alexa A special form of Rund's h-curvature tensor using R3-like Finsler space | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

A special form of Rund's h-curvature tensor using R3-like Finsler space

S. T. AVEESH

Department of Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, India. E-mail: [email protected]

S. K. NARASIMHAMURTHY

Department of Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, India. E-mail: [email protected]

H. G. NAGARAJA

Department of Mathematics, Bangalore University, Central College Campus, Bangalore 577001, Karnataka, India. E-mail: [email protected]

Pradeep KUMAR

Department of Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, India. E-mail: [email protected]

Received Date: September 22, 2008; Revised Date: January 12, 2009

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

The purpose of the present paper is to consider and study a special form of Rund's h-curvature tensor Ki ljk and Berwald's curvature tensor Hi ljk in an R3-like C-reducible Finsler space. In this paper, we modify the Rund's h-curvature tensor Ki ljk to special form by using some special Finsler spaces like C-reducible, R3-like Finsler spaces.

Introduction

Let Fn = (Mn, F) be an n-dimensional Finsler space with the fundamental function F(x, y). The Finsler Γv-connection RΓ, constructed from the Cartan connection CΓ, is called the Rund connection. Matsumoto de ned the curvature tensor RΓ and the concept of special form of Rund's curvature tensor Equation [5]. The author [9] have studied Finsler space with Rund's h-curvature tensor Equation of a special form. Here we extend the study of a special form of Rund's h-curvature tensor using R3-like Finsler space and obtain some results. In this paper, the range of indexes varies from 1 to n and the v-covariant and o-covariant derivatives are denoted by Equationj and EquationEquationj, respectively.

We use the following notations from [5,8]:

Equation    (1.1)

Definition 1.1 (see [1,2]). A Finsler space Fn (n > 3) is called R3-like, if the curvature tensor Rhijk is written in the form

Equation     (1.2)

where Equation is the tensor.

Definition 1.2 (see [2,3]). A Finsler space Fn is called C-reducible if it satis es the equation

Equation     (1.3)

where Equation.

Definition 1.3 (see[6]). A Finsler space Fn is called P-reducible if the torsion tensor Pijk is written as

Equation    (1.4)

where Equation andEquation.

The v-covariant derivative of P-reducible Finsler space is given by [9]

Equation    (1.5)

Definition 1.4 (see[7]). A Finsler space is called Landsberg Finsler space if Equation . We use the following identities from [5,7,9]:

Equation     (1.6)

where Equation is the (v) h-torsion tensor and the sux `0' means contraction with yi. The notation u(jk) denotes the interchange on indices j and k and substraction.

We use the following lemma in the next section.

Definition 1.5 (see [4]). If the equation Equation holds in Fn, then we haveEquation, with reference to the Moor frame (li, mi, ni), where v is a scalar.

Special form of Rund's h-curvature tensor Equation

Let Fn be a Finsler space with Rund's h-curvature tensor Equation of the special form [9]

Equation    (2.1)

where Equation are Finsler tensor elds. Consider h-curvature tensor of the form

Equation     (2.2)

Using equations (1.2) and (1.3) in (2.2), we get

Equation

By using some Finsler identities, the above equation can be written as

Equation

After simpli cation and the rearrange the terms, we get

Equation

In simple form, the above equation can be written as a special form of (2.1) as

Equation     (2.3)

where

Equation     (2.4)

Thus we state the following.

Theorem 2.1. In an R3-like, C-reducible Finsler space, the h-curvature tensor reduces to special form of Rund's h-curvature tensor (2.3).

Now we compare the Rund's curvature tensor and h-curvature tensor. Thus, from (2.1) and (2.2), we have

Equation    (2.5)

Contracting (2.5) with respect to yl, yk and using (1.1d), we get

Equation     (2.6)

Again contracting (2.6) with respect to i and j, we get

Equation     (2.7)

Now, we will nd D00. Consider Dij from the special form (2.3), and contract this with respect to i and j, and by using (1.1e), we have

Equation    (2.8)

Substituting (2.8) in (2.7), we have

Equation

Thus we state the following.

Theorem 2.2. If the Rund's h-curvature tensor has the special form (2.1), then the scalar curvature of the space is 2(n – 1)LF2.

Let us suppose that Fn is R3-like C-reducible Finsler space. Then, by using (1.3) and (2.3), the h-curvature tensor (2.2) can written as

Equation    (2.9)

where

Equation

Thus we have the following.

Theorem 2.3. In an R3-like C-reducible Finsler space, if the Rund's h-curvature tensor has the special form (2.3), then the Cartan h-curvature tensor Equation has the special form (2.9).

Using the special form of Rund's h-curvature tensor Equation in the Bianchi identity (1.6d), we get

Equation     (2.10)

Due to Lemma 1.1, equation (2.10) can be written as

Equation

Thus we have the following.

Theorem 2.4. If the Rund's h-curvature tensor Equation is of the special form (2.3), then both the tensor elds Aij and Dij are symmetric simultaneously.

It is also known that a Finsler space is Landsberg space with Equation. If Fn is Landsberg, then from (1.6a) and (1.6e), we get

Equation

Thus we can propose the following.

Corollary 2.5. If Fn is a Landsberg space and the Rund's h-curvature tensor Equation is of the form (2.3), then Cartan curvature tensor coincides with the Berwald's curvature tensor.

Now consider h-curvature tensor (1.6a) of the form

Equation

From equation (1.6e), the above equation can be written as

Equation    (2.11)

Suppose Fn is a P-reducible Finsler space, then by using (1.4), (1.5), (2.3), and (2.11), we have

Equation     (2.12)

where

Equation

Thus we have the following.

Theorem 2.6. In a P-reducible Finsler space, and the special form of Rund's h-curvature tensor Equation has the special form of Berwald's curvature tensor, then Equationis of the form (2.12).

Consider the Bianchi identity

Equation     (2.13)

Substituting (2.12) in (2.13), we get

Equation    (2.14)

Due to Lemma 1.1, equation (2.14) can be written as

Equation

Thus we have the following.

Theorem 2.7. If in a Finsler space Fn the h-curvature tensor Equation is of the form (2.12), then the tensor elds Tjk and Mkj both are simultaneously symmetric.

Acknowledgement

The authors are highly thankful to the referee for valuable comments and suggestions.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Recommended Conferences

  • 7th International Conference on Biostatistics and Bioinformatics
    September 26-27, 2018 Chicago, USA
  • Conference on Biostatistics and Informatics
    December 05-06-2018 Dubai, UAE
  • Mathematics Congress - From Applied to Derivatives
    December 5-6, 2018 Dubai, UAE

Article Usage

  • Total views: 11831
  • [From(publication date):
    August-2009 - May 24, 2018]
  • Breakdown by view type
  • HTML page views : 8048
  • PDF downloads : 3783
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2018-19
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri & Aquaculture Journals

Dr. Krish

[email protected]

1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2018 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
Leave Your Message 24x7