On Finsler spaces with the curvature tensors Phijk and Shijk satisfying special conditions

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.

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CONCURRENT π-VECTOR FIELDS AND ENERGY β-CHANGE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887809003904 ◽

2009 ◽

Vol 06 (06) ◽

pp. 1003-1031 ◽

Cited By ~ 5

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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Holomorphically projective mappings between generalized m-parabolic Kähler manifolds

Filomat ◽

10.2298/fil1715865p ◽

2017 ◽

Vol 31 (15) ◽

pp. 4865-4873 ◽

Cited By ~ 1

Author(s):

Milos Petrovic

Keyword(s):

Linear Systems ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Covariant Derivatives ◽

Linearly Independent ◽

Non Linear ◽

Non Linear Systems ◽

Curvature Tensors ◽

Derivatives Of

Generalized m-parabolic K?hler manifolds are defined and holomorphically projective mappings between such manifolds have been considered. Two non-linear systems of PDE?s in covariant derivatives of the first and second kind for the existence of such mappings are given. Also, relations between five linearly independent curvature tensors of generalized m-parabolic K?hler manifolds with respect to these mappings are examined.

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Projective Curvature Tensors of Some Special Manifolds with Non-symmetric Linear Connection

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01768-8 ◽

2021 ◽

Vol 18 (4) ◽

Author(s):

Miloš Z. Petrović ◽

Ana M. Velimirović

Keyword(s):

Linear Connection ◽

Projective Curvature ◽

Curvature Tensors

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On projective symmetries on Finsler spaces

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2021.101763 ◽

2021 ◽

Vol 77 ◽

pp. 101763

Author(s):

B. Lajmiri ◽

B. Bidabad ◽

M. Rafie-Rad ◽

Y. Aryanejad-Keshavarzi

Keyword(s):

Finsler Spaces

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SOME PROBLEMS IN DECOMPOSED BI-RECURRENT FINSLER SPACES

Demonstratio Mathematica ◽

10.1515/dema-1984-0105 ◽

1984 ◽

Vol 17 (1) ◽

Author(s):

N.K. Sharma ◽

Asha Srivastava

Keyword(s):

Finsler Spaces

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Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature

Comptes Rendus Mathematique ◽

10.1016/j.crma.2016.04.001 ◽

2016 ◽

Vol 354 (6) ◽

pp. 619-622

Author(s):

Ioan Bucataru ◽

Zoltán Muzsnay

Keyword(s):

Flag Curvature ◽

Finsler Spaces

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EXTENDED ABSOLUTE PARALLELISM GEOMETRY

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887808003235 ◽

2008 ◽

Vol 05 (07) ◽

pp. 1109-1135 ◽

Cited By ~ 12

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.

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