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.

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 The Β-Change by Finsler Metric of C-Reducible Finsler Spaces in Finsler Geometry

2019 ◽  
Vol 33 (1) ◽  
pp. 1-10
Author(s):  
Khageswar Mandal
Keyword(s):  
Finsler Space ◽  
Homogeneous Function ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Spaces

 This paper considered about the β-Change of Finsler metric L given by L*= f(L, β), where f is any positively homogeneous function of degree one in L and β and obtained the β-Change by Finsler metric of C-reducible Finsler spaces. Also further obtained the condition that a C-reducible Finsler space is transformed to a C-reducible Finsler space by a β-change of Finsler metric.

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.

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.

LEVI–CIVITA CONNECTION ON ALMOST COMMUTATIVE ALGEBRAS

2007 ◽  
Vol 04 (07) ◽  
pp. 1075-1085 ◽  
Author(s):  
F. NGAKEU
Keyword(s):  
Linear Connection ◽  
Commutative Algebras ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Definition Of ◽  
Torsion Free

Recently we introduced a new definition of metrics on almost commutative algebras. In this paper, we propose a coherent notion of compatible linear connection with respect to any almost commutative tensor and show that to every metric there corresponds a unique torsion-free compatible connection. This connection is called the Levi–Civita connection of the associated metric.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

scholarly journals Finsler Geometry for Two-Parameter Weibull Distribution Function

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Emrah Dokur ◽  
Salim Ceyhan ◽  
Mehmet Kurban
Keyword(s):  
Probability Density Function ◽  
Weibull Distribution ◽  
Probability Density ◽  
Density Function ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Cumulative Probability ◽  
Two Dimensional ◽  
Energy Potential ◽  
Two Parameter

To construct the geometry in nonflat spaces in order to understand nature has great importance in terms of applied science. Finsler geometry allows accurate modeling and describing ability for asymmetric structures in this application area. In this paper, two-dimensional Finsler space metric function is obtained for Weibull distribution which is used in many applications in this area such as wind speed modeling. The metric definition for two-parameter Weibull probability density function which has shape (k) and scale (c) parameters in two-dimensional Finsler space is realized using a different approach by Finsler geometry. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed which can be used in many real world applications. For future studies, it is aimed at proposing more accurate models by using this novel approach than the models which have two-parameter Weibull probability density function, especially used for determination of wind energy potential of a region.

Sign in / Sign up

Export Citation Format

Share Document