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.

scholarly journals On The Formulation of Bianchi Identity from Action Principle

2021 ◽  
Author(s):  
Shiladittya Debnath
Keyword(s):  
Curvature Tensor ◽  
Basic Property ◽  
Covariant Derivative ◽  
Bianchi Identity ◽  
Action Principle ◽  
Lie Derivative ◽  
General Covariance ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor

Abstract In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1st and 2nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’.

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 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.

scholarly journals Recurrence Decompositions in Finsler Space

2020 ◽  
Vol 1 (1) ◽  
pp. 77-86
Author(s):  
Adel M. A. Al-Qashbari
Keyword(s):  
Differential Geometry ◽  
Covariant Derivative ◽  
Finsler Space ◽  
Applied Mathematics ◽  
Torsion Tensor ◽  
Finsler Geometry ◽  
Second Order ◽  
Connected Space ◽  
Projective Curvature Tensor ◽  
Affinely Connected Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.

scholarly journals New Cartan’s tensors and pseudotensors in a generalized Finsler space

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 107-117
Author(s):  
Milica Cvetkovic ◽  
Milan Zlatanovic
Keyword(s):  
Covariant Derivative ◽  
Finsler Space ◽  
Differentiable Manifold ◽  
Curvature Tensors

In this work we defined a generalized Finsler space (GFN) as 2N-dimensional differentiable manifold with a non-symmetric basic tensor gij(x,x?), which applies that gij_?|m(x,x?)=0; ?=1,2. Based on non-symmetry of basic tensor, we obtained ten Ricci type identities, comparing to two kinds of covariant derivative of a tensor in Rund?s sense. There appear two new curvature tensors and fifteen magnitudes, we called ?curvature pseudotensors?.

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.

Quantitative parallel EELS spectrum imaging developed as a new tool in AEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1202-1203
Author(s):  
Gianluigi Botton ◽  
Gilles L'espérance
Keyword(s):  
Statistical Analysis ◽  
Spatial Resolution ◽  
Personal Computer ◽  
Systematic Study ◽  
Present Author ◽  
Chemical Elements ◽  
Detection Limits ◽  
Research Groups ◽  
Quantitative Performance ◽  
Spectrum Imaging

As interest for parallel EELS spectrum imaging grows in laboratories equipped with commercial spectrometers, different approaches were used in recent years by a few research groups in the development of the technique of spectrum imaging as reported in the literature. Either by controlling, with a personal computer both the microsope and the spectrometer or using more powerful workstations interfaced to conventional multichannel analysers with commercially available programs to control the microscope and the spectrometer, spectrum images can now be obtained. Work on the limits of the technique, in terms of the quantitative performance was reported, however, by the present author where a systematic study of artifacts detection limits, statistical errors as a function of desired spatial resolution and range of chemical elements to be studied in a map was carried out The aim of the present paper is to show an application of quantitative parallel EELS spectrum imaging where statistical analysis is performed at each pixel and interpretation is carried out using criteria established from the statistical analysis and variations in composition are analyzed with the help of information retreived from t/γ maps so that artifacts are avoided.

The observation of nano-voids in Au thin films

1987 ◽  
Vol 45 ◽  
pp. 366-367
Author(s):  
William Krakow
Keyword(s):  
Thin Films ◽  
Melting Point ◽  
Present Author ◽  
Stacking Faults ◽  
Damage Threshold ◽  
Ionic Species ◽  
Rare Gases ◽  
Chain Defects ◽  
Vacancy Chains ◽  
Au Thin Films

It has long been known that defects such as stacking faults and voids can be quenched from various alloyed metals heated to near their melting point. Today it is common practice to irradiate samples with various ionic species of rare gases which also form voids containing solidified phases of the same atomic species, e.g. ref. 3. Equivalently, electron irradiation has been used to produce damage events, e.g. ref. 4. Generally all of the above mentioned studies have relied on diffraction contrast to observe the defects produced down to a dimension of perhaps 10 to 20Å. Also all these studies have used ions or electrons which exceeded the damage threshold for knockon events. In the case of higher resolution studies the present author has identified vacancy and interstitial type chain defects in ion irradiated Si and was able to identify both di-interstitial and di-vacancy chains running through the foil.

Let all social partners in the social support process count: New perspectives on classical theory

2013 ◽  
Author(s):  
Liu-Qin Yang ◽  
Robert R. Wright ◽  
Liu-Qin Yang ◽  
Lisa M. Kath ◽  
Michael T. Ford ◽  
...  
Keyword(s):  
Social Support ◽  
Classical Theory ◽  
The Social ◽  
Social Partners
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