Proe. Indian Aead. Sei. (Math. Sci.), Voi. 105, No. 1, February 1995, pp. 33-40. �9 Printed in India. On infinitesimal h-conformal motions of Finsler metric H G NAGARAJA, C S BAGEWADI and H IZUMI* Department of Mathematics, Kuvempu University, B.R. Project 577 115, India *Sirahata 4-10-23, Fujisawa 251, Japan MS received 14 May 1993; revised 9 September 1994 Abstract. The conforraal theory of Finsler spaces was initiated by Knebelman in 1929 and lately Kikuchi 17] gave the conditions for a Finsler space to be conformal to a Minkowski space. However under the h-condition, the third author 14] obtained the conditions for a Finsler space to be h-conformal to a Minkowski space. The purpose of the paper is to investigate the infinitesimal h-conformal motions of Finsler metric and its application to an H-recurrent Finsler space. We obtain the following results. A. Theorem 2.1. If an HR-F. space is a Landsberg space, then the tensor F~, is recurrent. B. Proposition 3.3. An infinitesimal h-conformal motion satisfies LxG~ = Pg + VPiY'- YlP'" C. Proposition 3.6. An infinitesimal h-conformal motion satisfies LxP~k = pC~. D. Theorem 3.7. In order that an infinitesimal h-conformal motion preserves Landsberg spaces, it is necessary and sufficient that the transformation be an infinitesimal homothetic motion. E. Theorem 3.8. An infinitesimal h-conformal motion preserves *P-Finsler spaces. F. Theorem 3.10. An infinitesimal h-conformal motion preserves h-conformally flat Finsler spaces. G. Theorem 4.1. An infinitesimal homothetic motion preserves H-recurrent Finslcrspaces. H. Theorem 4.2. If an H-recurrent Finsler space admits an infinitesimal homothetic motion, then Lie derivatives of the tensor "~h#k and all its successive covariant derivatives by x ~ or y~ vanish. Keywords. Infinitesimal h-conforrnal motion; h-conformal tensor; infinitesimal homothetic motion. 1. Preliminaries 1.1 Berwald connection Let F, be an n-dimensional Finsler space with the Finsler metric F(x, y). The metric and angular metric tensors are given by go:= OiOjF2/2 and hij:= gij-li l j, where O, . yi ~i:--'~ ~Yi li:= OiF and !':= --'F 33 34 H G Nagaraja et ai We use the following: (a ) i . _ t ih ~j~.- ~O (c~tOjh + OjOkh -- Ohgjt), (b) G~k:= O~G~, " - OjG', G':= Gj.- �89 k. (1.1) Two types of covariant deiivatives for a vector X ~ are given by (a) X:~i .-'- dkX i + GihkX h, dr: = 0k -- Gk"0,.," (b) i . _~kX i, 0 Xi t . - ~t := ff~x ~, and the Cartan tensor is defined by C~t:= �89 This connection is known as the Berwald connection, which is not metrial, that is, Oo:k = -2Pot = -2Cot:mY m' ~':t"O -- ---kgPO" (cf. [2]) (1.2) When a Finsler space satisfies the condition P~t---0, the space is called a Landsbero space. The curvature tensor H~i k is defined by i . _ i m i HMk.-- dk Ghj + GhjGmk - jl k, (1.3) where i lk means the interchange of indices j and k in the foregoing terms. We see i . _ _ i i . _ _ i . _ _ i Hjk. - - Ho j t , Hk.-- Hot, H, j . - Hhji, (1.4) HiM~ = ~hH~t, i " , 3Hjk = OjHk - j l k , where the index 0 means the transvection by y. The Ricci identities are denoted by a) i i i m i m m i T,:j: t - Th:k: j = HmjkT h - T.,Hhj t -- HjtThlrn , (b) i ~ i , i ,, i . _ " ~ Th:jl k -- T,tt: j = GmjtT h - TmGh~t, GMk.-- OtGhj. In the theory of conformal transformation, the h-conformal tensor Fihjk is defined by ([4], (4.15)) i . _ i FMk.-- HMt -- _ _ 1 (n -1 ) (n -2 ) ~::._ gikHik (n -1 ) (n -2 ) ' 1 n - 2 (Hhi3~ + ghjH,,,kg"' --Jl k) + H(g,/~ - ghk3~) {lj(h~M, - M'h~, - (n - 2)h~Mk) - j l k} , (1.5) Mh := (Hmh -- Hhm)l m. 1.2 Lie derivative We consider an infinitesimal extended point transformation in a Finsler space generated by the vector X = vi(x)di, i.e. ~i = x ~ + vidt, ~i = yi + (0fi)y,/dt. (1.6) The well-known commutation formulae ([1], [5], [6], [8-10], etc.) involving Lie Infinitesimal h-conformal motions 35 and covariant derivatives are given by (a) L x Tj: k (L x Tj): k ,. i i . ,," , - = Tj Ask -- TmAjk - A~ OmTj, (b) ~ _ i ,. i - J Ik, LxHhj k - A,~:k + A~ G,k,. where m.__ m i . _ _ i i i h i m Aj . - LxG j , Ajk . - -Lx Gjk = V:j:k "~ HjkhO + GjkmV:o. In the usual way we raise or lower indices by means of the metric tensors go or go" (1.7) 2. An HR-F. space A Finsler space F. is said to be an H-recurrent Finsler space (denoted by an HR-F . space), if the Berwald curvature tensor H~j k satisfies the relation n~jk:m i i = KrnHhj k, Hhj k 50, (2.1) where Km is a nonzero vector. As yi = 0, we have :k H~: k = KkH~, Hij:k = KkHij and from (1.5) we obtain 2 i i, vhj ,k m- i l k ) F~jk: ~ = KmFhj k + ~__ 2(PhjmHtkg -- - H pit 2 s i ts i {ptSHtsghj~ k - g HtsP*jmJk (n -- 1)(n -- 2) it + l j ( f .M ,gh~ - M ip ,~. ) - i l k} . (2.2) Thus we have Theorem 2.1. I f an H R-F, space is a Landsberg space, then the tensor F~j k is recurrent. 3. An infinitesimal h-conformal motion 3.1 An i.c.m. The condition for an infinitesimal transformation (1.6) to be an infinitesimal conformal motion (denoted by an i.c.m.) is that there exists a function ~b ofx such that Lxgjk = 2~(X)gjk , Lxg jk = -2~b(x)g jk. (cf. [1], [5], etc.) (3.1) If the function ~b is a constant, the i.c.m. (3.1) is called an infinitesimal homothetic motion (denoted by an i.h.m.) and when ~b = 0, the (3.1) called an infinitesimal isometric motion (denoted by an i.i.m.). It is well known that an i.c.m. (3.1) satisfies LxC~k = 0 and LxY i = 0. We can easily see (a) LxC i = 0, Ci:= C~i, Lx C i= -2q~C i, (b) LxF = 49F, (c) Lx l i= -~1' , L J j = ~blj, (d) Lxh~ = O, Lx(gihgjk ) = O. (3.2) 36 H G Nagaraja et at Since the Lie derivative is commutative with at or ~h, we see from (1.1) (a) and (3.1) Transvecting the above equation by fyk we have L C; = y'y - �89 Differentiating (3.3) by yJ and f ' , we get (a) LxG~=B~'qb,, B~:=c~jBih=j)y h +6~yi -g 'hy j+ F2C~ h, (b) LxG ,= Using (3.2) we have PROPOSIT ION 3.1. An infinitesimal conformal motion satisfies the followin#: LxB ih = O*-~ LxB~. h = O*-* LxB~ = O. (3.3) (3.4) 3.2 An i.a.m. If an infinitesimal transformation (1.6) satisfies LxGjk = 0, then the transformation is called an infinitesimal affine motion (denoted by an i.a.m.). First, we shall show Theorem 3.2 ([1], (VII), Theorem 5.1). In order for an infinitesimal transformation be homothetic, it is necessary and sufficient that the transformation be conformal and affine motion at the same time. Proof. We see from (3.4) �9 I ih 0 = fykLxGjk = BoodPh = 2Bihc~h = F2(21il h -- gis)dph. Transvecting the above ecluation with 2I rk - gik, we have F2~k = O. Q.E.D. Remark.. This theorem was first proved by Takano (Japanese, 1952). 3.3 An i.h-c.m. If we impose the h-condition on the vector ~bj, i.e. FChijc~k = (~ lhu , ~)1 := Fch~)h. (cf. [4], w (3.5) n--1 the transformation is called an infinitesimal h-conformal motion (denoted by an i.h-c.m.). Because the function Ol(x) is proved to be a function ofx only (see [4], Lcmma 3.2), we get Infinitesimal h-conformal motions (a) F~kh ~ = - h i l l ' - h~l~, (b) F 2( ~k C)*)(~h = F ~kt F Ctj* dph) - F C)h dp,d,F = Fdh(dp,hj)- dp,h~l k = - dpt(hiJ' + h~lj + h~l~). Using the above calculations, we obtain , , , , ' r , A~ = LxG~ = pi6~ + p~6~ -- g~p - - ' - Lx~ = p~ +,~/ - y~p', Aj - FC~l~dp, ' pj(x,Y):=fb]+ dP , l j=(~ +~-~_ l j P:= Po=dPo + Ffb, �9 Hence we have 37 (3.6) (3.7) PROPOSITION 3.3. An infinitesimal h-conformal motion satisfies LxG~k = p?~ + pkr~- p'gj~ -- dp,l'tflk, L~G~ = p~ + ~y ' - y~p'. The vector pj is called an associated vector with a vector ~b i and satisfies the conditions: (a) FC~.kp , = dpxhjk, (h-condition) (b) Pilk:= dkP~ - C~p, = O. (Cartan's covariant derivative by yk) (3.8) A vector which satisfies (3.8) (a) (b) is called an h-vector. PROPOSITION 3.4 ([4], Proposition 3.4). Let vi(x,y) be a vector in a Finsler space. I f v~ satisfies the conditions vdk=O and FC~.~vh = vthj~, then the function v I and the vector *v~:= v~- vtli are independent of y. Here we shall show Lemma 3.5. We have 2 ." i " -- h~l~ i hjkli). f p OmCj~ = dpl(FC~k - hhl J - Proof. We see dkg 'h 2C~*, d.C}t=~m(g'hChjk)= 2 '''h'' -- '*~ C = - - - - ~"~m~'h jk -~" g m h jk , and using t~.Chjk = t~kChjm, (3.6) (a) and (3.8) (b), we get F~p "~. C~ = - 2F~1C~, + Fg 'h{~(FChj.p') - C~.~(rp')} = qbt(FC~h - h~l t - h~lj - hltl'). From (1.7) (a) we see LxC~,:,-(LxC~k): , = A~tC; - AT, C L - A~C~,- A?~,,C~.~. 38 H G Naoaraja et ai In consideration of LxC jk = 0 and transvecting the above equation by y~, we have L x P)k i i m m i m i = = -- AoOmC~k. LxCjk:o AmC ~ - A~ C~, k - A~ C~,~ m" i Substituting (3.7) into the above equation, we have i _ ' h~lj + h~ki i) 4,oC~ + F2p'O,,C~k. LxP~k - 4, t (hflk + + Using Lemma 3.5, we obtain = (4 ,0 + - ' - pC~k. (3.9) Thus we have PROPOSITION 3.6 An infinitesimal h-conformal motion satisfies Remark. If we denote the deformed tensor (of. I-l]) of Pjk with respect to an i.h-c.m. (1.6) by P~k' we see e jk = p i i i* + (PC)k) dt. This means that the deformed space of a Landsberg space (P~k = 0) is not necessarily a Landsberg space. However we can state the following. Theorem 3.7. In order that an infinitesimal h-conformal motion preserves Landsber9 spaces, it is necessary and sufficient that the transformation be an infinitesimal homothetic motion. Proof It is sufficient to show 4,i = 0. In fact. we have O=O~p=p~=(6, FCili\ (6, FC'l i\ 4,,= It is evident that the theorem holds. Q.E.D. 3.4 *P-Finsler space If the tensor *Pjk := P jk - 2Cjk vanishes, the space is called a *P-Finsler space (cf. [-3]). The *P-condition: Pjk = 2Cjk is invariant under any h-conformal change of Finsler metric. From (3.9) we have LxP j = pCj, Pj:= ej,. (3.10) Using (3.10) we see 0 = Lx(P ~ - 2Cj) = (p - Lx2)C j, Lx 2 = p. (3.11) This means Lx*Pjk = 0. Hence we have Theorem 3.8. An infinitesimal h-conformal motion preserves *P-Finder spaces. Infinitesimal h-conformal motions 39 3.5 An h-conformally flat Finder space If a Finsler space is h-conformal to a Minkowski space, the space is called an h-conformally flat Finder space. An h-conformally flat Finsler space is proved to be one of *P-Finsler space (cf. [4], (5.2)). Here we shall show Lemma 3.9. In a *P-Finsler space an infinitesimal h-conformal motion satisfies C'yi ~2,, 2,:= c5,2. (3.I2) Lx2h=O,, Lx*2j=~bj, *2j:= t~ n - i / Proof. Differentiating (3.11) w.r.t, yh we have Lx2 h = Ph- Next from (3.2) we see c'yj=o%kc.y (c'y,) Hence we have Lx*2j = 6~ n2 i p, = ~bj. Q.E.D. On the other hand, we know the theorem ([4], Theorem 6.6): The necessary and sufficient conditions for a Finsler space to be h-conformally flat are that i tgtl'Ijk = 0 and H~u = 0 and 2, is an h-vector, where 1-i~ k :~ i o ih ,~ Gj~ - - o jk "h, , ._ �9 ., i (3 .13) = dR - - Ho , ~m" Hhu. - *dtH~, k + HhkHmt - kll, *all m " The parameter H~k and the tensor H~t are invariant under an h-conformal transformation and these are independent of y. We shall show Theorem 3.10. An infinitesimal h-conformal motion preserves h-conformally flat Finsler spaces. Proof. It is sufficient to prove LxH~k = 0. We see LxB~ = 0 from Proposition 3.1. Moreover, we have from (3.4) (b) and (3.12) , _ , U,h .~ ~ _ B,~.h ,h LxHjk - Lx(G jk - "-'jk "~hJ - - j kwh - - Bjk~)h = O. It is easy to prove ]-l~k t = O. Q.E.D. 4. An in f in i tes ima l homothet ic mot ion in HR-F , , spaces In this section we shall consider an i.h.m, only, that is, Lxgij = 2cgu, Lx# u = - 2c0 u, c = constant. From Theorem 3.2 and (1.7) (b), we have LxH~jk = O. From (1.7) (a) and (2.1) we see i i _ _ LxHh jk : m = (LxHh jk ) :m - - O. i ' K i LxH,jk: m = Lx(KmHhjk) = (Lx m)Hhjk = 0, (4.1) 40 H G Naoaraja et al which means LxKm = 0 and l Lx(H~jk: ~ - K,,Hhik) = O. Thus we have Theorem 4.1. An infinitesimal homothetic motion preserves H-recurrent Finsler spaces and satisfies LxK m = O. An i.h.m. (4.1) satisfies Lxl~ = cl i, Lxl i = - cl i, Lxh~ = O. (4.2) F rom Propos i t ion 3.6 we see Lxe~ = O, LxP ji k = 2cP j~, LxP~ ~ = - 2cP~ J, LxM,=-cMh, LxM~=-3cM l, LxH=-2cH. (4.3) After some calculations we obta in LxF~j k = O. Moreover we see from (2.2), (4.2) and (4.3) Lx F~j~:. = (Lx r,,,)F~.ik + K.LxF~ih = O. Hence we have Theorem 4.2. I f an H-recurrent Finsler space admits an infinitesimal homothetic motion, then Lie derivatives of the tensor F~j~ and all its successive covariant derivatives w.r.t, x ~ or yi vanish. References [1] Yano K, The theory of lie derivatives and its applications (1957) (Amsterdam: North-Holland) [2] Rund H, The differential geometry of Finsler spaces (1959) (Berlin: Springer Verlag) [3] Izumi H, On *P-Finsler spaces, I, II, Mere. Defence Acad. Japan 16 (1976) 133-138; 17 (1977) 1-9 [4] Izumi H, Conformal transformations of Finsler spaces. II. An h-conformally fiat Finsler space, Tensor N.S. 34 (1980) 337-359 [5] Izumi H, On Lie derivatives in Finsler geometry, Syrup. on Finsler Geom., at Naruto, 1980. [6] Sinha R S, On projective motions in a Finsler space with recurrent curvature, Tensor N.S. 21 (1970) 124-126 [T] Kikuchi S, On the condition that a Finsler space be conformaUy fiat, Tensor N. S. 55 (1994) 97-100 [8] Pande H D and Kumar A, Special conformal motion in a special projective symmetric Finsler space, Lincei - Rend. Sci. fis. mat. e nat. 5g- Maggio (1975) 713-717 [9] Kumar A, On projective motion in recurrent Finsler spaces, Math. Phi. Sci. 12 (1978) 497-505 [101 Sinha R S and Chowdhury V S P, Projective motion in recurrent Finsler spaces, Bull. Cal. Math. Soc. 75 (1983) 289-294
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