Submetric classの面積空間におけるrecurrent curvature tensor のdecompositionについて II
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(2) (®2g|!C) ^l-^- Bgffi62^10^ Journal of Hokkaido University of Education (Section H C) Vol. 38, No. 1 October, 1987. On the decomposition of the recurrent tensor in an areal space of the submetric class II. Masao GAMA and IVtasaharu OYAMADA Mathematics Laboratory, Asahikawa College, Hokkaido University of Education Asahikawa 070. Submetric class ®®^^f3E§^^^'5 recurrent curvature tensor (D decomposition ^-^>l/^T II ^ • /J^Ojffl ^mmmis^w i w^s^^s. 0. Introduction. Let A w be an areal space of the submetric class with the normalized metric tensor g,j and the connection parameters F*'ji, and C[",, as defined by A. Kawaguchi and. K. Tandai [ 2 ] 1). A few years ago, one of the present authors [ 1 ] dealt with the decomposition of the recurrent curvature tensor in A (,m) .. The purpose of the present paper is to investigate the results published in [1,] in more detail.. The notations and terminology used in the present paper may be found by reference to the paper [ 1 ] . 1. Preliminaries. We exhibit as follows some properties concerning the curvature tensor. R hui, which will be needed later in the paper: (1. 1) R",,i,=-Rh^,. (1. 2 ) RhM+R"w+RhM=: O ,. (1. 3) R,w+R^+gy.^R\,,=Q where Rui,i=gia Raski ,. (1. 4) /3r,Rsr,,i=0. (1. 5) R"w | ;+ R"w \,+Rhw\ * +r*",,;: R\,,+rfsh,,,; : R\,, +v'tha; s j?a.,,= o.. 1) Numbers in brackets refer to the references at the end of the paper.. (27).
(3) 28 Masao GAMA and Masaharu OYAMADA In this paper we assume that A (,,"') is of recurrent curvature, that is, (1. 6 ) R''ijh I i=R",ji, K-i, Rl'{jk + 0 . Defining the Ricci tensor Rij and the scalar curvature. R in A M by (1. 7) Ru=R"w,. and (1. 8) R=gu Ru respectively , we have, by virtue of (1. 6 ) (1. 9) R.j\t=R,,Kt and (1. 10) R | i=RKt from which we find (1. 11) Ki= R-1 R | „ From (1. 2 ), we have (1. 12) Ru- Rj,+ R"w = 0.. From (1. 6) and (1. 11) , we have The necessary and sufficient condition for the recurrent cnrvature tensor R"uk to be a covariant constant is that the scalar curvature R is a covariant constant.. In the following paragraphs 2 and 3 , we consider two cases: Case 1, in which the recurrent curvature tensor R",ji! decomposes as follows, R"w= V" 5,,.,. and Case II, in which S.ji, decomposes as follows , 'IVA— <-'{ ^->jk-. 2 . Case I. In this paragraph we assume that the recurrent curvature tensor R"uk satisfying. (1. 6 ) decomposes in the form (2, 1) R".,,=V"S^ where V' and 5y/, are a contravariant vector and a covariant tensor respectively.. From (1. 1) and (1. 2), we have 'tjll ~ ^'ikj. and 2. 3) Sy;i+ S;;ti+ S/,,j= 0 respectively.. From (1. 4 ), we find (2. 4) 'Vap?S^= O where we put (2. 5) 'V =Vmpa,n.. From (1. 7) , (2 . 1) and (2 . 2 ),we have ( 2 . 6 ) R.j VJ= O from which we find ( 2 . 7 ) rank (7?,,) < n. From (1. 6) and (2 . 1) , we have. : 28;.
(4) On the decomposition of the recurrent tensor in an areal space of the submetric class II, 29. ( 2 . 8 ) V" Sw | ; = (V" K, - V" | ,) S,,,, from which we have ( 2 . 9 ) Sij/, | ( = Sijii Li where we put (2. 10) Li =Ki -vi (2. 11) Vt=v-1 v | ;. and (2. 12) (v)2=g,, V VJ. From ( 2 . 8) and (2 . 9 ), we have (2. is) y" |; = v" vi. From (1. 11) , (2. 9) and (2. 10) ,we have The necessafy and sufficient condition for S,j,, to be a covariant constant is that vi =R-1 R | ; .. From (2. 9) , (2. 10) and (2. 13) ,we have The necessary and sufficient condition for S,ji, \ ; =S,ji, Ki is that Vh \ i= 0 .. Putting (2. 14) V"K,=p, we have, by virtue of (2. 13) , (2. 15) Vt=p-1 K,, V"\ i from which, putting p= 1, we have Vt=Ht. which is the same result that we obtained in [ 1 ] . 3 . Case II. In this paragraph we assume that the tensor S,ji, in ( 2 . 1) decomposes in the form (3. 1) S,,,=y,S,,, where U, and S,-* are a covariant vector and a covariant tensor respectively.. From ( 2 . 2) and (2 . 3 ), we have (3. 2) S,,=- S,,. and ( 3 . 3 ) U, S,, + Uj 5,, + U, S,,= 0 respectively.. Putting (3. 4) 'U.= U.np': ,. we have, by virtue of ( 2 . 4) , (3. 1) and (3. 4) , (3. 5) 'V ' U,= 0.. From ( 2 . 9) and (3 . 1), we have (3. 6) <7,S,J,= (U,Lt-U.\,) S,, from which we have (3. 7) 5,J<=S,,,M, where we put (3. 8) Mi=Li-Ui. (29).
(5) 30 Masao GAMA and Masaharu OYAMADA ( 3 . 9 ) Ui=ir1 u | ;. and (3, 10) (u)2=gij U, U, .. From ( 3 . 6) , (3 . 7) and (3 . 8 ),we have (3. 11) C/..1 <= £/,^ .. From (2. 10) , (3. 7) and (3 . 8),we have The necessary and sufficient condition for Sji, to be a covariant constant is that K[=VI+ Ui , from which, by virtue of ( 3 . 6 ) and ( 3 . 11) , we have The necessary and sufficient condition for Sji, to be a covariant constant is that U.\ i=U.Lt.. Putting (3. 12) V" U,,=w ,. we have, by virtue of (1. 12) , (2. 1) and (3. 1) , ( 3 . 13) <»S,,= - (^,,- R,;). from which we have The necessary and sufficient condition for the Ricci tensor Rsj to be symmetric is that w= 0 .. From (2. 13) , (3. 11) and (3. 12) , we have ( 3 . 14) co} | i=WWi where we put (3. 15) wi=Vi+Ui. from which we find, by virtue of ( 2 . 12) and ( 3 . 8) , (3. 16) Mi=Ki-wi. Finally, we consider the case in which U, is equal to K,. In this case ,we have , by virtue. of (3. 3) , K, Sj.+Kj S,,+K, 5,,= 0. from which we have (3. 17) R"w\ i+R",H[j+ R",u\ ,<= 0.. From (1. 5) and (3. 17) ,we find (3. is) r*",,;: R\,i+r*",,; : R\,i+r*"n; : Ra .j,= o.. From (1. 3 ), it follows that g.J;:\ rR\,l= 0 from which we find. (3. 19) C;':| rR\,^ 0. Consequently, from ( 3 . 18) and ( 3 . 19) when Us is equal to K,, we have (3. 20) P"u,a,R\,i + P",,., °aR\,, •+P"il. °aRa^= 0. which is the same result that we obtained in [ 1 ] . From (1. 11) , ( 3 . 11) and the Ricci identity, we have (3. 21) S,,=RQ-1 (K.k,-W where we put. (30).
(6) On the decomposition of the recurrent tensor in an areal space of the submetric class II. 31. (3. 22) Q=-R ; : Va'K.. (3. 23) 'K.=Krpr^. (3. 24) k,=k-lk | , and (3. 25) (k)2=gijK,K,. Thus in this case the recurrent curvature tensor Rhijk can be written as. (3. 26) R"^=V"K. (K,k,-K,kj) RQ-\. REFERENCES [ 1 ] Gama, M. (1978) , On the decomposition of the recurrent tensor in an areal space of the submetric class. Journal of Hokkaido University of Education (Section II A) Vol. 28, No. 2 , p.77-80.. [ 2 ] Kawaguchi, A. and Tandai, K. (1952) , On areal spaces V. Normalized metric tensor and connection parameters in a space of the submetric class. Tensor, N.S. Vol. 2 , p.47—58.. ; 31;.
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