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.

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.

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

scholarly journals Killing Vector Fields in Generalized Conformalβ-Change of Finsler Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Mallikarjun Yallappa Kumbar ◽  
Narasimhamurthy Senajji Kampalappa ◽  
Thippeswamy Komalobiah Rajanna ◽  
Kavyashree Ambale Rajegowda
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Space ◽  
Killing Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finsler Spaces ◽  
Killing Vector Fields ◽  
Necessary And Sufficient

We consider a Finsler space equipped with a Generalized Conformalβ-change of metric and study the Killing vector fields that correspond between the original Finsler space and the Finsler space equipped with Generalized Conformalβ-change of metric. We obtain necessary and sufficient condition for a vector field Killing in the original Finsler space to be Killing in the Finsler space equipped with Generalized Conformalβ-change of metric.

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 On S-curvature of a homogeneous Finsler space with square metric

2020 ◽  
Vol 17 (02) ◽  
pp. 2050019
Author(s):  
Gauree Shanker ◽  
Sarita Rani
Keyword(s):  
Explicit Formula ◽  
Vector Fields ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Finsler Spaces ◽  
Invariant Vector ◽  
Homogeneous Finsler Space ◽  
The Mean ◽  
Invariant Vector Fields

The study of curvature properties of homogeneous Finsler spaces with [Formula: see text]-metrics is one of the central problems in Riemann–Finsler geometry. In this paper, the existence of invariant vector fields on a homogeneous Finsler space with square metric is proved. Further, an explicit formula for [Formula: see text]-curvature of a homogeneous Finsler space with square metric is established. Finally, using the formula of [Formula: see text]-curvature, the mean Berwald curvature of aforesaid [Formula: see text]-metric is calculated.

scholarly journals ON $S$-CURVATURE OF A HOMOGENEOUS FINSLER SPACE WITH RANDERS CHANGED SQUARE METRIC

2020 ◽  
pp. 673
Author(s):  
Sarita Rani ◽  
Gauree Shanker
Keyword(s):  
Explicit Formula ◽  
Vector Fields ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Finsler Spaces ◽  
Invariant Vector ◽  
Homogeneous Finsler Space ◽  
The Mean ◽  
Alpha Beta ◽  
Invariant Vector Fields

The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on a homogeneous Finsler space with Randers changed square metric has been proved. Further, an explicit formula for $S$-curvature of Randers changed square metric has been established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(\alpha, \beta)$-metric has been calculated. 

HOLOMORPHIC AND KILLING VECTOR FIELDS ON COMPACT BALANCED HERMITIAN MANIFOLDS

2000 ◽  
Vol 11 (01) ◽  
pp. 15-28 ◽  
Author(s):  
GEORGI GANCHEV ◽  
STEFAN IVANOV
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Hermitian Manifold ◽  
Killing Vector Fields ◽  
Chern Connection ◽  
Holomorphic Vector Fields ◽  
Hermitian Manifolds ◽  
Civita Connection ◽  
Levi Civita Connection

We extend the classical theorems for nonexistence of Killing and holomorphic vector fields on compact Kähler manifolds to compact balanced Hermitian manifolds. We express the obstructions to the existence of Killing and holomorphic vector fields on compact balanced Hermitian manifolds in terms of the Ricci tensors of the Levi–Civita connection as well as in terms of the Ricci tensors of the Chern connection. We show that every affine vector field with respect to the Chern connection on a compact balanced Hermitian manifold is holomorphic.

scholarly journals Shen's -process on Berwald connection

2020 ◽  
Vol 72 (8) ◽  
pp. 1134-1148
Author(s):  
M. Faghfouri ◽  
N. Jazer
Keyword(s):  
Explicit Form ◽  
Finsler Geometry ◽  
The Other ◽  
Chern Connection ◽  
Finsler Manifolds ◽  
Berwald Connection ◽  
Torsion Free

The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections.  Hence Tayebi–Najafi introduced two new processes called Shen's and -processes and showed that the Shen connection is obtained from the Chern connection by Shen's -process.  In this paper, we  study the Shen's - and -process on Berwald connection and introduce two new torsion-free connections in Finsler geometry.  Then, we obtain all of Riemannian and non-Riemannian curvatures of these connections.  Using it, we find the explicit form of -curvatures of these connections and prove that -curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian or Riemannian structures.  As an application, we consider compact Finsler manifolds and obtain ODEs.

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