CUBO A Mathematical Journal
Vol.14, No¯ 01, (81–91). March 2012

Special Recurrent Transformation in an NPR-Finsler Space

Anjali Goswami

Department of Mathematics

Jagannath Gupta Institute of Engineering and Technology

Sitapura, Jaipur, India

email: dranjaligoswami@rediffmail.com

ABSTRACT

In this paper, an infinitesimal transformation x̄i = xi + �vi (xj), where the vector

vi is recurrent has been considered in an NPR- Finsler space. Such transformation

is being called special recurrent transformation if the recurrence vector of the NPR-

Finsler space is Lie invariant. Besides different properties of such transformation, the

conditions for such transformation to be curvature collineation and an affine motion

have been obtained.

RESUMEN

En este art́ıculo se considera una transformación infinitesimal x̄i = xi +�vi (xj), donde

el vector vi es recurrente, en un espacio NPR- Finsler. Tal transformación se dice

transformación recurrente especial si el vector recurrente del espacio NPR- Finsler es

Lie invariante. Además se han obtenido diferentes propiedades de dicha transformación

y las condiciones para que ésta sea una colineación de curvatura y una moción af́ın.

Keywords and Phrases: NPR-Finsler space, recurrent vector fields, special recurrent transfor-

mation, curvature collineation, affine motion.

2010 AMS Mathematics Subject Classification: 53B40.



82 Anjali Goswami CUBO
14, 1 (2012)

1 Introduction

Let an n-dimensional Finsler space Fn be equipped with fundamental metric function F(x
k, ẋk),

metric tensor gij and Berwald connection G
i
jk. Covariant derivative of any tensor with respect to

Berwald connection is given by [6]

Bk T
i
j = ∂kT

i
j − (∂̇rT

i
j)G

r
k h ẋ

h + TrjG
i
kr − T

i
rG

r
j k (1.1)

where ∂k ≡ ∂∂ xk and ∂̇r ≡
∂

∂ ẋr
.

The commutation formulae for the operators Bk and ∂̇k are given by

∂̇jBkT
i
h − Bk∂̇jT

i
h = T

r
hG

i
jkr − T

i
rG

r
j kh, (1.2)

BjBkT
i
h − BkBjT

i
h = T

r
hH

i
jkr − T

i
rH

r
j kh − (∂̇rT

i
h)H

r
j k, (1.3)

where

Gijkh = ∂̇hG
i
jk, (1.4)

Hijkh = ∂jG
i
kh + G

i
hrjG

r
k + G

i
rjG

r
kh − j/k (1.5)

and

Hijk = H
i
jkhẋ

h. (1.6)

The symbol -j/k means the subtraction of the earlier terms after interchanging j and k. The tensor

Gijkh is symmetric in its lower indices and satisfies

Gijkhẋ
h = Gijhkẋ

h = Gihjkẋ
h = 0 (1.7)

while the Berwald curvature tensor Hijkh satisfies

(a) Hijkh = −H
i
kjh, (b) H

i
jkh = ∂̇hH

i
jk. (1.8)

The Berwald deviation tensor Hij is defined by

(a)Hij = H
i
jkẋ

k, (b) Hijk = 1/3∂̇kH
i
j − j/k. (1.9)

Pandey[2] proved that the relation between the normal projective curvature tensor Nijkh defined

by Yano [7] and the Berwald curvature tensor Hijkh is given by

Nijkh = H
i
jkh −

ẋi

n + 1
∂̇hH

r
j kr, (1.10)

Nrj kr = H
r
j kr. (1.11)

The relation between the tensors Nijkh and H
i
jk is given by

Nijkhẋ
h = Hijk. (1.12)



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Special Recurrent Transformation in an NPR-Finsler Space 83

2 An NPR-Finsler Space

An NPR-Finsler space was defined by P. N. Pandey [2] in 1980. It is a Finsler space whose normal

projective curvature tensor Nij k h satisfies

Bm N
i
j k h = λm N

i
j k h, (2.1)

where λm is a covariant vector called recurrence vector. This vector is atmost a point function,

i.e. independent of the directional arguments.

It was observed by P. N. Pandey [2] that the tensors Hij k and H
i
j are recurrent in NPR-Finsler

space. Thus in an NPR-Finsler space, we have

(a) Bm H
i
j k = λm H

i
j k, (b) Bm H

i
j = λm H

i
j. (2.2)

However, an NPR-Finsler space is not necessarily a recurrent Finsler space. Also, a recurrent

Finsler sapce is not necessarily an NPR-Finsler space. In another paper, P.N. Pandey [4] established

the following identities:

λm N
i
j k h + λj N

i
k m h + λk N

i
m j h = 0, (2.3)

λm H
i
j k h + λj H

i
k m h + λk H

i
m j h = 0, (2.4)

λm H
i
j k + λj H

i
k m + λk H

i
m j = 0, (2.5)

He further proved that in such space, the second Bianchi identity splits into the following identities:

Bm H
i
j k h + Bj H

i
k m h + Bk H

i
m j h = 0, (2.6)

Hrj k G
i
m h r + H

r
k m G

i
j h r + H

r
m j G

i
k h r = 0. (2.7)

Contracting the indices in (2.2b) and using Hii = (n − 1)H, we get

Bm H = λm H. (2.8)

Differentiating (2.8) covariantly with respect to xh and taking skew-symmetric part, we have

(Bh Bm − Bm Bh)H = Ah m H (2.9)

where Ahm = Bh λm − Bm λh.

Using (1.3) in (2.9), we have

− ∂̇r HH
r
h m = Ah m H, (2.10)

which after further covariant differentiation gives

− (Bk∂̇r H)H
r
h = (Bk Ah m)H. (2.11)

Using the commutation formula (1.2) and the equation (2.10), we get

Bk Ah m = λk Ah m (2.12)



84 Anjali Goswami CUBO
14, 1 (2012)

provided H is non-vanishing. If we multiply (2.10) with λk and take skew-symmetric part, we find

λk Ah m + λh Am k + λm Ak h = 0 (2.13)

provided H 6= 0. Thus, we find that the recurrence vector λm of an NPR-Finsler space satisfies
(2.12) and (2.13) provided H 6= 0.
In view of the commutation formula given by (1.2), we get

∂̇j Bm λk − Bm ∂̇jλk = −λr G
r
j m k

which due to the fact that the recurrence vector is independent of ẋi, gives

∂̇j Bm λk = −λr G
r
j m k. (2.14)

Taking skew-symmetric part of (2.14), we get

∂̇j Am k = 0. (2.15)

Now

∂̇j Bk Ah m − Bk ∂̇j Ah m = −Ar m G
r
jkh l − Ah r G

r
j k m (2.16)

which, in view of (2.12) and (2.15), gives

Ar m G
r
j k h + Ah r G

r
j k m = 0. (2.17)

3 A Recurrent Vector Field in An NPR-Finsler space

A vector field vi is called recurrent if it satisfies

Bk v
i = µk v

i. (3.1)

Differentiating (3.1) covariantly with respect to xj and using the commutation formula (1.3), we

get

Hijkh v
h = µjk v

i (3.2)

where µjk = Bj µk − Bk µj. The tensor µjk may or may not vanish. Let us consider the case

when µjk 6= 0. From (1.10) and (3.2), we find(
Nijkh +

ẋi

n + 1
∂̇h N

r
jkr

)
vh = µjkv

i. (3.3)

Differentiating (3.3) covariantly with respect to xm, and using (2.1) and (3.1), we have(
λm N

i
jkh +

ẋi

n + 1
Bm∂̇h N

r
jkr

)
vh = vi Bmµjk, (3.4)

which in view of (1.2), gives

λm

(
Nijkh +

ẋi

n + 1
∂̇h N

r
jkr

)
vh +

ẋi

n + 1

(
NrskrG

s
hmj + N

r
jsrG

s
hmk

)
vh = viBmµjk. (3.5)



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Special Recurrent Transformation in an NPR-Finsler Space 85

From (3.3) and (3.5), we get

(
λm µjk − Bm µjk

)
vi +

ẋi

n + 1
vh

(
Nrskr G

s
hmj + N

r
jsr G

s
hmk

)
= 0. (3.6)

Transvecting (3.6) by yi and using yi ẋ
i = F2, we get(

λm µjk − Bm µjk
)
yi v

i + F
2

n+1
vh

(
Nrskr G

s
hmj + N

r
jsr G

s
hmk

)
= 0

which implies

vh

n + 1

(
Nrskr G

s
hmj + N

r
jsr G

s
hmk

)
=

1

F2

(
Bm µjk − λm µjk

)
yi v

i . (3.7)

Using (3.7) in (3.6), we get(
λm µjk − Bm µjk

)
vi − lilr v

r
(
λm µjk − Bm µjk

)
= 0 (3.8)

where li = ẋi/F and lr = yr/F.

(3.8) may be rewritten as(
λm µjk − Bm µjk

) (
vi − li lr v

r
)
= 0.

This implies at least one of the conditions

(a) Bm µjk = λm µjk, (b) v
i = li lr v

r. (3.9)

Suppose that the condition (3.9 b)holds. Then the partial differentiation with respect to ẋh gives

0 = (∂̇hl
i)lrv

r + li(∂̇hlr)v
r. (3.10)

Using ∂̇hl
i = 1

F
(δih − l

ilh) and ∂̇hlr =
1
F
(ghr − lhlr) in (3.10),we find

0 = (δih − l
ilh)lrv

r + li(ghr − lhlr)v
r.

Contracting the indices i and h and using δii = n and l
rlr = 1, we get (n − 1)lrv

r = 0.

This implies lrv
r = 0 for n 6= 1. In view of lrvr = 0, (3.9 b) gives vi = 0, a contradiction. Therefore

(3.9b) can not be true. Hence, we have (3.9a). From (2.4) and (3.2), we may deduce

λm µjk + λj µkm + λk µmj = 0. (3.11)

This leads to:

Theorem 3.1. In an NPR-Finsler space admitting a recurrent vector field vi given by (3.1), the

tensor µjk either vanishes identically or is recurrent and satisfies the identity (3.11).

Differentiating (3.1) partially with respect to ẋj and using the commutation formula (1.2), we

get

Gijkr v
r = (∂̇j µk)v

i. (3.12)

Transvecting (2.17) by vj ẋm and using (3.12), we get

Arm v
r ẋm ∂̇k µh = 0. (3.13)



86 Anjali Goswami CUBO
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This gives at least one of the following conditions:

(a) Arm v
r ẋm = 0, (b) ∂̇k µh = 0. (3.14)

If (3.14a) holds, then its partial drivatives with respect to ẋk gives

Ark v
r = 0. (3.15)

Transvecting (2.13) by vk and using (3.15), we find

λk v
k Ahm = 0. (3.16)

Since Ahm 6= 0, we have
λk v

k = 0. (3.17)

Thus we have

Theorem 3.2. In an NPR-Fnsler space admitting a recurrent vector field vi characterized by

(3.1), we have at least one of the conditions (3.14b) and (3.17).

Suppose (3.14b) holds, then we have

∂̇jBk µm = −µr G
r
jkm. (3.18)

Taking skew-symmetric part of (3.18) with respect to the indices k and m, we get

∂̇jµkm = 0. (3.19)

Differentiating (3.19) covariantly with respect to xh and using commutation formula exhibitted by

(1.2) and the equation (3.9a), we find µrm G
r
kjh + µkr G

r
mjh = 0.

4 A Special Recurrent Transformation

An infinitesimal transformation

x̄i = xi + �vi(xj) (4.1)

where vi is a covariant vector field and � is an infinitesimal constant, is called a special recurrent

transformation if the vector field vi is recurrent and the transformation does not deform the

recurrence vector λm of the NPR-Finsler space, i.e. if the vector field v
i satisfies (3.1) and

£λm = 0 (4.2)

where £ is the operator of Lie differentiation with respect to the infinitesimal transformation (4.1).

The necessary and sufficient condition for (4.1) to be an affine motion is given by

£Gijk = 0. (4.3)



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Special Recurrent Transformation in an NPR-Finsler Space 87

Since every affine motion is a curvature collination, (4.3) implies

£Hijkh = 0. (4.4)

Operating (1.10) by the operator £ and using (4.4), we get

£Nijkh = −
ẋi

n + 1
£ ∂̇hH

r
jkr, (4.5)

Since the operators £ and ∂̇h are commutative, (4.5) becomes £N
i
jkh = −

ẋi

n+1
∂̇h £ H

r
jkr which

in view of (4.4), gives

£Nijkh = 0. (4.6)

Let us consider an NPR-Finsler space admitting an affine motion. Then we have (2.1), (4.3), (4.4)

and (4.6).

Operating (2.1) by the operator £ and using (4.6), we have

£Bm N
i
jkh = (£λm)N

i
jkh. (4.7)

In view of the commutation formula

£Bk T
i
j − Bk£T

i
j = T

r
j £G

i
rk − T

i
r £G

r
jk − (∂̇r T

i
j ) £G

r
ks ẋ

s (4.8)

and equations (4.3) and (4.6), the equation (4.7) gives (4.2) for Nijkh 6= 0. Thus, we obsereve
that every affine motion generated by a recurrent vector field in an NPR-Finsler space is a special

recurrent transformation. Now, we wish to discuss its converse problem.

Let us consider a special recurrent transformation (4.1) in an NPR-Finsler space. This transfor-

mation is characterized by (3.1) and (4.2). In view of theorem (3.2), we have at least one of the

equations (3.14b) and (3.17). If (3.14b) does not hold, we must have (3.17), i.e. L = λr v
r = 0.

We shall divide the special recurrent transformations in two classes according as L 6= 0 and L = 0.
A special recurrent transformation is called of first kind if L 6= 0 while it is called of second kind
if L = 0.

Let us consider a special recurrent transformation of the first kind. For such transformation L 6= 0.
Therefore in view of Theorem (3.2), the vector field µk must be a point function, i.e. ∂̇j µk = 0.

Expanding the left hand side of equation (4.2) with the help of the formula

£T ij = v
r Br T

i
j − T

r
j Br v

i + T ir Bj v
r + (∂̇rT

i
j ) Bs v

r ẋs, (4.9)

we get

vr Br λm + Lµm = 0. (4.10)

Also

Bm L = Bm (λr v
r) = vr Bm λr + Lµm . (4.11)

Using (4.10) in (4.11), we have

vr Ark + Bm L = 0. (4.12)



88 Anjali Goswami CUBO
14, 1 (2012)

Differentiating (2.3) covariantly with respect to xp and using (2.1), we have

(Bpλm)N
i
jkh + (Bpλj)N

i
kmh + (Bpλk)N

i
mjh = 0. (4.13)

Transvecting (4.13) by vp and using (4.10), we get

µmN
i
jkh + µjN

i
kmh + µkN

i
mjh = 0. (4.14)

Differentiating (2.11) and (2.13) covariantly with respect to xp and then multiplying by vp, we get

(vp Bp λk)Ahm + (v
p Bp λh)Amk + (v

p Bp λm)Akh = 0,

and (vp Bp λk)µhm + (v
p Bp λh)µmk + (v

p Bp λm)µkh = 0,

which imply

µkAhm + µhAmk + µmAkh = 0 (4.15)

and

µkµhm + µhµmk + µmµkh = 0 (4.16)

since L 6= 0.
This proves the following:

Theorem 4.1. An NPR-Finsler space admitting a special recurrent transformation admits the

identities (4.14), (4.15) and (4.16) provided L 6= 0.

The commutation formula for the operators £ and Bk in case of the recurrence vector λm is

given by

£Bkλm − Bk£λm = −λr£G
r
mk,

which, in view of (4.2), gives

£Bkλm = −λr£G
r
mk. (4.17)

Taking skew-symmetric part of (4.17), we get

£Amk = 0. (4.18)

Transvecting (4.14) by ẋh and using (1.12), we get

µmH
i
jk + µjH

i
km + µkH

i
mj = 0. (4.19)

Now £Hijk = LH
i
jk + µH

i
jkrv

r − µrH
r
jkv

i + µjH
i
rkv

r + µkH
i
jrv

r.

Transvecting (4.19) by vm and using (3.2) in the above equation, we get

£Hijk = (L + µmv
m)Hijk + (µµjk − µrH

r
jk)v

i.

This shows that £Hijk = 0 if

L + µmv
m = 0 and µµjk − µrH

r
jk = 0. (4.20)

We know that £Hijk = 0 is equivalent to £H
i
jkh = 0.

Therefore we have:



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Special Recurrent Transformation in an NPR-Finsler Space 89

Theorem 4.2. A special recurrent transformation of the first kind is a curvature collineation if

(4.20) holds.

The Lie derivative of Gijk is given by

£Gijk = Bj Bkv
i + Himjkv

m + GijkrBsv
rẋs, (4.21)

which in the present case is given by

£Gijk = (Bjµk + µjµk)v
i + Himjkv

m , (4.22)

for Gijkrv
r = ∂̇jµkv

i = 0.

Differentiating (2.4) covariantly with respect to xp and transvecting by vp, we ge

(vpBpλm)H
i
jkh + (v

pBpλj)H
i
kmh + (v

pBpλk)H
i
mjh = 0.

Using (4.10) in it, we find

µmH
i
jkh + µjH

i
km + µkH

i
mj = 0 (4.23)

for L 6= 0.
Transvecting (2.4) and (4.23) by vm and adding, we get

(λk + µk)H
i
mjhv

m − (λj + µj)H
i
mkhv

m = 0.

From this we may conclude

Himjhv
m = φ(λj + µj)X

i
h. (4.24)

for some tensor Xih. Therefore

£Gijk = (Bjµk + µjµk)v
i + φ(λj + µj)X

i
k. (4.25)

From this we find that the special recurrent transformation is affine motion if

(Bj µk + µjµk)v
i = −φ(λj + µj)X

i
k.

Now we consider a special recurrent transformation of the second kind (L = 0). Transvecting (2.5)

by vm and using L = λmv
m = 0, we get

λjH
i
kmv

m + λkH
i
mjv

m = 0.

This is possible only when

Himkv
m = λkX

i (4.27)

for some vector field Xi. Since yiH
i
jk = 0, yiX

i = 0.

£Hijk, in view of (2.2), (3.1) and (3.17), becomes

£Hijk = µH
i
jkrv

r − Hrjkµrv
i + µjH

i
rkv

r + µkH
i
jrv

r (4.28)

where µ = µkẋ
k.

Using (3.2) and (4.9) in (4.10), we get

£Hijk = (µµjk − µrH
r
jk)v

i + (µjλk − µkλj)X
i. (4.29)



90 Anjali Goswami CUBO
14, 1 (2012)

This shows that £Hijk = 0 if

(a) µrH
r
jk = µµjk (b) µj = ψλj, (4.30)

where ψ is a scalar. Also £Hijk = 0 if and only if £H
i
jkh = 0.

This leads to

Theorem 4.3. A special recurrent transformation of the second kind in an NPR-Finsler space is

a curvature collineation if (4.30) holds.

In view of (4.21), we have

£Gijk = (Bjµk + µjµk + µ∂̇jµk)v
i + Himjkv

m (4.31)

which gives

£Gijk = (Bjµk + µjµk + µ∂̇jµk)v
i + λjX

i
k (4.32)

where Xik = ∂̇kX
i.

This shows that a special recurrent transformation of the second kind is an affine motion if

(Bjµk + µjµk + µ∂̇jµk)v
i = −λjX

i
k. (4.33)

Transvecting this equation by ẋk, we get

(Bjµk + µjµk)ẋ
kvi = −λjX

i. (4.34)

Transvecting this equation by yi, we have

(Bjµk + µjµk)ẋ
k = 0 (4.35)

for yiv
i 6= 0 and yiXi = 0.

Using (4.35) in (4.34), we get Xi = 0. Therefore Xik = 0.

Using Xik = 0 in equation (4.33), we get

Bjµk + µjµk + µ∂̇jµk = 0. (4.36)

Thus (4.33) implies (4.36). Conversely if (4.36) holds, its skew symmetric part gives

µjk = Bjµk − Bkµj = 0. (4.37)

Using this in (3.2) we get Hijkhv
h = 0, which implies Himjkv

m = 0.

Therefore Xik = 0.

Hence we conclude:

Theorem 4.4. A special recurrent transformation of the second kind in an NPR-Finsler space is

an affine motion if Bjµk + µjµk + µ∂̇jµk = 0.

Received: September 2010. Revised: May 2011.



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Special Recurrent Transformation in an NPR-Finsler Space 91

References

[1] Pandey, P. N., A recurrent Finsler manifold admitting special transformations, Progress of

Mathematics, 13 (1979), 85-98.

[2] Pandey, P. N., On NPR-Finsler manifold, Ann. Fac. Kinshasha, 6 (1980), 65-77.

[3] Pandey, P. N., Affine motion in a recurrent Finsler manifold, Ann. Fac. Kinshasha, 6 (1980),

51-63.

[4] Pandey, P. N., Some identities in an NPR-Finsler manifold, Proc. Nat. Acad. Sci. (India), 51

(1981), 105-109.

[5] Pandey, P. N., Certain types of affine motion in a Finsler manifold, Colloquium Mathematicum,

49 (1985), 243-252.

[6] Rund, H., The Differential Geometry of Finsler spaces, Springer-Verlag, Berlin, 1959.

[7] Yano, K., The theory of Lie derivatives and its applications, North Holland Publ. Co., Ams-

terdam, 1957.


	Introduction
	An NPR-Finsler Space
	A Recurrent Vector Field in An NPR-Finsler space
	A Special Recurrent Transformation