In this work we define Quasi-Recurrent Weyl spaces and examine the hypersurfaces of
them

CANFES, Elif Özkara

UYSAL, S. Aynur

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODIC\ POLYTECH!\1CASER. TRA!\SP. ESG. VOL. 25. NO. 1-2, PP. 21-25 (1997) QUASI-RECURRENT WEYL SPACES El if Ozkara CA.0:FES and S. Aynur UYSAL Istanbul Technical University Faculty of Science and Lette~s 80626 ?vIaslak. Istanbul Turkey Abstract In this work we define Quasi-Recurrent \Neyl spaces and examine the hypersurfaces of them. Keywords: Weyl spaces. Quasi-Recurrent \Veyl Spaces. 1. Introduction An 71-dimensional manifold vVn is said to be a Weyl space if it has a confor-mal metric tensor gij and a symmetric connection V k satisfying the com-patibility condition given by the equation (1) where Tk denotes a covariant vector field and V kgij denotes the usual co-variant derivative. Under renormalization of the fundamental tensor of the form (2) the complementary vector Tk is transformed by the law 1; = Ti + 8i In A . (3) where A is a function. Writing (1) out in full, we have where r~l are the connection coefficients of the form (4) 22 2. O. CAXFES ilnd S. A. !.:YSAL A quantity A is called a satellite wit h \\-eight {p} of the tensor %. if it admits a transformation of the form under the renorrnalization (2) of the metric tensor gij. The prolonged covariant derivati\'e of a satellite A of Lhe tensor gij with \,-eight {p} is defined by [1] = \;,,_-1 - pTI,A . (5) RDIARE 1 The p,o!onged covariant deri\'atiye preserves t he weight. The cun'atme tensor of the \Ye~'l connection is defined by (6) and the Hicci tensor Rij of the Weyl connection is R Rnl ij = ijm' (7) Since the \Vevl connection is not metric. the Ricci tensor Rii is not necessar-ily symmetri~. In fact. R[;j: = nV[;TjJ . We remark that i(TJ is a gradient. then the space is Riemannian. It is eas)' to see that the covariant curvature tensor R!ijh is a satellite of.9ij \yith \\'eight {2}. The Bianchi identity for the \Veyl space is. by [2J ~ Rh '~Rh I ~ Rh - 0 I ,jk T 'k ilj T . J ikl - . (8) 2. Quasi-Recurrent \iVeyl Spaces A non-flat We~·l space vV,,(g;j, h) will be called quasi-recurrent ((QRIV)" in short) if the curvature tensor satisfies the follo\\'ing condition for some non-zero cO\'ariant vector field Ok (# h) A (QRI'V)r, manifold can be Weyl recurrent. i.e, it can. beside (9). satisfy (10) \Ve examine the spaces satisfying (9) but not satisfying (10), Recurrent \Veyl spaces have been examined in [2], \Ve note that 0" is a satellite of g,) Ql.:ASJ-HECC"?REST WEYL SP.:'.CES 23 with weight {O}. By multiplying (9) by yil and summing up with respect to i and l we get Similarly transvectin o' (11\ bv qjk we obtain ... b \ ; ... ~. "" R -)0 R -L 0'- (R k -L Rh -L Ri!: I Rki ) Vs 1 - SI... s. I .s I - 51.. T .. is . (12) where R;. = Rs.iyk j and R!:s = Rjsgjk. Hence the scalar curvature of the (QR1+-)" satisfies (12). By changing the indices j and k in (11) and subtracting the resulting equation from the one obtained therefrom \\'e obtain ( 13) where FJk = R[ji:]' In fact this is the relation between the complementary vector Yi, and the recurrent vector Ok. 3. Hypersurfaces of Quasi-Recurrent vVeyl Spaces Let W,,(gij, h) be a hypersurface. with coordinates 11; (i = 1. 2.···, n) of a \Yeyl space IF,,+1 (gab. Ye) \\'ith coordinates .r 2 (a = 1.? .... , n + 1). The metrics of rv" and H',,+! are connected by the relations (i.j=1.2 ... ·.n: a.b 1.2 ... ·.11+1). (Ll) where .1'~ denotes the covariant derivatii'e of .2''1 with respect to [li. It is easy to see that the prolonged covariant deri~'ative of a satellite .4. relative to HT" and TFn+l' are related by (k=1.2.···.n: c=1.2,···,n+1). (1.5) Let n 2 be the contravariant components of the vector field of HTn+! normal to Wn \\'hich is normalized by the condition ( 16) The moving frame {.r~, n2} in HT". reciprocal to the moving frame {.rr. n2} is defined by the relations O. (17) 24 E. 6. G..>,:-:FES and S. A. UYSAL Remembering that the weight of xi is {O}, the prolonged covariant derivative of xi with respect to uk is found as ( 18) \vhere "":ik is the second fundamental form. It can be shown that i.;.)ik is a satellite of gij with weight {I}. The follO\ving t\VO relations, which are respectively the generalization of Gauss and :YIainardi-Codazzi equations. are obtained in [2] ( 19) (20) where Rdbce is the covariant curvature tensor of lrVn+1 and 0.pijk is the Sylvesterian of i.;.)ij defined by 0.pijk = i.;.)pji.;.)ik - i.;.)pk"'·i.i' THEOREM 1 A hypersurface Wn of (QRlrV)n+l satisfies the following. 2os(Rijki - Slijkl) + oi(Rsjl-:l - 0.sjkd + +OJ(Riskl - 0.iskl) + odRijsl - 0.ijsl) + +91(R jjks - 0.ijks) + Rabcd'Vs(x'!;r;x'k:rT). (21) Proof 1 By taking the prolonged covariant derivative of (19) we have r; R··· - -\'; 0·· -L r; (R .'). a. bed, R. r; (.a. bed) Vs lJkl- s··'Jkl I Vs abed Xi·rjXkXI T aocdVs·Ti·rj.rkxl . Moreover. '" R '" 0 '0 fR. ). a. bed e Vs ijk! = V s·-;ji:1 + ve\ aocd,Xi XjXIJi.Ts + By using (9), .\Iainardi-Codazzi equations and (15), we obtain the result. THEORBl 2 A simply connected hypersurface of (Q R,\;V)n+l is Rieman-man. Proof 2 If we change the indices k, l. s cyclically in (21), we obtain t\\·o more equations. :\amely 201-:{ Rijls - 0.ijls) + 0i (Rkjis - 0.kjls) + +OJ (Rikls - 0.ik!s) + ol(R ,jks 0.ijks) + +Os(Rijlk 0.ijlk) + Rabcd"\:'Ic(x,!xj:rfx;), (22) 2o!(R;jsk - 0.ijsk) + oi(Rljsk - 0.ljsk) + +OJ(Rilsk 0.ilsk) + Os(Rijlk 0.ijlk) + +odR;jsl - 0.ijsl) + Rabcd"\:'I(X,!X~X~x~). (23) Ql·ASI.RECURRENT WEYL SPACES 25 Adding these three equations and using Mainardi-Codazzi equations and Bianchi identities. we find (24) Since 0; i- 0 we have R!jsk + Rkjls + Rsjk/ = 0 . (2.5 ) From the first Bianchi identity we get R(/j)sk + R(kj)/s + R(sj)k! = 0 . (26) Hence. the result follows from the fact that R(ij)sk = 2g ij T[s.kJ. A hypersurface TV" of W,,+l is called totally geodesic if Wij = o. There-fore we have the follmving theorem. THEORE:vl 3 If the simply connected hypersurface of (QRW),,+l is totally geodesic, then the hypersurface is Riemannian quasi-recurrent. Proof 3 Follows from (21) and the Theorem 2. References [1] HL.-\vATY. V. (1949): Theorie d'immersion cl'une Wm dans W n . Ann. Soc. P%n. :'vfath .. Vo!. 21. pp. 196- 206. [2] CA:\'FES. O. E. OZOEGER. A.: Some Applications of Prolonged Covariant Differen-tiation in Weyl Geometry. (to appear in Journal of Geometry). 

QUASI-RECURRENT WEYL SPACES

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QUASI-RECURRENT WEYL SPACES

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