research article on a special form of () -torsion tensor...

Research ArticleOn a Special Form of (ℎ) ℎ]-Torsion Tensor 119875

119894119895119896in Finsler Space

Brijesh Kumar Tripathi1 and K B Pandey2

1Department of Mathematics LE College Morbi Gujarat 363642 India2Department of Mathematics KNIT Sultanpur 228118 India

Correspondence should be addressed to K B Pandey kunjbiharipandey05gmailcom

Received 11 May 2016 Revised 18 August 2016 Accepted 29 August 2016

Academic Editor Willi Freeden

Copyright copy 2016 B K Tripathi and K B Pandey This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A special form of (ℎ) ℎ]-torsion tensor was introducedwhichmay be considered generalization of119875lowast-Finsler space and119875-reducibleFinsler space and then some properties of this space were studied We also introduce connection and give some case and conditionof torsion tensor 119879119894

119895119896

1 Introduction

Let 119872119899 be an 119899-dimensional differentiable manifold and119879119872119899 be its tangent bundle The manifold 119872119899 is covered

by neighborhoods (119880) in each 119880 of which we have a localcoordinate system (119909119894) A tangent vector at a point 119909 = (119909119894)of 119880 is written as 119910119894(120597120597119909119894)119909 and we have a local coordinatesystem (119909119894 119910119894) of 119879119872119899 over 119880

In paper [1] let 119865119899 (119899 ge 1) be an 119899-dimensional Finslerspace with metric function 119871(119909 119910) There are five kinds offunction 119871(119909 119910) There are five kinds of torsion tensors in thetheory of Finsler space based on Cartonrsquos connection out ofwhich

119875119894119895119896= 119910ℎ

119875ℎ119894119895119896

119862119894119895119896=1

4

1205973

1198713

120597119910119894

120597119910119895

120597119910119896

(1)

as (]) ℎ]-torsion tensor and (ℎ) ℎ-Torsion tensor are ofgreat important tensors for the present study where 119875

ℎ119894119895119896is

h]-curvature tensor In Finsler geometry based on Cartanrsquosconnection there are three kinds of covariant differentiationsdenoted as |

119894and v-covariant differentiation denoted as |119894

An 119899 (119899 ge 3)-dimensional Finsler space 119865119899 is said to be asemi-119862-reducible Finsler space whose Cartanrsquos tensor 119862

119894119895119896is

written as

119862119894119895119896=

119901

119899 + 1119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895 +

119902

1198622119862119894119862119895119862119896 (2)

where 1198622 = 119892119894119895119862119894119862119895and scalars satisfy 119901 + 119902 = 1 Moreover

if scalars 119901 and 119902 are constants 119865119899 is said to be 119862-reducibleFinsler space with constants coefficients

A Special semi-119862-reducible Finsler space has been intro-duced by Ikeda [2] as follows

An 119899 (119899 ge 3)-dimensional Finsler space 119865119899 is said to be aSpecial semi-119862-reducible Finsler space (in short we call SSR-Finsler space) [1 2] whose ℎ (ℎ])-torsion tensor119862

119894119895119896is written

as

119862119894119895119896=

1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895

minus3

(119899 minus 2) 1198622119862119894119862119895119862119896

(3)

Various interesting forms of these tensors have been studiedby many ([3ndash7] ) two of them are 119862-reducible Finsler

Hindawi Publishing CorporationJournal of MathematicsVolume 2016 Article ID 3694017 5 pageshttpdxdoiorg10115520163694017

2 Journal of Mathematics

space and a Special semi-119862-reducible Finsler space ([1 2]) inwhich the torsion tensor 119862

119894119895119896 respectively is in the forms

119862119894119895119896=

1

119899 + 1(119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895) (4)

119862119894119895119896=

1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895

minus3

(119899 minus 2) 1198622119862119894119862119895119862119896

(5)

where ℎ119894119895is angularmetric tenser and119862

119894= 119862119894119895119896119892119895119896 where 119892119895119896

is reciprocal of the metric tensor 119892119895119896

Izumi ([4 5]) introduced 119875lowast-Finsler space in which 119875119894119895119896

is of the form

119875119894119895119896= 120582119862119894119895119896 (6)

where 120582 is the scalar homogeneous function on 119879119872 of zerodegree in 119910119894 In 119875-reducible Finsler space the tensor 119875

119894119895119896is the

form [8]

119875119894119895119896=

1

119899 + 1(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895) (7)

where 119869119894= 119862119894|0= 119862119894|119895119910119894 A Finsler space with 119875

119894119895119896= 0 is

called a Landsbergs space [9] If 119862119894119895119896|ℎ

= 0 then 119865119899 is calledBerwaldrsquos affinely connected space ([10 11])

Rund [11] introduced a special form of torsion tensor 119875119894119895119896

as follows

119875119894119895119896= 120582119862119894119895119896+ 119886119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895 (8)

where120582 = 120582(119909 119910) is a scalar homogeneous function on119879119872ofdegree 1 and 119886

119894= 119886119894(119909) is a homogeneous function of degree

0 with respect to 119910119894 He then studied some properties of 119865119899satisfying (8) The present author introduced a more generalformof (8) and studies some properties of119865119899 satisfying it [12]

We quote the following lemmas which will be used in thepresent paper

Lemma 1 (see [6]) If the curvature tensor119875119894119895119896

of a119862-reducibleFinsler space vanishes then the space vanishes and then thespace is Berwaldrsquos affinely connected space

Lemma 2 (see [13]) A Finsler space 119865119899is locally Minkowskian

if h-curvature tensor 119877ℎ119894119895119896= 0 and 119862

119894119895119896|ℎ= 0

Definition 3 (see [1]) A Finsler connection FΓ is defined astried (119865119894

119895119896(119909 119910) 119881

119894

119895119896(119909 119910)) as ℎ-connection and V-connection

which are components of a tensor field of (1 2)-type Thetensor 119863 of component 119863119894

119895is called deflection tensor of FΓ

Therefore119863119894119895= 119873119894

119895and 119910119903119881119894

119903119895= 0 are desirable conditions for

a Finsler connectionLet 119872 be an 119899-dimensional 119862infin modified by 119879

119909119872 (we

mean the tangent space at 119909 isin 119872) and by 119879119872 0 (we meanthe slit tangent bundle of119872)

A Finsler metric on119872 is a function 119871 119879119872 rarr [0infin)

which has the following properties(i) 119871 is119862infin 119900119899 119879119872 0

(ii) 119871 is positively homogeneous function of degree 1 on119879119872

(iii) For each 119910 isin 119879119909119872 the metric tensor 119892

119894119895and the

angular metric tensor ℎ119894119895are respectively given by

119892119894119895=1

2

1205972

1198712

120597119910119894

120597119910119895

ℎ119894119895= 119871

1205972

119871

120597119910119894

120597119910119895

(9)

The angular metric tensor ℎ119894119895can also be written in terms of

the normalized element of support

119897119894=1

119871119892119894119895119910119894

119910119895

ℎ119894119895= 119892119894119895minus 119897119894119897119895

(10)

(see [14]) For119910 isin 119879119909119872 0 Cartanrsquos tensor vector is defined

as

119862119894= 119892119895119896

119862119894119895119896 (11)

According to Deickersquos theorem 119862119894= 0 is the necessary

and sufficient condition for 119865119899 to be Riemannian Let 119865119899 =(119872119899

119871) be a Finsler space for 119910 isin 119879119909119872 0 We define

Matsumoto torsions of 119862-reducible and Special semi-119862-reducible Finsler space respectively as follows

119872119894119895119896= 119862119894119895119896minus

1

119899 + 1(119862119894ℎ119895119896+ 119862119895ℎ119894119896+ 119862119896ℎ119894119895)

119872119894119895119896= 119862119894119895119896minus

1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894

+3

(119899 minus 2) 1198622

119862119894119862119895119862119896

(12)

A Finsler space 119865119899 is said to be 119862-reducible if119872119894119895119896= 0 and is

Special semi-119862-reducible

if

119872119894119895119896= 0 (13)

Next we define a tensor

119871119894119895119896= 119862119894119895119896|119897119910119897

(14)

where ldquo|rdquo means h-covariant differentiation withrespect to Cartanrsquos connection

A Finsler space 119865119899 is called a Landsberg space if 119875119894119895119896= 0

or equivalently 119871119894119895119896= 119862119894119895119896|ℎ

= 0Define

119869119894= 119892119895119896

119871119894119895119896 (15)

A Finsler space is said to be weakly Landsberg space if 119869119894= 0

[15]

Journal of Mathematics 3

It is obvious that every 119862-reducible Finsler space is 119875-reducible but the converse is not true

In paper [1] define

119872119894119895119896= 119875119894119895119896minus

1

119899 + 1[119869119894ℎ119894119895+ 119869119895ℎ119895119896+ 119869119896ℎ119894119895] (16)

where

119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) minus 119887lowast

119862119894119862119895119862119896 (17)

Let 119861 = minus119887lowast hence

119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (18)

119869119895= 119892119894119896

119875119894119895119896 (19)

where 120582 119860 and 119861 are some scalar function homogeneous ofdegree 1 and 119886

119894rsquos are homogeneous of degree zero It is obvious

that 119865119899 is a 119875-reducible Finsler space if119872119894119895119896= 0

The purpose of the present paper is to study 119865119899 satisfying(18)

If 119865119899 is a Landsberg space then 119875119894119895119896= 0 hence from (18)

119862119894119895119896= minus119860

120582(119886119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) minus119861

120582119862119894119862119895119862119896 (20)

where

119886119894= minus

120582

119860 (119899 minus 2)119862119894

1198611198622

120582=

3

119899 minus 2

(21)

Corollary 4 A Landsbergs space satisfying (18) is a Specialsemi-119862-reducible Finsler space

Since for 119865119899 to be Landsberg space 119875119894119895119896ℎ= 0 therefore from

Lemma 1 and Corollary 4

Corollary 5 A Landsbergs space satisfying (18) is Berwaldrsquosaffinely connected space if 119861 = 0

In view of Lemma 2 and Corollary 5 one has the following

Corollary 6 If Landsbergs space satisfying (18) has vanishingh-curvature tensor that is 119877

119894119895119896ℎ= 0 then it is locally

Minkowskian

Special Forms of 119875119894119895119896 Let 119865119899 be a Finsler space satisfying (18)

A Finsler space with 119875119894119895119896

of given form reduces to 119875lowast-Finslerspace when 119860 = 0 and 119861 = 0 while it reduces to 119875-reducibleFinsler space when 120582 = 0 and 119861 = 0 and 119860119886

119894= (1(119899 + 1))119869

119894

By definition from (18) we can write

119871119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (22)

Contracting by 119892119894119895 we get

119869119896= (120582 + 119861119862

2

) 119862119896+ 119860 (119899 + 1) 119886

119896

119886119896=

1

(119899 + 1)119860119869119896minus120582 + 119861119862

2

(119899 + 1)119860119862119896

(23)

By replacing (23) into (22)

119871119894119895119896= 120582119862119894119895119896+

1

119899 + 1[119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895]

minus120582 + 119861119862

2

(119899 + 1)[119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895]

+ 119861119862119894119862119895119862119896

(24)

or

119871119894119895119896minus

1

119899 + 1(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895) = 120582 [119862

119894119895119896

minus1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895

+3

(119899 minus 2) 1198622

119862119894119862119895119862119896]

(25)

where

(119899 + 1

119899 minus 2) = 1 +

119861

1205821198622

3

119899 minus 2=119861

1205821198622

(26)

Hence we have the following

Theorem 7 The Matsumoto torsion of 119875-reducible Finslerspace119872

119894119895119896andMatsumoto torsion of Special semi-119862-reducible

Finsler space119872119894119895119896

are related by

119872119894119895119896= 120582119872

119894119895119896 (27)

Corollary 8 A Finsler space 119865119899 satisfying (18) is a weaklyLandsberg space if

119886119894= minus

120582 + 1198611198622

(119899 + 1)119860119862119894 (28)

The notation of stretch curvature denoted by Σℎ119894119895119896

was intro-duced byBerwald as generalization of Landsberg curvature [10]in which

Σℎ119894119895119896

fl 2 [119871ℎ119894119895|119896minus 119871ℎ119894119896|119895] (29)

A Finsler space is said to be stretch space if Σℎ119894119895119896= 0

Again taking h-covariant derivative of (22) and thencontracting by 119910ℎ we get

119871119894119895119896|ℎ119910ℎ

= (120582 + 1205822

) 119862119894119895119896+ (120582119860119886

119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895

+ (119861119862119896+ 119861119871119896) 119862119894119862119895+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896

(30)

where we put 120582 = 120582|ℎ 119860 = 119860

|ℎ and 119861 = 119861

|ℎ


Suppose that 119865119899 is stretch space then119871119894119895119896|ℎminus 119871119894119895ℎ|119896

= 0 (31)

By contacting (30) with 119910119896 we obtain

119871119894119895119896|ℎ119910ℎ

= 0 (32)

From (32) and (30) we have

119862119894119895119896= minus

1

(120582 + 1205822)

[(120582119860119886119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895+ (119861119862

119896+ 119861119869119896) 119862119894119862119895

+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896]

(33)

Contacting by (33) by 119892119895ℎ

119862119896= minus

1

(120582 + 1205822)

[(119899 + 1) (120582119860119886119896+ 119860119886119896+ 119860119886119896)

+ (119861119862119896+ 119861119871119896) 1198622

+ 2119861119869119862119862119896]

(34)

whence

120582119860119886119896+ 119860119886119896+ 119860119886119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1)119862119896

minus1198611198622

(119899 + 1)119869119896

(35)

Substituting (35) into (33) we get

119862119894119895119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1) (120582 + 1205822)

(119862119894ℎ119895119896+ 119862119895ℎ119896119895

+ 119862119896ℎ119894119869) +

1198611198622

(119899 + 1) (120582 + 1205822)

(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895)

+

(minus119861)

(120582 + 1205822)

119862119894119862119895119862119896+

(minus119861)

(120582 + 1205822)

(119869119894119862119895119862119896+ 119869119895119862119894119862119896

+ 119869119896119862119895119862119894)

(36)

From (36) it follows that 119865119899 is a semi-119862-reducible Finslerspace if it is a weakly Landsberg space

Therefore we have the following

Theorem 9 Let a Finsler space 119865119899 satisfying (18) be a stretchspace then it is a Special semi-119862-reducible Finsler space if it isa weakly Landsberg space

2-Connection A connection connects with tengent spaces oftwo points of manifold The 1198993 quantities 119871119894

119895119896are connection

coefficients if

119882119894

119895= 119871119894

119895119896

119889119909119896

119889119905

(37)

(see [16])

Connection 119871119894119895119896

is uniquely expressible as the sum of thesymmetric connections and the torsion tensor [12]

119871119894

119895119896= 119878119894

119895119896+ 119879119894

119895119896 (38)

where

119878119894

119895119896=1

2(119871119894

119895119896+ 119871119894

119896119895) (39a)

is symmetric connection

119871119894

119895119896= 119875119894

119895119896+ 119879119894

119895119896science 119875119894

119895119896= 119878119894

119895119896 (39b)

A connection119871119894119895119896is called symmetric connection if119871119894

119895119896= 119871119894

119896119895

Torsion tensor for symmetric connection science is

119879119894

119895119896=

119871119894

119895119896minus 119871119894

119896119895

2 (40)

119879119894

119896119895= minus119879119894

119895119896 (41)

that is 119879119894119895119896is a skew-symmetric tensor

Five kinds of torsion tensors [17] are as follows

119879 = (119879119894

119895119896) (ℎ) ℎ-torsion

119881 = (119881119894

119895119896) (ℎ) ℎ]-torsion

1198771

= (119877119894

119895119896) (]) ℎ-torsion

1198751

= (119875119894

119895119896) (]) ℎ]-torsion

1198781

= (119878119894

119895119896) (]) ]-torsion

(42)

It is noted that ]-connection (119881119894119895119896) also plays a role of torsion

tensor and

119879119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895(43)

(see [9])

119875119894

119895119896= 119896119873119894

119869minus 119865119894

119896119895 (44)

119878119894

119895119896= 119881119894

119895119896minus 119881119894

119896119895 (45)


119871119894

119895119896minus 119871119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (46)


119878119894

119895119896minus 119879119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (47)

From (39b) and (43) we have

119871119894

119895119896minus 119875119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895

119875119894

119895119896= 119871119894

119895119896minus 119865119894

119895119896+ 119865119894

119896119895

(48)

For Cartonrsquos connection (ℎ) torsion 119879119894119895119896= 0


Hence from (47) we have

119878119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (49)

Using (45)

119881119894

119895119896minus 119881119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (50)

Also for (ℎ) ℎ-torsion 119878119894119895119896= 0 from (47)

119879119894

119894119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (51)

119881119894

119895119896= 119881119894

119896119895 (52)

Then tensor119863 of component119863119894119895is called the deflection tensor

119865Therefore

119863119894

119895= 119873119894

119895

119910119903

119881119894

119903119895= 0

(53)

Put 119903 = 119896 we have

119910119896

119881119895

119896119895= 0 (54)

Using (52) we have

119910119896

119881119894

119895119896= 0 997904rArr

119910119903

119881119895

119896119903= 0

(55)

Theorem 10 For Cartanrsquos connection (h) h tensor 119878119894119895119896

anddeflection FT 119881119894

119895119896is symmetric as (52)

Competing Interests

The authors declare that they have no competing interests

References

[1] H Wosoughi ldquoOn a special form of (V) HV-torsion tensor119875119894119895119896

in finsler spacesrdquo Cumhuriyet University Faculty of ScienceJournal vol 36 no 4 2015

[2] F Ikeda ldquoOn special semi C-reducible Finsler spacerdquo TensorNS vol 66 pp 95ndash99 2005

[3] S Basco and I Papp ldquoPlowast-Finsler spaces with vanishing Douglastensorrdquo Acta Academiae Paedagogicae Agriensis Mathematicaevol 25 pp 91ndash95 1998

[4] H Izumi ldquoOn P-Finsler spacesrdquo I-Memoirs of the DefenceAcademy Japan vol 16 no 4 pp 133ndash138 1976

[5] H Izumi ldquoOn P-Finsler space IIrdquo Memoirs of the DefenceAcademy Japan vol 17 no 1 pp 1ndash9 1977

[6] M Matsumoto ldquoOn C-reducible Finsler spacerdquo Tensor vol 24pp 29ndash37 1972

[7] M Matsumoto ldquoProjective Randers change of P-reducibleFinsler spacerdquo Tensor NS vol 59 pp 6ndash11 1998

[8] M Matsumoto and H Shimada ldquoOn Finsler spaces with thecurvature tensors P

ℎ119894119895119896and S

ℎ119894119895119896satisfying special conditionsrdquo

Reports on Mathematical Physics vol 12 no 1 pp 77ndash87 1977

[9] V V Wagner ldquoA generalization non holonomic manifolds inFinslerian spacerdquo Abh Tscherny state Univ Saratow I vol 14no 2 pp 67ndash96 1938

[10] L Berwald ldquoUber paraallelu ubertragung in Raumen mit all-gemeiermassbstimmungrdquoDeutscheMathematiker-Vereinigungvol 34 pp 213ndash220 1926

[11] H RundDifferential Geometry of Finsler Spaces Springer 1959[12] H Wosoughi ldquoOn generalization of the Landsburg spacerdquo

Research Journal of the Recent Sciences vol 2 no 8 pp 63ndash652013

[13] M Matsumoto ldquoOn some transformation of locallyMinkowskian spacesrdquo Tensor vol 22 pp 103ndash111 1971

[14] M Matsumoto Foundation of the Finsler Geometry and SpecialFinsler Spaces Keiseisha Press Saikawa Otsu Japan 1986

[15] Z Shen Differential Geometry of Spray and Finsler SpacesKluwer Academic Dordrecht The Netherlands 2001

[16] D K Pandey and S K Sukla Differential Geometry and TensorAnalysis Prakashan Kendra Lucknow India 2013

[17] T N Pandey and V K Chaubey ldquoTheory of Finsler space with(120574 120573) metricsrdquo Bulletin of the Transilvania University of BrasovMathematics Informatics Physics vol 4 no 53 pp 43ndash56 2011

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


space and a Special semi-119862-reducible Finsler space ([1 2]) inwhich the torsion tensor 119862

119894119895119896 respectively is in the forms

119862119894119895119896=

1

119899 + 1(119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895) (4)

119862119894119895119896=

1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894119895

minus3

(119899 minus 2) 1198622119862119894119862119895119862119896

(5)

where ℎ119894119895is angularmetric tenser and119862

119894= 119862119894119895119896119892119895119896 where 119892119895119896

is reciprocal of the metric tensor 119892119895119896

Izumi ([4 5]) introduced 119875lowast-Finsler space in which 119875119894119895119896

is of the form

119875119894119895119896= 120582119862119894119895119896 (6)

where 120582 is the scalar homogeneous function on 119879119872 of zerodegree in 119910119894 In 119875-reducible Finsler space the tensor 119875

119894119895119896is the

form [8]

119875119894119895119896=

1

119899 + 1(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895) (7)

where 119869119894= 119862119894|0= 119862119894|119895119910119894 A Finsler space with 119875

119894119895119896= 0 is

called a Landsbergs space [9] If 119862119894119895119896|ℎ

= 0 then 119865119899 is calledBerwaldrsquos affinely connected space ([10 11])

Rund [11] introduced a special form of torsion tensor 119875119894119895119896

as follows

119875119894119895119896= 120582119862119894119895119896+ 119886119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895 (8)

where120582 = 120582(119909 119910) is a scalar homogeneous function on119879119872ofdegree 1 and 119886

119894= 119886119894(119909) is a homogeneous function of degree

0 with respect to 119910119894 He then studied some properties of 119865119899satisfying (8) The present author introduced a more generalformof (8) and studies some properties of119865119899 satisfying it [12]

We quote the following lemmas which will be used in thepresent paper

Lemma 1 (see [6]) If the curvature tensor119875119894119895119896

of a119862-reducibleFinsler space vanishes then the space vanishes and then thespace is Berwaldrsquos affinely connected space

Lemma 2 (see [13]) A Finsler space 119865119899is locally Minkowskian

if h-curvature tensor 119877ℎ119894119895119896= 0 and 119862

119894119895119896|ℎ= 0

Definition 3 (see [1]) A Finsler connection FΓ is defined astried (119865119894

119895119896(119909 119910) 119881

119894

119895119896(119909 119910)) as ℎ-connection and V-connection

which are components of a tensor field of (1 2)-type Thetensor 119863 of component 119863119894

119895is called deflection tensor of FΓ

Therefore119863119894119895= 119873119894

119895and 119910119903119881119894

119903119895= 0 are desirable conditions for

a Finsler connectionLet 119872 be an 119899-dimensional 119862infin modified by 119879

119909119872 (we

mean the tangent space at 119909 isin 119872) and by 119879119872 0 (we meanthe slit tangent bundle of119872)

A Finsler metric on119872 is a function 119871 119879119872 rarr [0infin)

which has the following properties(i) 119871 is119862infin 119900119899 119879119872 0

(ii) 119871 is positively homogeneous function of degree 1 on119879119872

(iii) For each 119910 isin 119879119909119872 the metric tensor 119892

119894119895and the

angular metric tensor ℎ119894119895are respectively given by

119892119894119895=1

2

1205972

1198712

120597119910119894

120597119910119895

ℎ119894119895= 119871

1205972

119871

120597119910119894

120597119910119895

(9)

The angular metric tensor ℎ119894119895can also be written in terms of

the normalized element of support

119897119894=1

119871119892119894119895119910119894

119910119895

ℎ119894119895= 119892119894119895minus 119897119894119897119895

(10)

(see [14]) For119910 isin 119879119909119872 0 Cartanrsquos tensor vector is defined

as

119862119894= 119892119895119896

119862119894119895119896 (11)

According to Deickersquos theorem 119862119894= 0 is the necessary

and sufficient condition for 119865119899 to be Riemannian Let 119865119899 =(119872119899

119871) be a Finsler space for 119910 isin 119879119909119872 0 We define

Matsumoto torsions of 119862-reducible and Special semi-119862-reducible Finsler space respectively as follows

119872119894119895119896= 119862119894119895119896minus

1

119899 + 1(119862119894ℎ119895119896+ 119862119895ℎ119894119896+ 119862119896ℎ119894119895)

119872119894119895119896= 119862119894119895119896minus

1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119895+ 119862119896ℎ119894

+3

(119899 minus 2) 1198622

119862119894119862119895119862119896

(12)

A Finsler space 119865119899 is said to be 119862-reducible if119872119894119895119896= 0 and is

Special semi-119862-reducible

if

119872119894119895119896= 0 (13)

Next we define a tensor

119871119894119895119896= 119862119894119895119896|119897119910119897

(14)

where ldquo|rdquo means h-covariant differentiation withrespect to Cartanrsquos connection

A Finsler space 119865119899 is called a Landsberg space if 119875119894119895119896= 0

or equivalently 119871119894119895119896= 119862119894119895119896|ℎ

= 0Define

119869119894= 119892119895119896

119871119894119895119896 (15)

A Finsler space is said to be weakly Landsberg space if 119869119894= 0

[15]



In paper [1] define

119872119894119895119896= 119875119894119895119896minus

1

119899 + 1[119869119894ℎ119894119895+ 119869119895ℎ119895119896+ 119869119896ℎ119894119895] (16)

where

119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) minus 119887lowast

119862119894119862119895119862119896 (17)


119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (18)

119869119895= 119892119894119896

119875119894119895119896 (19)






119862119894119895119896= minus119860

120582(119886119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) minus119861

120582119862119894119862119895119862119896 (20)

where

119886119894= minus

120582

119860 (119899 minus 2)119862119894

1198611198622

120582=

3

119899 minus 2

(21)








Minkowskian




119894= (1(119899 + 1))119869

119894


119871119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (22)


119869119896= (120582 + 119861119862

2

) 119862119896+ 119860 (119899 + 1) 119886

119896

119886119896=

1

(119899 + 1)119860119869119896minus120582 + 119861119862

2

(119899 + 1)119860119862119896

(23)


119871119894119895119896= 120582119862119894119895119896+

1

119899 + 1[119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895]

minus120582 + 119861119862

2

(119899 + 1)[119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895]

+ 119861119862119894119862119895119862119896

(24)

or

119871119894119895119896minus

1

119899 + 1(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895) = 120582 [119862

119894119895119896

minus1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895

+3

(119899 minus 2) 1198622

119862119894119862119895119862119896]

(25)

where

(119899 + 1

119899 minus 2) = 1 +

119861

1205821198622

3

119899 minus 2=119861

1205821198622

(26)




Finsler space119872119894119895119896

are related by

119872119894119895119896= 120582119872

119894119895119896 (27)


119886119894= minus

120582 + 1198611198622

(119899 + 1)119860119862119894 (28)



Σℎ119894119895119896

fl 2 [119871ℎ119894119895|119896minus 119871ℎ119894119896|119895] (29)



119871119894119895119896|ℎ119910ℎ

= (120582 + 1205822

) 119862119894119895119896+ (120582119860119886

119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895

+ (119861119862119896+ 119861119871119896) 119862119894119862119895+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896

(30)

where we put 120582 = 120582|ℎ 119860 = 119860

|ℎ and 119861 = 119861

|ℎ



= 0 (31)


119871119894119895119896|ℎ119910ℎ

= 0 (32)


119862119894119895119896= minus

1

(120582 + 1205822)

[(120582119860119886119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895+ (119861119862

119896+ 119861119869119896) 119862119894119862119895

+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896]

(33)


119862119896= minus

1

(120582 + 1205822)

[(119899 + 1) (120582119860119886119896+ 119860119886119896+ 119860119886119896)

+ (119861119862119896+ 119861119871119896) 1198622

+ 2119861119869119862119862119896]

(34)

whence

120582119860119886119896+ 119860119886119896+ 119860119886119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1)119862119896

minus1198611198622

(119899 + 1)119869119896

(35)


119862119894119895119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1) (120582 + 1205822)

(119862119894ℎ119895119896+ 119862119895ℎ119896119895

+ 119862119896ℎ119894119869) +

1198611198622

(119899 + 1) (120582 + 1205822)

(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895)

+

(minus119861)

(120582 + 1205822)

119862119894119862119895119862119896+

(minus119861)

(120582 + 1205822)

(119869119894119862119895119862119896+ 119869119895119862119894119862119896

+ 119869119896119862119895119862119894)

(36)






coefficients if

119882119894

119895= 119871119894

119895119896

119889119909119896

119889119905

(37)

(see [16])

Connection 119871119894119895119896


119871119894

119895119896= 119878119894

119895119896+ 119879119894

119895119896 (38)

where

119878119894

119895119896=1

2(119871119894

119895119896+ 119871119894

119896119895) (39a)


119871119894

119895119896= 119875119894

119895119896+ 119879119894

119895119896science 119875119894

119895119896= 119878119894

119895119896 (39b)


119895119896= 119871119894

119896119895


119879119894

119895119896=

119871119894

119895119896minus 119871119894

119896119895

2 (40)

119879119894

119896119895= minus119879119894

119895119896 (41)



119879 = (119879119894

119895119896) (ℎ) ℎ-torsion

119881 = (119881119894

119895119896) (ℎ) ℎ]-torsion

1198771

= (119877119894

119895119896) (]) ℎ-torsion

1198751

= (119875119894

119895119896) (]) ℎ]-torsion

1198781

= (119878119894

119895119896) (]) ]-torsion

(42)


tensor and

119879119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895(43)

(see [9])

119875119894

119895119896= 119896119873119894

119869minus 119865119894

119896119895 (44)

119878119894

119895119896= 119881119894

119895119896minus 119881119894

119896119895 (45)


119871119894

119895119896minus 119871119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (46)


119878119894

119895119896minus 119879119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (47)


119871119894

119895119896minus 119875119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895

119875119894

119895119896= 119871119894

119895119896minus 119865119894

119895119896+ 119865119894

119896119895

(48)




119878119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (49)

Using (45)

119881119894

119895119896minus 119881119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (50)


119879119894

119894119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (51)

119881119894

119895119896= 119881119894

119896119895 (52)


119865Therefore

119863119894

119895= 119873119894

119895

119910119903

119881119894

119903119895= 0

(53)

Put 119903 = 119896 we have

119910119896

119881119895

119896119895= 0 (54)

Using (52) we have

119910119896

119881119894

119895119896= 0 997904rArr

119910119903

119881119895

119896119903= 0

(55)




Competing Interests


References










ℎ119894119895119896and S



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











In paper [1] define

119872119894119895119896= 119875119894119895119896minus

1

119899 + 1[119869119894ℎ119894119895+ 119869119895ℎ119895119896+ 119869119896ℎ119894119895] (16)

where

119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) minus 119887lowast

119862119894119862119895119862119896 (17)


119875119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119896119894+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (18)

119869119895= 119892119894119896

119875119894119895119896 (19)






119862119894119895119896= minus119860

120582(119886119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) minus119861

120582119862119894119862119895119862119896 (20)

where

119886119894= minus

120582

119860 (119899 minus 2)119862119894

1198611198622

120582=

3

119899 minus 2

(21)








Minkowskian




119894= (1(119899 + 1))119869

119894


119871119894119895119896= 120582119862119894119895119896+ 119860 (119886

119894ℎ119895119896+ 119886119895ℎ119894119896+ 119886119896ℎ119894119895) + 119861119862

119894119862119895119862119896 (22)


119869119896= (120582 + 119861119862

2

) 119862119896+ 119860 (119899 + 1) 119886

119896

119886119896=

1

(119899 + 1)119860119869119896minus120582 + 119861119862

2

(119899 + 1)119860119862119896

(23)


119871119894119895119896= 120582119862119894119895119896+

1

119899 + 1[119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895]

minus120582 + 119861119862

2

(119899 + 1)[119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895]

+ 119861119862119894119862119895119862119896

(24)

or

119871119894119895119896minus

1

119899 + 1(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895) = 120582 [119862

119894119895119896

minus1

119899 minus 2119862119894ℎ119895119896+ 119862119895ℎ119896119894+ 119862119896ℎ119894119895

+3

(119899 minus 2) 1198622

119862119894119862119895119862119896]

(25)

where

(119899 + 1

119899 minus 2) = 1 +

119861

1205821198622

3

119899 minus 2=119861

1205821198622

(26)




Finsler space119872119894119895119896

are related by

119872119894119895119896= 120582119872

119894119895119896 (27)


119886119894= minus

120582 + 1198611198622

(119899 + 1)119860119862119894 (28)



Σℎ119894119895119896

fl 2 [119871ℎ119894119895|119896minus 119871ℎ119894119896|119895] (29)



119871119894119895119896|ℎ119910ℎ

= (120582 + 1205822

) 119862119894119895119896+ (120582119860119886

119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895

+ (119861119862119896+ 119861119871119896) 119862119894119862119895+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896

(30)

where we put 120582 = 120582|ℎ 119860 = 119860

|ℎ and 119861 = 119861

|ℎ



= 0 (31)


119871119894119895119896|ℎ119910ℎ

= 0 (32)


119862119894119895119896= minus

1

(120582 + 1205822)

[(120582119860119886119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895+ (119861119862

119896+ 119861119869119896) 119862119894119862119895

+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896]

(33)


119862119896= minus

1

(120582 + 1205822)

[(119899 + 1) (120582119860119886119896+ 119860119886119896+ 119860119886119896)

+ (119861119862119896+ 119861119871119896) 1198622

+ 2119861119869119862119862119896]

(34)

whence

120582119860119886119896+ 119860119886119896+ 119860119886119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1)119862119896

minus1198611198622

(119899 + 1)119869119896

(35)


119862119894119895119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1) (120582 + 1205822)

(119862119894ℎ119895119896+ 119862119895ℎ119896119895

+ 119862119896ℎ119894119869) +

1198611198622

(119899 + 1) (120582 + 1205822)

(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895)

+

(minus119861)

(120582 + 1205822)

119862119894119862119895119862119896+

(minus119861)

(120582 + 1205822)

(119869119894119862119895119862119896+ 119869119895119862119894119862119896

+ 119869119896119862119895119862119894)

(36)






coefficients if

119882119894

119895= 119871119894

119895119896

119889119909119896

119889119905

(37)

(see [16])

Connection 119871119894119895119896


119871119894

119895119896= 119878119894

119895119896+ 119879119894

119895119896 (38)

where

119878119894

119895119896=1

2(119871119894

119895119896+ 119871119894

119896119895) (39a)


119871119894

119895119896= 119875119894

119895119896+ 119879119894

119895119896science 119875119894

119895119896= 119878119894

119895119896 (39b)


119895119896= 119871119894

119896119895


119879119894

119895119896=

119871119894

119895119896minus 119871119894

119896119895

2 (40)

119879119894

119896119895= minus119879119894

119895119896 (41)



119879 = (119879119894

119895119896) (ℎ) ℎ-torsion

119881 = (119881119894

119895119896) (ℎ) ℎ]-torsion

1198771

= (119877119894

119895119896) (]) ℎ-torsion

1198751

= (119875119894

119895119896) (]) ℎ]-torsion

1198781

= (119878119894

119895119896) (]) ]-torsion

(42)


tensor and

119879119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895(43)

(see [9])

119875119894

119895119896= 119896119873119894

119869minus 119865119894

119896119895 (44)

119878119894

119895119896= 119881119894

119895119896minus 119881119894

119896119895 (45)


119871119894

119895119896minus 119871119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (46)


119878119894

119895119896minus 119879119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (47)


119871119894

119895119896minus 119875119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895

119875119894

119895119896= 119871119894

119895119896minus 119865119894

119895119896+ 119865119894

119896119895

(48)




119878119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (49)

Using (45)

119881119894

119895119896minus 119881119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (50)


119879119894

119894119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (51)

119881119894

119895119896= 119881119894

119896119895 (52)


119865Therefore

119863119894

119895= 119873119894

119895

119910119903

119881119894

119903119895= 0

(53)

Put 119903 = 119896 we have

119910119896

119881119895

119896119895= 0 (54)

Using (52) we have

119910119896

119881119894

119895119896= 0 997904rArr

119910119903

119881119895

119896119903= 0

(55)




Competing Interests


References










ℎ119894119895119896and S



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 0 (31)


119871119894119895119896|ℎ119910ℎ

= 0 (32)


119862119894119895119896= minus

1

(120582 + 1205822)

[(120582119860119886119894+ 119860119886119894+ 119860119886119894) ℎ119895119896

+ (120582119860119886119895+ 119860119886119895+ 119860119886119895) ℎ119894119896

+ (120582119860119886119896+ 119860119886119896+ 119860119886119896) ℎ119894119895+ (119861119862

119896+ 119861119869119896) 119862119894119862119895

+ (119869119894119862119895+ 119869119869119862119894) 119861119862119896]

(33)


119862119896= minus

1

(120582 + 1205822)

[(119899 + 1) (120582119860119886119896+ 119860119886119896+ 119860119886119896)

+ (119861119862119896+ 119861119871119896) 1198622

+ 2119861119869119862119862119896]

(34)

whence

120582119860119886119896+ 119860119886119896+ 119860119886119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1)119862119896

minus1198611198622

(119899 + 1)119869119896

(35)


119862119894119895119896=

minus (120582 + 1205822

+ 1198611198622

+ 2119861119869119862)

(119899 + 1) (120582 + 1205822)

(119862119894ℎ119895119896+ 119862119895ℎ119896119895

+ 119862119896ℎ119894119869) +

1198611198622

(119899 + 1) (120582 + 1205822)

(119869119894ℎ119895119896+ 119869119895ℎ119896119894+ 119869119896ℎ119894119895)

+

(minus119861)

(120582 + 1205822)

119862119894119862119895119862119896+

(minus119861)

(120582 + 1205822)

(119869119894119862119895119862119896+ 119869119895119862119894119862119896

+ 119869119896119862119895119862119894)

(36)






coefficients if

119882119894

119895= 119871119894

119895119896

119889119909119896

119889119905

(37)

(see [16])

Connection 119871119894119895119896


119871119894

119895119896= 119878119894

119895119896+ 119879119894

119895119896 (38)

where

119878119894

119895119896=1

2(119871119894

119895119896+ 119871119894

119896119895) (39a)


119871119894

119895119896= 119875119894

119895119896+ 119879119894

119895119896science 119875119894

119895119896= 119878119894

119895119896 (39b)


119895119896= 119871119894

119896119895


119879119894

119895119896=

119871119894

119895119896minus 119871119894

119896119895

2 (40)

119879119894

119896119895= minus119879119894

119895119896 (41)



119879 = (119879119894

119895119896) (ℎ) ℎ-torsion

119881 = (119881119894

119895119896) (ℎ) ℎ]-torsion

1198771

= (119877119894

119895119896) (]) ℎ-torsion

1198751

= (119875119894

119895119896) (]) ℎ]-torsion

1198781

= (119878119894

119895119896) (]) ]-torsion

(42)


tensor and

119879119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895(43)

(see [9])

119875119894

119895119896= 119896119873119894

119869minus 119865119894

119896119895 (44)

119878119894

119895119896= 119881119894

119895119896minus 119881119894

119896119895 (45)


119871119894

119895119896minus 119871119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (46)


119878119894

119895119896minus 119879119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (47)


119871119894

119895119896minus 119875119894

119895119896= 119865119894

119895119896minus 119865119894

119896119895

119875119894

119895119896= 119871119894

119895119896minus 119865119894

119895119896+ 119865119894

119896119895

(48)




119878119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (49)

Using (45)

119881119894

119895119896minus 119881119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (50)


119879119894

119894119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (51)

119881119894

119895119896= 119881119894

119896119895 (52)


119865Therefore

119863119894

119895= 119873119894

119895

119910119903

119881119894

119903119895= 0

(53)

Put 119903 = 119896 we have

119910119896

119881119895

119896119895= 0 (54)

Using (52) we have

119910119896

119881119894

119895119896= 0 997904rArr

119910119903

119881119895

119896119903= 0

(55)




Competing Interests


References










ℎ119894119895119896and S



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878119894

119895119896= 2 (119865

119894

119895119896minus 119865119894

119896119895) (49)

Using (45)

119881119894

119895119896minus 119881119894

119896119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (50)


119879119894

119894119895= 2 (119865

119894

119895119896minus 119865119894

119896119895) (51)

119881119894

119895119896= 119881119894

119896119895 (52)


119865Therefore

119863119894

119895= 119873119894

119895

119910119903

119881119894

119903119895= 0

(53)

Put 119903 = 119896 we have

119910119896

119881119895

119896119895= 0 (54)

Using (52) we have

119910119896

119881119894

119895119896= 0 997904rArr

119910119903

119881119895

119896119903= 0

(55)




Competing Interests


References










ℎ119894119895119896and S



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on a special form of () -torsion tensor...

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