scholarly journals On generalized P-reducible Finsler manifolds

2018 ◽  
Vol 16 (1) ◽  
pp. 718-723
Author(s):  
Seyyed Mohammad Zamanzadeh ◽  
Behzad Najafi ◽  
Megerdich Toomanian
Keyword(s):  
Vector Field ◽  
Finsler Geometry ◽  
Finsler Manifolds ◽  
Cartan Torsion ◽  
Mean Cartan Torsion

AbstractThe class of generalized P-reducible manifolds (briefly GP-reducible manifolds) was first introduced by Tayebi and his collaborates [1]. This class of Finsler manifolds contains the classes of P-reducible manifolds, C-reducible manifolds and Landsberg manifolds. We prove that every compact GP-reducible manifold with positive or negative character is a Randers manifold. The norm of Cartan torsion plays an important role for studying immersion theory in Finsler geometry. We find the relation between the norm of Cartan torsion, mean Cartan torsion, Landsberg and mean Landsberg curvatures of the class of GP-reducible manifolds. Finally, we prove that every GP-reducible manifold admitting a concurrent vector field reduces to a weakly Landsberg manifold.

scholarly journals The connections of pseudo-Finsler spaces

2014 ◽  
Vol 11 (07) ◽  
pp. 1460025 ◽  
Author(s):  
E. Minguzzi
Keyword(s):  
General Relativity ◽  
Conservation Law ◽  
Finsler Geometry ◽  
The Other ◽  
Finsler Spaces ◽  
Dynamical Equations ◽  
Nonlinear Connection ◽  
Cartan Torsion ◽  
The Mean ◽  
Mean Cartan Torsion

We give an introduction to (pseudo-)Finsler geometry and its connections. For most results we provide short and self-contained proofs. Our study of the Berwald nonlinear connection is framed into the theory of connections over general fibered spaces pioneered by Mangiarotti, Modugno and other scholars. The main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald's, Cartan's or Chern–Rund's. In this way it becomes easy to compare them and see the advantages of one connection over the other. Since we introduce two soldering forms we are able to characterize the notable Finsler connections in terms of their torsion properties. As an application, the curvature symmetries implied by the compatibility with a metric suggest that in Finslerian generalizations of general relativity the mean Cartan torsion vanishes. This observation allows us to obtain dynamical equations which imply a satisfactory conservation law. The work ends with a discussion of yet another Finsler connection which has some advantages over Cartan's and Chern–Rund's.

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.

On the conformal scalar curvature equations on Finsler manifolds

2016 ◽  
Vol 14 (01) ◽  
pp. 1750008
Author(s):  
Neda Shojaee ◽  
Morteza MirMohammad Rezaii
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Yamabe Problem ◽  
Finsler Manifolds ◽  
Yamabe Flow ◽  
Conformal Deformations

In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

scholarly journals Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Shin-ichi Ohta
Keyword(s):  
Vector Field ◽  
Ricci Curvature ◽  
Splitting Theorem ◽  
Gradient Vector ◽  
Busemann Function ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Straight Line ◽  
Weighted Ricci Curvature ◽  
Gradient Vector Field

AbstractWe investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger–Gromoll–Lichnerowicz Splitting Theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

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.

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 a Yamabe Type Problem in Finsler Geometry

2017 ◽  
Vol 60 (2) ◽  
pp. 253-268
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Constant Scalar Curvature ◽  
Conformal Invariants ◽  
Type Problem ◽  
Conformal Class ◽  
Standard Sphere ◽  
Yamabe Constant ◽  
Cartan Torsion

AbstractIn this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results

2013 ◽  
Vol 65 (6) ◽  
pp. 1401-1435 ◽  
Author(s):  
Wei Zhao ◽  
Yibing Shen
Keyword(s):  
Isoperimetric Inequality ◽  
Comparison Theorem ◽  
Finsler Geometry ◽  
Finsler Manifolds ◽  
Volume Comparison

AbstractIn this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger–Kazdan inequality and Santalá's formula in Finsler geometry. Based on these, we derive a Berger–Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

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.

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.

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