scholarly journals ON FINSLERIZED ABSOLUTE PARALLELISM SPACES

2013 ◽  
Vol 10 (07) ◽  
pp. 1350029 ◽  
Author(s):  
NABIL L. YOUSSEF ◽  
AMR M. SID-AHMED ◽  
EBTSAM H. TAHA
Keyword(s):  
Geometric Structure ◽  
Vector Fields ◽  
Finsler Geometry ◽  
Absolute Parallelism ◽  
Finsler Structure ◽  
New Space ◽  
Curvature Tensors ◽  
Physical Phenomena

The aim of this paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism (GAP)-space. The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized absolute parallelism (parallelizable) space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and their interrelation is carried out. Five connections are introduced and their torsion and curvature tensors derived. Some special Finslerized parallelizable spaces are singled out. One of the main reasons to introduce this new space is that both absolute parallelism and Finsler geometries have proved effective in the formulation of physical theories, so it is worthy to try to build a more general geometric structure that would share the benefits of both geometries.

scholarly journals EXTENDED ABSOLUTE PARALLELISM GEOMETRY

2008 ◽  
Vol 05 (07) ◽  
pp. 1109-1135 ◽  
Author(s):  
NABIL. L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Tangent Bundle ◽  
Special Form ◽  
Vector Fields ◽  
A Priori ◽  
Building Blocks ◽  
Absolute Parallelism ◽  
Linearly Independent ◽  
Geometric Objects ◽  
Directional Argument ◽  
Curvature Tensors

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.

PARALLEL π-VECTOR FIELDS AND ENERGY β-CHANGE

2011 ◽  
Vol 08 (04) ◽  
pp. 753-772 ◽  
Author(s):  
A. SOLEIMAN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Geometry ◽  
Linear Connection ◽  
Free Form ◽  
Cartan Connection ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Berwald Connection ◽  
Curvature Tensors

The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text] being a parallel π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: The Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

scholarly journals CONCURRENT π-VECTOR FIELDS AND ENERGY β-CHANGE

2009 ◽  
Vol 06 (06) ◽  
pp. 1003-1031 ◽  
Author(s):  
NABIL L. YOUSSEF ◽  
S. H. ABED ◽  
A. SOLEIMAN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Geometry ◽  
Linear Connection ◽  
Free Form ◽  
Cartan Connection ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Berwald Connection ◽  
Curvature Tensors

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text]; [Formula: see text] being a concurrent π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection, and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

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.

scholarly journals THE ACCELERATING EXPANSION OF THE UNIVERSE AND TORSION ENERGY

2007 ◽  
Vol 22 (31) ◽  
pp. 5709-5716 ◽  
Author(s):  
M. I. WANAS
Keyword(s):  
Geometric Structure ◽  
Repulsive Force ◽  
Fine Tuning ◽  
Tuning Parameter ◽  
Absolute Parallelism ◽  
Accelerating Expansion ◽  
Torsion Energy ◽  
Expansion Of The Universe ◽  
Curvature And Torsion ◽  
The Universe

In the present work, it is shown that the problem of the accelerating expansion of the Universe can be directly solved by applying Einstein geometrization philosophy in a wider geometry. The geometric structure used to fulfil the aim of the work is a version of Absolute Parallelism geometry in which curvature and torsion are simultaneously non vanishing objects. It is shown that, while the energy corresponding to the curvature of space- time gives rise to an attractive force, the energy corresponding to the torsion indicates the presence of a repulsive force. A fine tuning parameter can be adjusted to give the observed phenomena.

scholarly journals Suitable Spaces for Shape Optimization

2021 ◽  
Author(s):  
Kathrin Welker
Keyword(s):  
Shape Optimization ◽  
Explicit Expression ◽  
Geometric Structure ◽  
Covariant Derivative ◽  
Optimization Problems ◽  
Optimization Techniques ◽  
Poincaré Metric ◽  
Shape Hessian ◽  
New Space ◽  
Definition Of

AbstractThe differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov–Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to $$H^{1/2}$$ H 1 / 2 -shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of $$H^{1/2}$$ H 1 / 2 -shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.

scholarly journals On equitorsion concircular tensors of generalized Riemannian spaces

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 463-471 ◽  
Author(s):  
Milan Zlatanovic ◽  
Irena Hinterleitner ◽  
Marija Najdanovic
Keyword(s):  
Vector Fields ◽  
Metric Tensor ◽  
The Subject ◽  
Curvature Tensors ◽  
Riemannian Spaces

In this paper we consider concircular vector fields of manifolds with non-symmetric metric tensor. The subject of our paper is an equitorsion concircular mapping. A mapping f : GRN?GRN? is an equitorsion if the torsion tensors of the spaces GRN and GRN? are equal. For an equitorsion concircular mapping of two generalized Riemannian spaces GRN and GRN, we obtain some invariant curvature tensors of this mapping Z?,? = 1,2,... 5, given by equations (3.14, 3.21, 3.28, 3.31, 3.38). These quantities are generalizations of the concircular tensor Z given by equation (2.5).

ENERGY β-CONFORMAL CHANGE IN FINSLER GEOMETRY

2012 ◽  
Vol 09 (04) ◽  
pp. 1250029 ◽  
Author(s):  
A. SOLEIMAN
Keyword(s):  
Vector Field ◽  
Finsler Geometry ◽  
Linear Connection ◽  
Finsler Manifold ◽  
Free Form ◽  
Cartan Connection ◽  
Chern Connection ◽  
Conformal Change ◽  
Berwald Connection ◽  
Curvature Tensors

The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ([Formula: see text] with [Formula: see text]; [Formula: see text] being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

scholarly journals The Finsler geometry of groups of isometries of Hilbert Space

1987 ◽  
Vol 42 (2) ◽  
pp. 196-222 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Orthogonal Group ◽  
Finsler Geometry ◽  
Operator Norm ◽  
Full Description ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Finsler Structure ◽  
Groups Of Isometries

AbstractThe paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

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