theory ◽  
Riemannian Metric ◽  
Moving Frames ◽  
Theory Of Relativity ◽  
Relativistic Dynamics ◽  
Metric Tensor ◽  
Minkowski Space Time ◽  
Physical Phenomena

The invariance of the speed of light with respect to any inertial observational frame leads to a surprisingly large number of unusual results that defy common intuition. Chief among these are time dilation, length contraction, and loss of simultaneity. The Lorentz transformation intermixes space and time, but an overarching structure is provided by the metric tensor of Minkowski space-time. The pseudo-Riemannian metric supports 4-vectors whose norms are invariants, independent of any observational frame. These invariants constitute the proper objects of reality to study in the special theory of relativity. Relativistic dynamics defines the equivalence of mass and energy, which has many applications in nuclear energy and particle physics. Forces have transformation properties between relatively moving frames that set the stage for a more general theory of relativity that describes physical phenomena in noninertial frames.

scholarly journals PSEUDO-FINSLERIAN SPACE–TIMES AND MULTIREFRINGENCE

2010 ◽  
Vol 19 (07) ◽  
pp. 1119-1146 ◽  
Author(s):  
JOZEF SKÁKALA ◽  
MATT VISSER
Keyword(s):  
Quantum Theory ◽  
Finsler Space ◽  
Conformal Factor ◽  
Riemannian Metric ◽  
Space Time ◽  
Low Energy ◽  
Analog Model ◽  
Birefringent Crystal ◽  
Fresnel Equation ◽  
Finslerian Space

Ongoing searches for a quantum theory of gravity have repeatedly led to the suggestion that space–time might ultimately be anisotropic (Finsler-like) and/or exhibit multirefringence (multiple signal cones). Multiple (and even anisotropic) signal cones can be easily dealt with in a unified manner, by writing down a single Fresnel equation to simultaneously encode all signal cones in an even-handed manner. Once one gets off the signal cone and attempts to construct a full multirefringent space–time metric the situation becomes more problematic. In the multirefringent case we shall report a significant no-go result: in multirefringent models there is no simple or compelling way to construct any unifying notion of pseudo-Finsler space–time metric, different from a monorefringenent model, where the signal cone structure plus a conformal factor completely specifies the full pseudo-Riemannian metric. To throw some light on this situation we use an analog model where both anisotropy and multirefringence occur simultaneously: biaxial birefringent crystal. But the significance of our results extends beyond the optical framework in which (purely for pedagogical reasons) we are working, and has implications for any attempt at introducing multirefringence and intrinsic anisotropies to any model of quantum gravity that has a low energy manifold-like limit.

scholarly journals AN AP-STRUCTURE WITH FINSLERIAN FLAVOR I: THE PRINCIPAL IDEA

2009 ◽  
Vol 24 (22) ◽  
pp. 1749-1762 ◽  
Author(s):  
M. I. WANAS
Keyword(s):  
Geometric Structure ◽  
Building Blocks ◽  
The Other ◽  
Metric Tensor ◽  
Nonlinear Connection ◽  
Absolute Parallelism ◽  
Cartan Type ◽  
Linear Connections

A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structure are assumed to be functions of position and direction. A nonlinear connection emerges naturally and is defined in terms of the building blocks of the structure. Two linear connections, one of Berwald type and the other of the Cartan type, are defined using the nonlinear connection of the FAP. Both linear connections are nonsymmetric and consequently admit torsion. A metric tensor is defined in terms of the building blocks of the structure. The condition for this metric to be a Finslerian one is obtained. Also, the condition for an FAP-space to be an AP-one is given.

Indefinite Finsler Spaces and Timelike Spaces

1970 ◽  
Vol 22 (5) ◽  
pp. 1035-1039 ◽  
Author(s):  
John K. Beem
Keyword(s):  
Triangle Inequality ◽  
Finsler Space ◽  
Sufficient Conditions ◽  
Partial Ordering ◽  
Two Dimensional ◽  
Finsler Spaces ◽  
Metric Tensor ◽  
Timelike Surface ◽  
Lorentz Manifolds ◽  
Riemannian Spaces

In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.

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