Submetric classの面積空間におけるrecurrent curvature tensorのdecompositionについて
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 28, No. 2 February 1978. WMK±W^ (^ 2 ^A) ^ 28 ^ ^ 2 ^ Bg% 53 ^ 2 H. On the Decomposition of the Recurrent Tensor. in an Areal Space of the Submetric Class. Masao GAMA MMhematics Laboratory, Asahikawa College, Hokkaido University of Education. Asahikawa 070. Submetric class ^M^tSI^ ^ ^ (t & recurrent curvature tensor (7) decomposition ^•^>^^X. m m ^ WMK±Wn\ ^-TOTO^. 1. Introduction. LetA(n be an areal space of the submetric class whose metric tensor. gij, connection parameters F^ and CJ°a were defined by A. Kawaguchi and K. Tandai [3]2) K. Takano [ 4 ] studied the decomposition of the recurrent curvature tensor in an af finely connected space. In this paper we assume that A(nm) is of recurrent curvature, that is,. (1.1) R^\t=RwKi, Rh^O, where R^=nh-k-nh;aar^+ Ua F^ i*h __ T^*h.a 7-i*aj_ r^*a i-i*h \3) ik,J ~1 ik~,al aj'Ti ik-l. aj ) ,. Ki is the recurrence vector and \i denotes the covariant derivative with respect to-v'defined in [1]. The purpose of the present paper is to extend the work of K. Takano [4] to the case of an areal space An of the submetric class which is of recurrent curvature. 2. First we assume that the recurrent curvature tensor R^ decomposes in the form. (2.1).) R^=VhSw, (2.1).) VhKH=l, where V and S uk are a contravariant vector and a covariant tensor respectively.. The curvature tensor R^ has the following properties :. 1) Latin indices run from 1 to n and Greek indices from 1 to m. 2 ) Numbers in brackets refer to the references at the end of the paper. 3 ) ,j and ;ca denote 9/9xJ and ol/o'^ respectively, r\} = F'a'jp'i and for repeated indices the so-called summation convention will be used.. (77).
(3) M. GAMA. (2.2) Rijk= ~ Rikj,. (2.3) R^+RW+RW=O, (2.4) Rwk+Rwk+ghi;aaRSjk=0, where Rhijk=gzaR^k and RSjk=R^kpba,. (2.5) Rijk\ I + ^;A(|j + RiU\ k i* A. a D a j_ r^*h.a j^a _L T-i*h.a jya ij',a-Kaki^~-i ik,a.Fictlj ~V 1 a ~,aK.ajk— 'J.. From (2.1)a), (2.1),), (2.2) and (2.3), we have (2.6)a) Sijk = — Sikj; (2.6)&) Sijk + >S',A;+ Sftzj = 0. From (1.1), (2.1)a) and (2.1)6), we have (2.7) Sijk\i=-{Ki~ Hi)Sijk,. (2.8) Vh\i=VHHi, where. (2.9) Hi=KaVa\i Thus, we have. Proposition 1. If the recurrent curvature tensor R^k characterized by (1.1) decomposes in the form (2.1) a), V and Sijk become a recurrent vector and tensor, which are given by (2.7) and. (2.8) respectively, V satisfying (2.1)t>) and Hi being defined by (2.9). 3. Next we assume that the tensor S uk in (2.1)a) decomposes in the form. (3.1) S^=K,Sjk. In this case the curvature tensor R^k reduces to. (3.2) Rkk=KiVhS,k. From (2.6)a), (2.6)&) and (3.1), we have (3.3),) S,k=-Skj, (3.3).) KiSjk+K,Skz+KkSi,=Q.. Differentiating (3.1) covariantly with respect to xl and taking (2.1)&), (2.7) and (2.9) into account, we have. (3.4) S,k\i=SjkKi. Thus, we have. Proposition 2. If the tensor Sijk defined by (2.1)a) decomposes in the form (3.1), the tensor Sjk becomes a recurrent tensor which is given by (3.4).. From (1.1), (3.2) and (3.4), we have. (3.5) {K,Vh)\,==0 which, by virtue of (2.D&) and (2.9), is equivalent to. (78).
(4) On the Decomposition of the Recurrent Tensor in an Areal Space. (3.6) Ki\i=-KiHi. From (3.5) and Ricci's formula derived from it , we have h\.a^a. ',aL[a— U ,. where ,a— 770 V A& Qa= V ~l\.bPa.. From (2.4) and (3.2), we have (3.7) gaiVaKHJrgaHVaKiJrgHi;aaqaa= 0 ,. from which we have by virtue of (3.5) (3.8) C},^=0. Contracting (3.7) with ghi and p^plsges, we have CHH,W=-I and pEaqae=Q respectively.. From (1.1) and (3.3)&), we have. (3.9) RMt+R^+Rhj\k=0, from which we have by virtue of (2.5) (3.10) {HW^+P^; aaSu+H!l-aaS^qaa= 0 . From (3.8) and (3.10), we have \j, aSki~^Pik, aiSijJrPil, °iSjk)q^= 0 , where Pihj, "a is a curvature tensor in A(n") defined in [2 ] as follows : ih a — /"i* A.ff _ /^h a|._i_/-'/i^^c^*b.ff iJi a —i ij , a ^;, a\j i <^ i ,bpp 1 cj,a.. From (3.6), we have Ki\i\ m~ Ki\ m\l = Ki[Hm\l ~ H i\ m). On the other hand, from Ricci's formula, (2.1)&) and (3.2), we have Ki\i\m~ Ki\m\l= ~ KiSlm~ Ki',a.qaSlm.. Accordingly by virtue of (2.1)i,) we have Hi\m-H^i=(l-K,Vt;W)Sim, from which we have (3.11) S im'=\{Hi\m—Hm\l),. (79).
(5) M.GAMA. where (3.12) \-l=l-KiVt;W. Thus we have Proposition 3. If the recurrent curvature tensor R^-/, decomposes in the form (3.2), the skew-. symmetric tensor Sim is given by (3.11), V , Hi. and A being defined by (2.1)&), (2.9) and (3.12) respectively.. In particular, if V satisfies the relation KiV ,^q'S=0, R^-k is characterized by. R^=KiVhWk-H^}. REFERENCES [ 1 ] Gama, M. (1965), On areal spaces of the submetric class. Tensor, N.S. Vol. 16, p.262-268. [ 2 ] Gama, M. (1966), On areal spaces of the submetric class HI. Tensor, N.S. Vol. 17, p.79-85. [ 3 ] Kawaguchi, A. and Tandai, K. (1952), On areal spaces V. Normalized metric tenser and connection parameters in a space of the submetric class. Tensor, N.S. Vol. 2, p.47-58. [ 4] Takano, K. (1967), Decomposition of curvature tensor in a recurrent space. Tensor, N.S. Vol. 18, p.343-347.. (80).
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