Abstract: The paper contains a study of an infinitesimal projective motion in a Finsler space Fn(n > 2), which leaves invariant the skew- symmetric part of the covariantderivative of projective deviation tensor and it is proved that either the Finsler space Fn(n > 2) admitting such projective motion is a space of scalar curvature or the infinitesimal projective motion is necessarily an affine motion. It is established that an infinitesimal projective motion in a projectively flat as well as in a non- Riemannian symmetric Finsler space of dimension greater than 2, is necessarily anaffine motion while a symmetric Finsler space Fn(n > 2) admitting a non- affine projective motion is a Riemannian space of constant Riemannian curvature. An infinitesimal projective motion in a Finsler space Fn(n > 2) of recurrent projective deviation tensor is an affine motion if the projective motion leaves invariant the recurrence vector of the space. It is further proved that such result also holds in case of projective recurrent and recurrent Finsler space of dimension greater than 2

Kumari, Sweta

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 31, No 1 (2016), 237–244GENERALIZATION OF CERTAIN RESULTS ON PROJECTIVE MOTION INA FINSLER SPACEP.N. Pandey and Sweta Kumari∗Abstract. The paper contains a study of an infinitesimal projective motion in a Finslerspace Fn(n > 2), which leaves invariant the skew-symmetric part of the covariant deriva-tive of projective deviation tensor and it is proved that either the Finsler space Fn(n > 2)admitting such projective motion is a space of scalar curvature or the infinitesimal projec-tivemotion is necessarily an aﬃnemotion. It is established that an infinitesimal projectivemotion in a projectively flat as well as in a non-Riemannian symmetric Finsler space ofdimension greater than 2, is necessarily an aﬃne motion while a symmetric Finsler spaceFn(n > 2) admitting a non-aﬃne projective motion is a Riemannian space of constantRiemannian curvature. An infinitesimal projective motion in a Finsler space Fn(n > 2) ofrecurrent projective deviation tensor is an aﬃne motion if the projective motion leavesinvariant the recurrence vector of the space. It is further proved that such result alsoholds in case of projective recurrent and recurrent Finsler space of dimension greaterthan 2.Keywords: Finsler space, Projective motion, projectively symmetric space, recurrentspace, projective recurrent space.1. IntroductionK. Yano and T. Nagano [1] discussed an infinitesimal projective motion in ann-dimensional Riemannian space Vn(n > 2) which leaves invariant the covariantderivative of Weyl’s projective curvature tensor. It was proved by them that if thespace is not of constant curvature then the projectivemotion is necessarily an aﬃnemotion. They also discussed an infinitesimal projective motion in a symmetricRiemannian space Vn(n > 2) and proved that such space is of constant curvature ifthe projective motion is non-aﬃne. P. N. Pandey [2] extended these results to ann-dimensional Finsler space Fn which leaves invariant the covariant derivative ofWeyl’s projective tensor Wijkh. He also generalized these results by considering aninfinitesimal projective motion which leaves invariant the covariant derivative ofthe projective deviation tensor. This condition is certainly weaker than the earlier.Received October 12, 2015; Accepted December 11, 20152010 Mathematics Subject Classification. 53B40∗Corresponding author.237238 P.N. Pandey and S. KumariSimilar results were also found by P. N. Pandey [2] for a projectively symmetricFinsler space which is more general than a symmetric Finsler space. The aim ofthe present paper is to generalize the results of the first author (P. N. Pandey [2])by considering an infinitesimal projective motion which leaves invariant the skew-symmetric part of the covariant derivation tensor. We also aim to prove certainresults for recurrent and projective recurrent Finsler spaces of dimension greaterthan 2, admitting an infinitesimal projective motion.2. PreliminariesLet us consider a Finsler space Fn(n > 2) of dimension n having F as a metricfunction satisfying the requisite conditions [3]. Let us denote the componentsof corresponding metric tensor, Berwald’s connection parameters, componentsof Berwald’s curvature tensor and components of projective curvature tensor byi j,Gijk,Hijkh and Wijkh respectively. Hijkh and Wijkh are positively homogeneous ofdegree 0 in yi and are skew-symmetric in their last two lower indices. Followingtensors are obtained after transvectionof curvature andprojective curvature tensorsby the directional arguments yi of the line element (xi, yi):(a) Hikh = Hijkhyj, (b) Hih = Hikhyk,(c)Wikh =Wijkhyj, (d) Wih =Wikhyk.(2.1)The tensors Hih and Wih are called deviation tensor and projective deviationtensor respectively.The above tensors are also related by(a)Hijkh = ∂˙ jHikh, (b)Hikh =13(∂˙kHih − ∂˙hHik),(c)Wijkh = ∂˙ jWikh, (d)Wikh =13(∂˙kWih − ∂˙hWik)(e)Wih = Hih −Hδih −yin + 1(∂˙rHrh − ∂˙hH)(2.2)where H is scalar curvature defined by H = 1(n−1)Hii and ∂˙r ≡ ∂∂yr .The directional diﬀerential operator ∂˙k and Berwald covariant diﬀerential oper-ator Bh satisfy the commutation formula:(2.3) ∂˙kBhTij −Bh∂˙kTij = TrjGikhr − TirGrkhj ,whereTij denote the components of an arbitrary tensor of type (1, 1) andGikhr = ∂˙rGikhare components of a symmetric tensor i.e. Gijkh = Gikhj = Gihjk. A Finsler space Fn iscalled symmetric or projectively symmetric according as [4](2.4) BmHijkh = 0, Hijkh  0.Generalization of Certain Results on Projective Motion in a Finsler Space 239or(2.5) BmWijkh = 0, Wijkh  0.A Finsler space Fn is called recurrent or projectively recurrent according as(2.6) BmHijkh = λmHijkh, Hijkh  0.or(2.7) BmWijkh = λmWijkh, Wijkh  0.The vector λm( 0) is called recurrence vector. It was established by P. N. Pandey[5] that the recurrence vector of a recurrent space is independent of directionalarguments provided the scalar curvature, H  0. The tensors Wikh and Wih of arecurrent as well as a projectively recurrent Finsler space are recurrent, i.e.(2.8) (a)BmWikh = λmWikh, (b)BmWih = λmWih.Let us consider an infinitesimal transformation T : Fn → Fn such that T(xi) = x˜i and(2.9) x˜i = xi + vi(xj)where  is an infinitesimal constant. Let us denote the operator of Lie- diﬀerentia-tion with respect to the above transformation by £. The commutation formula forthe operators £ and Bm is given by(2.10) £BmTij −Bm£Tij = Trj£Girm − Tir£Grjm − (∂˙rTij)£Grmhyhwhere Tij are components of an arbitrary tensor of type (1, 1).The transformation (2.9) is known as an aﬃne motion or a projective motion if itpreserves the parallelism of vectors or geodesics respectively. The necessary andsuﬃcient condition for the transformation (2.9) to be an aﬃne motion is(2.11) £Gijk = 0while the necessary and suﬃcient condition for the transformation (2.9) to be aprojective motion is(2.12) £Gijk = yipjk + pjδik + pkδij,where(2.13) (a) pj = ∂˙ jp, (b) pjk = ∂˙ j∂˙kp,and p is a scalar invariant which is positively homogeneous in yi of degree 1. Thevector pj and the tensor pjk satisfy(2.14) (a) pjyj = p, (b) pjkyk = 0.Every aﬃne motion is a projective motion trivially. A non-aﬃne projective motionis characterized by (2.12) and p  0.240 P.N. Pandey and S. Kumari3. Projective Motion in Symmetric and Projectively Symmetric Finsler SpacesLet a Finsler space Fn(n > 2) admits an infinitesimal projective motion. Then, wehave (2.12) together with (2.13). The integrability condition [3] of (2.12) is given by(3.1) £Wijkh = 0.Transvecting (3.1) by yj and using (2.1c), we get(3.2) £Wikh = 0.Again transvecting (3.2) by yk and using (2.1d), we get(3.3) £Wih = 0.Replacing Tij in the commutation formula (2.10) by Wij and using (3.3), we have(3.4) £BmWij =Wrj£Girm −Wir£Grjm − (∂˙rWij)£Grmsyswhich in view of (2.12) yields(3.5) £BmWij = yiWrjprm + prWrjδim − pjWim − 2pmWij − (∂˙mWij)p.Taking skew-symmetric part of (3.5) with respect to the indices m and j, we obtain£(BmWij −B jWim) = yiWrjprm − yiWrmprj + pr(Wrjδim −Wrmδij)+(pjWim − pmWij) − p(∂˙mWij − ∂˙ jWim).(3.6)Under the assumption that the left hand side of (3.6) equals to zero, i.e. £(BmWik −BkWim) = 0, it follows that(3.7) yiWrjprm − yiWrmprj + pr(Wrjδim −Wrmδij) + (pjWim − pmWij) − 3pWimj = 0.Transvecting (3.7) by ym and using (2.14a), we conclude(3.8) yiprWrj − 4pWij = 0.Transvecting (3.8) by pi and using (2.14a), we get(3.9) 3pprWrj = 0.This implies at least one of the following conditions:(3.10) (a) p = 0, (b) prWrj = 0.If (3.10a) holds, the projective motion is an aﬃne motion. If (3.10a) does not hold,(3.10b) must hold. Using (3.10b) in (3.8), we get pWij = 0, which implies Wij = 0 forp  0. Z. I. Szabo [6] and P. N. Pandey [7] proved that a Finsler space Fn(n > 2)whose projective deviation tensor vanishes identically, is a Finsler space of scalarcurvature. Therefore, the space considered is a Finsler space of scalar curvature.Thus, the next statement holds.Generalization of Certain Results on Projective Motion in a Finsler Space 241Theorem 3.1. A Finsler space Fn(n > 2) admitting an infinitesimal projective motionwhich leaves invariant the skew-symmetric part of the covariant derivative of projectivedeviation tensor, i.e. £(BmWik −BkWim) = 0, is a space of scalar curvature or the projectivemotion is an aﬃne motion.Let us consider a Finsler space Fn(n > 2) which satisfies BmWik = 0,Wik  0.Then, the condition £(BmWik − BkWim) = 0 is trivially satisfied. The Finsler spaceconsidered cannot be of scalar curvature because a Finsler space of scalar curvatureis projectively flat, which gives Wih = 0. Therefore, in view of the above theorem,the next result can be stated.Theorem 3.2. If the projective deviation tensor Wih of a Finsler space Fn(n > 2) satisfiesBmWih = 0, an infinitesimal projective motion in such space is necessarily an aﬃne motion.Since, the projective deviation tensor of a projective symmetric space satisfiesBmWih = 0, it is possible to conclude:Theorem 3.3. An infinitesimal projectivemotion in a projectively symmetric spaceFn(n >2) is necessarily an aﬃne motion.Let Fn(n > 2) be a symmetric Finsler space characterised by (2.4). If it admitsan infinitesimal projective motion, then we have (2.12). On transvecting (2.4) by yjand using (2.1a), we get(3.11) BmHikh = 0.Diﬀerentiating (3.11) partially with respect to yj and using (2.3) , we get(3.12) HrkhGijmr −HirhGrjmk −HikrGrjmh = 0.Transvecting (3.12) by yk and using (2.1a) and (2.1b), we get(3.13) HrhGijmr −HirGrjmh = 0.Transvecting (3.11) by yk and using (2.1b), we get(3.14) BmHih = 0.Diﬀerentiating (3.14) partially with respect to yk and using (2.3) and (3.13), we have(3.15) Bm∂˙kHih = 0.Contracting the indices i and h in equations (3.14) and (3.15), one can verify(3.16) BmH = 0 and Bm∂˙kH = 0.242 P.N. Pandey and S. KumariContracting the indices i and h in equation (3.15), we conclude(3.17) Bm∂˙rHrh = 0.Diﬀerentiating (2.2 e) covariantly with respect to xm and using (3.14), (3.15), (3.16)and (3.17), we get(3.18) BmWih = 0.which implies(3.19) £BmWih = 0.Taking skew-symmetric part of (3.19) with respect to the indices m and h, the nextfollows:(3.20) £(BmWih −BhWim) = 0.In view of Theorem 3.1, either the projectivemotion is an aﬃnemotion or the spaceis of scalar curvature. P. N. Pandey [8] established that a symmetric Finsler spaceFn of scalar curvature is a Riemannian space of constant Riemannian curvature.Hence, we conclude:Theorem 3.4. An infinitesimal projectivemotion in a non-Riemannian symmetric Finslerspace Fn(n > 2) is necessarily an aﬃne motion.Theorem 3.5. A symmetric Finsler space Fn(n > 2) admitting a non-aﬃne projectivemotion is a Riemannian space of constant Riemannian curvature.4. Projective Motion in Recurrent and Projective Recurrent Finsler SpacesLet us consider a Finsler space Fn(n > 2) whose projective deviation tensorWihis recurrent. Such Finsler space is characterized by (2.8b) andWih  0. Suppose thatit admits an infinitesimal projective motion (2.9). Then, we have (2.12) togetherwith (2.13). Operating equation (2.8b) by the Lie diﬀerential operator £, we find£BmWih = £(λmWih). This implies(4.1) £BmWih = (£λm)Wih.If £λm = 0, equation (4.1) implies £BmWih = 0. Therefore, £(BmWih −BhWim) = 0.In view of Theorem 3.1, this implies that either the infinitesimal projective motionis an aﬃne motion or the space is of scalar curvature. The space considered cannot be of scalar curvature for the projective deviation tensor Wih of a space ofscalar curvature necessarily vanishes which contradicts our assumption Wih  0.Therefore, the projective motion is necessarily aﬃne.This leads to:Generalization of Certain Results on Projective Motion in a Finsler Space 243Theorem 4.1. An infinitesimal projectivemotion in a Finsler space Fn(n > 2) of recurrentprojective deviation tensor, which leaves the recurrence vector invariant, is necessarily anaﬃne motion.A projective recurrent Finsler space Fn(n > 2) necessarily satisfies (2.8b) [9]. There-fore, in view of Theorem 4.1, we have:Corollary 4.1. An infinitesimal projective motion in a projective recurrent Finsler spaceFn(n > 2) is an aﬃne motion if the infinitesimal projective motion leaves the recurrencevector of the projective recurrent space invariant.Let us consider a recurrent Finsler space Fn characterised by (2.6). Transvecting(2.6) by yj and using (2.1a), we have(4.2) BmHikh = λmHikhTransvecting (4.2) by yk and using (2.1b), we get(4.3) BmHih = λmHih.Contracting the indices i and h in (4.3), we get(4.4) BmH = λmHwhereHrr = (n−1)H. Diﬀerentiating (2.2e) covariantly with respect to ym and using(4.3) and (4.4), we have(4.5) BmWih = λm(Hih −Hδih) −yin + 1(Bm∂˙rHrh −Bm∂˙rH)Using the commutation formula exhibited by (2.3) and using (4.3), (4.4) and (2.2e),we get(4.6) BmWih = λmWih −yin + 1(HrsGshmr −HshGrrms).P. N. Pandey [5] proved that a recurrent space admits the identity (3.13). Thus,equation (4.6) gives (2.8 b). Therefore, in view of Theorem 4.1, we have the nextresult.Theorem 4.2. An infinitesimal projective motion in a recurrent Finsler space Fn(n > 2)which leaves the recurrence vector invariant is necessarily an aﬃne motion.244 P.N. Pandey and S. KumariREFERENCES[1] K. Yano and T. Nagano, Some theorems on projective and conformal transforma-tions, Nederl. Akad. Wetensch. Proc. Ser. A. 60= Indag. Math., 19 (1957), 451-458.[2] P. N. Pandey, Projective motion in a symmetric and projectively symmetric Finslermanifold , Proc. Nat. Acad. Sci., 54(A) (1984), 274-278.[3] H. Rund, The diﬀerential Geometry of Finsler spaces, Springer- Verlag, (1959).[4] R. B. Misra, A projective symmetric Finsler space,Math. Z., 126 (1972), 143-153.[5] P. N. Pandey, A note on recurrence vector, Proc. Nat. Acad. Sci., 50 (A) (1980), 6-8.[6] Z. I. Szabo, Ein Finslerscher Raum ist gerade dann von skalarer Kru¨mmung, wennseine Weylsche Projektivkru¨mmung verschwindet, Acta Sci. Math. Szeged, 39 (1977),163-168.[7] P. N. Pandey, On a Finsler space of zero projective curvature,Acta. Math. acad. Sci.Hungar, 39(4)(1982), 387-388.[8] P. N. Pandey, On Some Finsler spaces of scalar curvature, Prog. of Maths., 18(1) (1984),41-48.[9] P. N. Pandey and R. B. Misra, Projective recurrent Finsler manifolds I , PublicationsMathematicae Debrecen, 28(3-4) (1981), 191-198.S. KumariDepartment of MathematicsUniversity of AllahabadAllahabad- 211002, Indiajhasweta735@gmail.comP.N. PandeyDepartment of MathematicsUniversity of AllahabadAllahabad- 211002, Indiapnpiaps@gmail.com

GENERALIZATION OF CERTAIN RESULTS ON PROJECTIVE MOTION IN A FINSLER SPACE

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GENERALIZATION OF CERTAIN RESULTS ON PROJECTIVE MOTION IN A FINSLER SPACE

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