ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES

2013 ◽  
Vol 10 (09) ◽  
pp. 1350044
Author(s):  
M. RAFIE-RAD
Keyword(s):  
Principal Curvature ◽  
Finsler Geometry ◽  
Linear Operators ◽  
Curvature Operator ◽  
Riemann Curvature ◽  
Principal Curvatures ◽  
Underlying Space ◽  
Algebraic Properties ◽  
Tangent Manifold ◽  
Randers Metrics

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM0. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature [Formula: see text] may be broken into two operators [Formula: see text] and [Formula: see text]. The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of [Formula: see text] and [Formula: see text].

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

scholarly journals Some properties of shift operators on algebras generated by $*$-polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 206-212
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Countable System ◽  
Shift Operators ◽  
Underlying Space ◽  
Algebra Of Functions ◽  
Algebras Of Holomorphic Functions ◽  
Proper Subalgebra

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

SPECIAL PROJECTIVE ALGEBRA OF RANDERS METRICS OF CONSTANT S-CURVATURE

2012 ◽  
Vol 09 (04) ◽  
pp. 1250034 ◽  
Author(s):  
M. RAFIE-RAD
Keyword(s):  
Vector Fields ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Lie Bracket ◽  
Finite Dimensional ◽  
Projective Vector ◽  
Projective Algebra ◽  
Local Characterization ◽  
Randers Metrics

The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most [Formula: see text]. Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

Geodesic orbit Randers metrics on spheres

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shaoxiang Zhang ◽  
Zaili Yan
Keyword(s):  
Finsler Geometry ◽  
Riemannian Metrics ◽  
Complete Classification ◽  
Invariant Metrics ◽  
Navigation Data ◽  
Randers Metrics

AbstractStudying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.

scholarly journals Palm Vein Recognition Algorithm using Multilobe Differential Filters

2019 ◽  
Author(s):  
Екатерина Сафронова ◽  
Ekaterina Safronova ◽  
Елена Павельева ◽  
Elena Pavelyeva
Keyword(s):  
Principal Curvature ◽  
Spatial Configuration ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Gaussian Kernel ◽  
Branch Points ◽  
Principal Curvatures ◽  
Feature Maps ◽  
Vein Recognition ◽  
Palm Vein

In this article the new algorithm for palm vein recognition using multilobe differential filters is proposed. After palm vein image preprocessing vein structure is detected based on principal curvatures. The image is considered as a surface in a three-dimensional space. Some vein points are selected using the maximum principal curvature values, and the other vein points are found from starting points by moving along the direction of minimum principal curvature. Multilobe differential filters are used to extract feature maps for vein images. These filters are flexible in terms of basic lobe choice and spatial configuration of lobes. The multilobe differential filters used in the article simulate vein branch points, and Gaussian kernel is used as the basic lobe. The normalized root-mean-square error is applied for image matching. Experimental results using CASIA multi-spectral palmprint image database demonstrate the effectiveness of the proposed method. The value of EER=0.01693 is obtained.

Principal curvatures and parallel surfaces of wave fronts

2019 ◽  
Vol 19 (4) ◽  
pp. 541-554 ◽  
Author(s):  
Keisuke Teramoto
Keyword(s):  
Principal Curvature ◽  
Singular Points ◽  
Principal Curvatures ◽  
Wave Fronts ◽  
Geometric Invariants ◽  
Parallel Surfaces

Abstract We give criteria for which a principal curvature becomes a bounded C∞-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.

scholarly journals The Existence of Two Homogeneous Geodesics in Finsler Geometry

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 850
Author(s):  
Zdeněk Dušek
Keyword(s):  
Finsler Geometry ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Homogeneous Geodesics ◽  
Homogeneous Geodesic ◽  
Randers Metrics

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible.

ON COMPLEX RANDERS METRICS

2010 ◽  
Vol 21 (08) ◽  
pp. 971-986 ◽  
Author(s):  
BIN CHEN ◽  
YIBING SHEN
Keyword(s):  
Finsler Geometry ◽  
Randers Metric ◽  
Holomorphic Curvature ◽  
Randers Metrics

A characteristic for a complex Randers metric to be a complex Berwald metric is obtained. The formula of the holomorphic curvature for complex Randers metrics is given. It is shown that a complex Berwald Randers metric with isotropic holomorphic curvature must be either usually Kählerian or locally Minkowskian. The Deicke and Brickell theorems in complex Finsler geometry are also proved.

scholarly journals A class of Randers metrics of scalar flag curvature

2020 ◽  
Vol 31 (13) ◽  
pp. 2050114
Author(s):  
Xinyue Cheng ◽  
Li Yin ◽  
Tingting Li
Keyword(s):  
Necessary Conditions ◽  
Finsler Geometry ◽  
Classification Problem ◽  
Flag Curvature ◽  
Finsler Metrics ◽  
Randers Metrics

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that [Formula: see text] is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

scholarly journals Dupin's indicatrix: a tool for quantifying periclinal folds on maps

2003 ◽  
Vol 140 (6) ◽  
pp. 721-726 ◽  
Author(s):  
RICHARD J. LISLE
Keyword(s):  
Remote Sensing ◽  
Aspect Ratio ◽  
Principal Curvature ◽  
Quantitative Assessment ◽  
Tangent Plane ◽  
New Method ◽  
Principal Curvatures ◽  
Remote Sensing Images ◽  
Geometrical Figure ◽  
Anticlastic Curvature

The elliptical and hyperbolic outcrop patterns characteristic of periclinal folds can be used to classify structures according to different curvature attributes. Elliptical patterns indicate domal-basinal structures with synclastic curvature, that is, principal curvatures of the same sign. Hyperbolic patterns are diagnostic of anticlastic curvature (saddle-like structures). Such outcrop geometries are geological examples of Dupin's indicatrix, the geometrical figure obtained by sectioning a curved surface on a plane parallel and almost coincident with the tangent plane. The aspect ratio of Dupin's indicatrix is theoretically related to the ratio of the principal curvature values for the part of the structure being considered. This new method allows quantitative assessment of structures on maps and on remote sensing images. Illustrations are given from Wyoming, USA, and Yorkshire, England.

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