(1)ON CURVATURE INHERITING SYMMETRY IN FINSLER SPACE C

ON CURVATURE INHERITING SYMMETRY IN FINSLER SPACE

C. K. Mishra and Gautam Lodhi

Abstract. K. L. Duggal[3] has stuided curvature Inheritance symmetry Rie- mannian space with application of fluid space time. Ricci curvature inheriting symmetry of semi-Riemannian manifolds were introduced by K. L. Duggal and R.

Sharma[7]. In this paper we have study on curvature inheritance symmetry and Ricci-Inheriting symmetry in Finsler space and investigated some results.

2000Mathematics Subject Classification: 53B40.

Keyword and phrases: Finsler space, curvature & relative curvature tensor, cur- vature inheritance, Ricci inheritance, projective motion, CI projective motion.

1.Introduction and Preliminaries

We consider an n-dimensional Finsler space (F n) in which the curvature tensor field due to Berwald’s are given by

H_jkⁱ = ∂²Gⁱ

∂x^k∂x˙^j − ∂²Gⁱ

∂x^j∂x˙^k +Gⁱ_kr∂G^r

∂x˙^j −Gⁱ_rj∂G^r

∂x˙^k, (1) H_jkhⁱ = ∂Gⁱ_jk

∂x^h − ∂Gⁱ_jh

∂x^k +G^r_jhGⁱ_rk−G^r_jkGⁱ_rh+Gⁱ_rjh∂G^r

∂x˙^k −Gⁱ_rjk∂G^r

∂x˙^h, (2) where

( (a) H_jkhⁱ = ^∂H_∂x˙^khij ,

(b) Gⁱ_jkh= ^∂G_∂x˙^ikhj . (3) The commutation formulae involving the curvature tensor are given by

T_(h)(k)−T_(k)(h) = ∂T

∂x˙ⁱH_hkⁱ , (4)

T_ij(h)(k)−T_ij(k)(h)=−∂Tij

∂x˙^rH_hk^r −TrjH_ihk^r −TirH_jhk^r , (5) T_jkh(l)(m)ⁱ −T_jkh(l)(m)ⁱ = −∂T_jkhⁱ

∂x˙^r H_lm^r +T_jkh^r H_rlmⁱ −T_rkhⁱ H_jlm^r (6)

−T_jrhⁱ H_klm^r −T_jkrⁱ H_hlm^r . The curvature tensor satisfies the following relations.



































(a) H_jkhⁱ =−H_jhkⁱ , (b) H_jkhⁱ = ˙∂jH_khⁱ , (c) H_kh=H_ikhⁱ , (d) Hk=H_ikⁱ , (e) H_j^j = (n−1)H,

(f) ∂˙_lH_jkhⁱ x˙^j = ˙∂_lH_jkhⁱ x˙^l = 0, (g) H_jkhⁱ x˙^h=H_jkⁱ ,

(h) H_jkⁱ x˙^h =H_jⁱ, (i) H_jⁱx˙^j = 0.

(7)

The relative curvature tensor in a Finsler space is defined as[1].

K˜_jkhⁱ = ∂Γ^∗i_jk

∂x^h +∂Γ^∗i_jk

∂x˙ⁱ

∂ξⁱ

∂x^h

!

− ∂Γ^∗i_jh

∂x^k +∂Γ^∗i_jh

∂x˙^l

∂ξ^l

∂x^k

!

(8) +Γ^∗i_mhΓ^∗m_jk + Γ^∗i_mkΓ^∗m_jh .

The commutation formula involving the relative curvature tensor is given by T_ij;kh−T_ij;hk =−T_irK˜_jkhⁱ −TrjK˜_ikh^r , (9) whereTij is an arbitrary tensor field and ; is a covariant δ derivative.

The relative curvature tensor satisfies the following relations.

( (a) K˜_jkhⁱ =−K˜_jhkⁱ ,

(b) K˜ij = ˜K_ijl^l . (10)

Let an infinitesimal transformation

¯

xⁱ =xⁱ+εvⁱ(x^j), (11)

be generated by a vector field vⁱ(x^j) independent of directional arguments (de- pendent of positional coordinatesxⁱonly). This transformation is called infinitesimal because of infinitesimal constantε appearing in (11).

The Lie derivatives of a vector fieldXⁱ, connection parametersGⁱ_jk and curvature tensor H_jkhⁱ are given by[2] .

£Xⁱ=v^jxⁱ_(j)−X^jv_(j)ⁱ + ˙∂_jXⁱv^j_(l)x˙^l, (12)

£Gⁱ_jk =vⁱ_(j)(k)+v^hH_hjkⁱ +Gⁱ_jkhv^h_(l)x˙^l, (13)

£H_jkhⁱ = v^lH_jkh(l)ⁱ −H_jkh^l vⁱ_(l)+H_lkhⁱ v^l_(j)+H_jlhⁱ v_(k)^l + (14) H_jklⁱ v^l_(h)+ ˙∂_lH_jkhⁱ v_(m)^l x˙^m.

The process of Lie differentiation and other differentiation of any arbitrary tensor and connection coefficient Gⁱ_jk are connected by

∂˙l(£T_jkⁱ )−£( ˙∂lT_jkⁱ ) = 0, (15)

£{(T_jⁱ)_(k)} −(£T_jⁱ)_(k)=£Gⁱ_klT_j^l−£G^l_kjT_lⁱ−∂˙_hGⁱ_kT_j^h, (16)

(£Gⁱ_jh)_(k)−(£Gⁱ_kh)_(j)=£H_hjkⁱ + (£G^r_kl)Gⁱ_rjhx˙^l−(£G^r_jl)Gⁱ_rhkx˙^l. (17) As indicated by K. Takano[4] the covariant vector field may assume any one of following alternative forms.

v_(l)ⁱ = 0, (18)

vⁱ_(l)=cδⁱ_j, (19) v_(l)ⁱ =βδⁱ_j. (20) Accordingly, the vector field vⁱ is respectively called as a contra vector field, concurrent vector field and special concircular vector field. Where vⁱ is a scalar function and c being a non zero constant.

Special Finsler Spaces

Definition 1.In F n, the Berwald’s curvature tensor satisfies the relation

H_jhk(l)ⁱ =λ_lH_jhkⁱ , (21)

is called recurrent Finsler space denoted by HR−F_n[5].

and

H_jhkⁱ 6= 0, (22)

whereλ_l is non zero scalar vector independent of the directional arguments.

Definition 2.In F_n, the covariant derivative of Berwald’s curvature tensor is zero at every point, is called a symmetric Finsler space[6].

H_jhk(l)ⁱ = 0. (23)

2.Curvature inheritance and Ricci Inheritance symmetry

The curvature inheritance(CI) defined as an infinitesimal transformation with respect to which the Lie derivative of Berwald’s curvature tensor H_jkhⁱ satisfies a relation of the form

£H_jkhⁱ =αH_jkhⁱ . (24)

Similarly, we have define the curvature inheritance symmetry satisfies by relative curvature tensor of the form

£K˜_jkhⁱ =αK˜_jkhⁱ . (25) Whereα=α(x) is a scalar function. A subcase of CI is the well Known symmetry curvature collineation (£H_jkhⁱ = 0) denoted by CC, whenα = 0. In the sequel we say that CI is a proper if α6= 0.

We also define Ricci-Inheritance(RI) Symmetry

£H_jk =αH_jk, (26)

and for relative curvature tensor

£K˜_jk =αK˜_jk, (27)

If α follows the invariance property, its Berwald’s covariant derivative vanish.

α_(k)= 0. (28)

Let us set

£g_ij =h_ij. (29)

Wheregij is a metric tensor defined Rund[1].

g_ij(x,x) =˙ 1

2∂˙_i∂˙_jF²(x,x).˙ (30) Theorem 1.A necessary condition for a vector vⁱ to defined a CI is

(hij)_(h)(k)−(hij)_(k)(h)= ∂hij

∂x˙^rH_hk^r . (31) Proof. The curvature tensor satisfies the following identity

g_rjH_ihk^r +g_riH_jhk^r = 0. (32) Taking the Lie derivative of (32) and using (24), (29) and (32), we get

h_rjH_ihk^r +h_riH_jhk^r = 0. (33) Using (33) in commutation formula (5), we obtain(31).

Theorem 2.A necessary condition for a vectorvⁱ to defined a CI is (for relative curvature tensor).

h_ij;hk−h_ij;kh= 0. (34)

Proof: The relative curvature tensor satisfies the identity

g_rjK˜_ihk^r +g_riK˜_jhk^r = 0. (35) Taking the Lie derivative of (35) and using (25), (35), we get

h_rjK˜_ihk^r +h_riK˜_jhk^r = 0. (36) Using (36) in commutation formula (9), we obtain (34).

The infinitesimal transformation is said to be an affine motion if it satisfies the condition

£Gⁱ_jh= 0. (37)

Applying (24) and (37) in (17), we get

H_hjkⁱ = 0. (38)

Hence we have

Theorem 3.InFn,the necessary condition is that curvature inheriting symmetry admitted an affine motion, space is flat.

Contracting equation (38) with respect to indices iand k, we obtain

H_hj = 0. (39)

Corollary 1.In F_n, the necessary and sufficient condition that Ricci curvature inheritance symmetry admitted an affine motion, then space is Ricci flat.

Now we considered the consequence of CI characterized by the equation (24).

Its successive transvection with the directional coordinates, and using (7)(g), (7)(h), and (7)(i), we obtain

a) £H_jkⁱ =αH_jkⁱ ,

b) £H_jⁱ =αH_jⁱ. (40) The partial differentiation of (40) with respect to directional coordinates, and applying equation (7)(b) and (15), we get (24).

Thus we say that, either of the conditions expressed by (24) and (40) are equiv- alent.

Accordingly we state

Theorem 4.The necessary and sufficient conditions for an infinitesimal trans- formation to define CI.

3.Relation between special Finsler spaces, special vector field and curvature inheritance.

We consider an infinitesimal transformation generated by vector fieldvⁱsatisfying the conditions (18), (19) and (20).

In view of (18), the equation (14) becomes

£H_jkhⁱ =v^lH_jkh(l)ⁱ . (41)

Applying (23) in (41), we get

£H_jkhⁱ = 0. (42)

Hence we have

Theorem 5.A symmetric Finsler space with contra vector field admits no CI other then CC.

Using (18), (21) and (24) in (14), we have

αH_jkhⁱ =v^lλlH_jkhⁱ . (43) Let us assume that a vectorv^l is orthogonal to vectorλl,that is

v^lλ_l= 0. (44)

In view of (44), the equation (43) reduces to

αH_jkhⁱ = 0. (45)

which immediately reduces to

α= 0. (46)

In view of (22).

Using (46) in (24), we get

£H_jkhⁱ = 0. (47)

Conversely, If the above equation (46) is true, the equation (43) yields

v^lλ_lH_jkhⁱ = 0. (48) Since H_jkhⁱ is non zero inHR−F_n,which implies

v^lλ_l= 0, (49)

which shows that a vectorv^l is orthogonal to vectorλ_l. Accordingly we state

Theorem 6.In HR−F_n, the necessary and sufficient condition for CI becomes CC is that the vector v^l is orthogonal to vector λ_l.

In view of (19) the equation (14) becomes

£H_jkhⁱ =v^lH_jkh(l)ⁱ + 2cH_jkhⁱ . (50)

Applying (23) in (50), we have

£H_jkhⁱ = 2cH_jkhⁱ . (51)

From equation (24) and (51), we get

α

2 =c. (52)

Using (52) in (51), we obtain

£H_jkhⁱ =αH_jkhⁱ . (53)

Hence we have

Theorem 7.A symmetric Finsler space with concurrent vector field admits CI.

Using equation (20) and (7)(f), in (14), we have

£H_jkhⁱ =v^lH_jkh(l)ⁱ + 2βH_jkhⁱ , (54)

by virtue of (23), the equation (54) becomes

£H_jkhⁱ = 2βH_jkhⁱ . (55)

Conversely, If the above equation (55) is true, the equation (54) reduces to

v^lH_jkh(l)ⁱ = 0. (56)

Since, v^l is non zero vector, it implies

H_jkh(l)ⁱ = 0. (57)

which shows that the space is symmetric.

Thus we state

Theorem 8.A symmetric Finsler space with special concircular vector field, the necessary and sufficient condition for the space admits CI is that the space is sym- metric.

3.CI projective motion

In this section we shall discuss the possibilities of an infinitesimal transformation to be simultaneously CI projective motion we drive an explicit expression which admitting a CI projective motion and obtain it possibilities in symmetric Finsler space.

The condition of projective motion in infinitesimal transformation of the form

£Gⁱ_kj = 2δⁱ_(kp_j)+ ˙xⁱp_kj, (58)

(a) pk= ˙∂kp,

(b) Gⁱ_kjx˙^j =Gⁱ_k. (59)

where p(x,x) is homogeneous scalar function of degree one in ˙˙ xⁱ. The consequent equation of projective motion is given by

£Gⁱ_k=δⁱ_kp+ ˙xⁱp_k, (60) (While driving (4.3), we have used (4.2)(b) and homogeneous properties of p.)

According theorem(4) the equation (40) gives a condition equivalent to that given by (24), therefore for simplicity (40) may be taken as defining equation for CI.

Applying (58), (60) and (40)b in commutation formula (16) and using homogeneous properties of H_jⁱ,we obtain

£{(H_jⁱ)_(k)} −H_jⁱ(α)_(k)−α(H_jⁱ)_(k) = H_j^l(p_lδⁱ_k+ ˙xⁱp_kl) (61)

−(p_jH_kⁱ + 2p_kH_jⁱ+p∂˙_kH_jⁱ).

Contracting the equation (61) with respect to indicesi, j and using (7)i, we have

£{(H)_(k)} −H(α)_(k)−α(H)_(k)=−(2Hp_k+p∂˙_kH). (62) Transvecting the equation (62) by ˙xⁱ and using homogeneous properties ofpand H, we get

p= 1 4

(α)_(k)+α(H)_(k)

H −£{(H)_(k)} H

. (63)

Thus we state

Theorem 9.The Complete integral of CI projective motion in a manifold of non zero curvature is given by (63).

For a symmetric Finsler space vanishing the covariant derivative of non zero curvature tensor the equation (63) reduces to

p= 1

4(α)_(k), (64)

In the view of (28) the equation (64) immediately reduces to

p= 0. (65)

Thus, the projective motion under the consideration reduces to an affine motion.

we, therefore, have the

Theorem 10.There exist non- trivial CI projective motion in a symmetric Finsler manifold.

References

[1] H. Rund, The Differential geommetry of Finsler spaces, Springer-Verlag, (1959).

[2] K. Yano, The theory of Lie derivative and its applications, North-Holland, Amsterdam, (1957).

[3] K.L. Duggal, Curvature inheriting symmetry in Riemannian space with ap- plication of fluid space time, J. Math . Phys. 33(9), (1992).

[4] K.Takano, Affine motion in non-Riemannian K*-spaces I, II, III (with M.

Okumara), IV,V, Tensor, N. S., 11(1961), 137-143.

[5] R. N. Sen,Finsler space of recurrent curvature tensors, Tensor, N.S., 19(1968), 291-199.

[6] R. B. Misra,A symmetric Finsler space, Tensor, N. S., 24(1972), 346-350.

[7] K.L. Duggal and R. Sharma, Ricci curvature inheriting symmetry of semi Riemannian manifolds, Contemporary Mathematics 170, (1994).

C. K. Mishra

Department of Mathematics and Statistics Dr. R.M.L. Avadh University

Faizabad -224001(U.P.)India

email:[email protected] Gautam Lodhi

Department of Mathematics

Babu Banarasi Das Group of Educational Institutions Lucknow-227105(U.P.) India

email:lodhi [email protected]