On Pseudo-Union Curves in a Hypersurface of a Weyl Space

Transcription

1 Intenational Mathematical Foum, Vol., 07, no. 6, HIKARI Ltd, On Pseudo-Union Cues in a Hypesuface of a Weyl Space Nil Kofoğlu Beykent Uniesity Faculty of Science and Lettes Depatment of Mathematics Ayazağa-Maslak, Istanbul, Tukey Copyight c 07 Nil Kofoğlu. This aticle is distibuted unde the Ceatie Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, poided the oiginal wok is popely cited. Abstact In this pape, fistly we hae obtained the diffeential equation of pseudo-union cues and then we hae defined pseudo-union cues in W n. Secondly, we hae expessed pseudo-asymptotic cues and pseudogeodesic cues in W n. Finally, we hae gien elation among these cues and by means of this elation, necessay theoems hae been expessed. Mathematics Subject Classification: 53B5 Keywods: Weyl space, the pseudo-union cues Intoduction A manifold with a confomal metic g ij and a symmetic connection k satisfying the compatibility condition k g ij T k g ij = 0 ) is called a Weyl space, which will be denoted by W n g i j, T k ). The ecto field T k is named the complementay ecto field. Unde a enomalization of the metic tenso g ij in the fom g ij = λ g ij )

2 78 Nil Kofoğlu the complementay ecto field T k is tansfomed by the law T k = T k + k lnλ 3) whee λ is a scala function [3]. The coefficients Γ i kl of the symmetic connection k ae gien by Γ i kl = { i } g im ) g kl mk T l + g ml T k g kl T m. 4) If unde the tansfomation ), the quantity A is changed accoding to the ule à = λ p A 5) then A is called a satellite of g ij with weight { p }. The polonged deiatie and polonged coaiant deiatie of A ae, espectiely defined by [, ] and k A = k A pt k A 6) k A = k A pt k A. 7) Let W n g ij, T k ) be n-dimensional Weyl space and W n+ g ab, T c ) be n + )- dimensional Weyl space i, j, k =,,.., n; a, b, c =,,.., n + )). Let x a and u i be the coodinates of W n+ g ab, T c ) and W n g ij, T k ), espectiely. The metics of W n g ij, T k ) and W n+ g ab, T c ) ae connected by the elations g ij = g ab x a i x b j 8) whee x a i is the coaiant deiatie of x a with espect to u i. The polonged coaiant deiatie with espect to u k and x c ae k and c, espectiely. These ae elated by the conditions k A = x c k c A k =,,..., n; c =,,..., n + ). 9) Let the nomal ecto field n a of W n g ij, T k ) be nomalized by the condition g ab n a n b =. The moing fame { x i a, n a} and its ecipocal { x i a, n a} ae connected by the elations [3] n a n a =, n a x a i = 0, n a x i a = 0, x a i x j a = δ j i. 0)

3 Pseudo-union cues 783 Since the weight of x a i is { 0 }, the polonged coaiant deiatie of x a i, elatie to u k, is gien by [3] k x a i = k x a i = w ik n a, ) whee w ik ae the coefficients of the second fundamental fom of W n g ij, T k ). On the othe hand, it is easy to see that the polonged coaiant deiatie of n a is gien by k n a = w kl g il x a i. ) By means of 0), the polonged coaiant deiatie of x j a is found to be [8] k x j a = Ω j k n a. 3) Let i i, =,,..., n) be the contaaiant components of the ecto field in W n g ij, T k ). Suppose that the ecto fields =,,..., n) ae nomalized by the conditions g ij i j =. The ecipocal ecto fields ae defined by the elations [6] i j = δ i j, i s i = δ s i, j,, s =,,..., n ). 4) The polonged coaiant deiaties of the ecto field and its ecipocal ae, espectiely, gien by [7] k i = s T k s i, k i = s i. 5) T k s Let a and i be the contaaiant components of the ecto field elatie to W n+ g ab, T c ) and W n g ij, T k ), espectiely. Denoting the components elatie to W n+ g ab, T c ) and W n g ij, T k ) by a and i, we hae [8] a = x a i i, a = x i a i. 6) If K is the nomal cuatue of W n g ij, T k ) in the diection of, we hae K = w ij i j. 7) Since the weight of w ij is { } and that of i is { }, K is a satellite of g ij of { }.

4 784 Nil Kofoğlu The quantities p η = T p k k, p =,,..., n) 8) ae called the geodesic cuatue of the lines of the net,,..., n ) elatie to W n g ij, T k ) [7]. The ecto fields c i = i η p p i,, p =,,..., n) 9) ae called the geodesic ecto fields of the net,,..., n ) elatie to W n g ij, T k ) [7]. If the components of the geodesic ecto fields elatie to W n+ g ab, T c ) ae denoted by c a, then we hae [8] c c a = c a = w ik i k ) n a + c i x a i. 0) Since the net,,..., n ) is othogonal, we hae [7] T k = 0, p T k + T k = 0 p). ) p Peliminaies Let C : x i = x i s) i =,,..., n) be a cue in W n, whee s is its ac length and is the tangent ecto field of C at the point P. Let λ be a unit ecto field in W n+ and λ a be the contaaiant components of λ. λ is a conguence of unit ecto fields. It can be expessed as λ a = x a i w i + zn a a =,,..., n + ) ) whee w i ae the contaaiant components of the ecto field w with espect to W n and z is a scala. Since g ab λ a λ b = we hae o g ij w i w j + z = 3) z = g ij w i w j. 4)

5 Pseudo-union cues 785 Let N a be the contaaiant components of a unit ecto field which satisfies the conditions: it is linealy dependent on λ and and it is othogonal to [4]. Hence and g ab N a N b = b =,,..., n + ) 5) g ab N a b = 0. 6) On the othe hand, we know that, the elation between c a and c i is as follows: c a = c i x a i + w ij i j )n a j =,,..., n) 7) whee c a and c i ae the geodesic ecto fields with espect to W n+ and W n, espectiely; n a ae the contaaiant components of a unit ecto field nomal to W n and w ij ae the coefficients of the second fundamental fom of W n. Since N a is linealy dependent on λ and, it can be witten as N a = αλ a + β a 8) whee α and β ae scalas. Multiplying 8) by g ab N b, we get whee g ab a N b = 0, o g ab N a N b = = αg ab λ a N b 9) α = Multiplying 8) by g ab b, we obtain whee g ab a b, o Fom 8), 30) and 3), we hae g ab λ a N b. 30) g ab N a b = 0 = αg ab λ a b + β 3) αg ab λ a b β = αg ab λ a N. 3) b N a = g cd λ c N d λa g cd λ c d g cd λ c N d a c, d =,,..., n + ) 33)

6 786 Nil Kofoğlu o N a = x a i w i + zn a) g cd x c j w j + zn c) k x d k a g cd x c j w j + zn c) 34) N d o N a = x a i w i + zn a g jk w j k a zg cd n c N d 35) whee d = k x d k, g cdx c jx d k = g jk, g cd n c x d k = 0 and g cdx c jn d = 0. o Using a = i x a i and 4), we get x a N a i w i g jk w j k ) i + zn a = ± 36) gij w i w j g cd n c N d x a N a i w i g jk w j k ) i + zn a = ± p =,,..., n) δ pj g 37) ipw i w j g cd n c N d whee g ij = δ p j g ip o x a N a i w i g jk w j k ) i + zn a = ± g pm g jm g ip w i w j g cd n c N d m =,,..., n) 38) whee δ p j = gpm g jm, o x a N a i w i g jk w j k ) i + zn a = ± n = p m 39) g jm g ip w i w j g cd n c N d whee g pm = n = p m, o x a N a i w i g jk w j k ) i + zn a = ±. 40) p g jm g ip w i w j m The plus sign in 40) is to be taken when z > 0, the minus sign when z < 0. Thus 40) will educe to N a = n a, when λ is linealy dependent on and n a ; that is w i = k i, k being any constant diffeent fom unity.

7 Pseudo-union cues 787 o Using 40), we hae ) n a = N a p g jm g ip w z i w j + x a w j i g jk m z k i wi z n a = N a ) p g jm g ip w z i w j + x a i g jk ρ j k i ρ i m 4) 4) whee ρ j = wj z. If we wite 4) in 7), we get o c a = c i x a i + K [ N a ) ] p g jm g ip w z i w j + x a i g jk ρ j k i ρ i 43) m ] c a = x a i [c i + Kg jk ρ j k i Kρ i N a + K p g jm g ip w z i w j 44) m whee K = w ij i j is the nomal cuatue of C. whee and Fom 44), we hae Fom 45): c a = x a i p i + K N a 45) p i = c i + K g jk ρ j k i K ρ i 46) K = K p g jm g ip w z i w j. 47) m Definition. Fo conenience, c a in 45) is called elatie fist cuatue ecto field of C at P in W n+ and p i is called elatie fist cuatue ecto field of C at P in W n. K g = g ij p i p j is called elatie fist cuatue of C at P in W n.

8 788 Nil Kofoğlu 3 Pseudo Union Cues Definition 3. The totally pseudo-geodesic suface is defined by a and c a. Let µ be a unit ecto field in the diection of the cue of conguence of cues, one cue of which passes though each point of W n. µ a, in geneal, ae not nomal to W n and it can be specified by whee t i and ae paametes [5]. Then we hae and o With the help of 48), 49) and 50), we get µ a = t i x a i + N a 48) g ab µ a µ b = 49) g ab x a i N b = 0. 50) = g ab µ a µ b = g ab t i x a i + N a) t j x b j + N b) = g ij t i t j + g ij t i t j = 5) whee g ab x a i x b j = g ij and g ab N a N b =. If the pseudo-geodesic in W n+ in the diection of the cue of the conguence with coaiant components µ a is to be a pseudo-geodesic of the totally pseudo-geodesic suface, then it is necessay that µ a be a linea combination of a and c a, theefoe µ a = α a + βc a 5) whee α and β scalas. Fom 45), 48) and 5), we get whee a = x a i i. t i x a i + N a = αx a i i + β x a i p i + K N a) 53) Multiplying 53) by g ab x b j and summing fo a and b, we obtain g ij t i = αg ij i + βg ij p i 54)

9 Pseudo-union cues 789 whee g ab x a i x b j = g ij and g ab N a x b j = 0. Multiplying 54) by j, we get g ij t i j = α + βg ij p i j 55) whee g ij i j =. Fom 46), we obtain g ij p i j = g ij c i j + K g tk ρ t k g ij i j K g ij ρ i j p = g ijη i j + Kg tk ρ t k Kg ij ρ i j p = g ij p T m m p i j p =, 3,..., n) g ij p i j = 0 56) whee g ij i j = and g ij p i j = 0 Using 56) in 55), we hae p =, 3,..., n). α = g ij t i j. 57) Multiplying 53) by g ab N b and summing fo a and b, we get = βk o β = K whee g ab x a i N b = 0 and g ab N a N b =. Witing 57) and 58) in 54), we obtain ) g ij t i = g kh t k h g ij i + g ij p i h =,,..., n). K 58) 59) Multiplying 59) by g jm and summing fo j, we obtain ) δi m t i = g kh t k h δi m i + δi m p i K ) t m = g kh t k h m + 60) p m K whee g jm g ij = δ m i. Fom 60), we hae t m = g kh t k h ) m + K p m 6)

10 790 Nil Kofoğlu o l m = ) g kh l k h m + p m K 6) whee tm = l m, o p m + Kg kh l k h m Kl m =0 m =,,..., n) ) p m + K g kh l k h m l m =0. 63) Equation 63) is the diffeential equation of the pseudo-union cues. The solutions of the n equations 63) detemine the pseudo-union cues in W n to that conguence. Let us denote the left hand side of 63) by η m : ) η m = p m K l m g kh l k h m = p m K m = 0. 64) whee m = l m g kh l k h m. η m ae called the contaaiant components of the pseudo-union cuatue ecto field. Definition 3. Pseudo-union cue is defined as a cue whose pseudounion cuatue ecto field is a null ecto field: η m = 0. If φ = µ a, N b ), then cosφ = g ab µ a N b whee g ab µ a µ b = and g ab N a N b =. Since g ab µ a N b =, cosφ = is obtained. Since g ij t i t j = = sin φ, we hae g ij l i l j t = g i t j ij = sin φ cos φ = tan φ. If α = i, l j ), then cosα = g ij i l j gij l i l j o cosαtanφ = g ij i l j whee g ij i j i t =. Fom cosαtanφ = g ij j, we hae cosαcosφtanφ = cosαsinφ = g ij i t j. The magnitude K u of the ecto field η k is K u =g kh η k η h ) ) =g kh p k Kl k + Kg im l i m k p h Kl h + Kg im l i m h =g kh p k p h K g kh p k l h + K gkh l k l h K g im l i m ) 65) =K g K g kh p k l h + K tan φ K cos αtan φ =K g K g kh p k l h + K sin αtan φ whee g kh p k h = 0, g kh k h = and K g = g kh p k p h.

11 Pseudo-union cues 79 Multiplying 54) by p j, we hae whee g ij i p j = 0 and K g = g ij p i p j, o g ij t i p j = βk g 66) β = g ijt i p j K g 67) Witing 57) and 67) in 54), we get g ij t i = g kh t k h )g ij i + g kht k p h g ij p i. 68) Multiplying 68) by t j and summing fo i and j, we obtain K g ) g ij t i t j = g kh t k h gkh t k p h) + sin φ =cos αsin φ + sin φsin αk g = g kh t k p h) sinφsinαk g =g kh t k p h. K g gkh t k p h) K g 69) Fom 69), we hae tanφsinαk g = g kh l k p h. 70) Using 70) in 65), we get K u =K g KtanφsinαK g + K sin αtan φ ) = K g Ksinαtanφ 7) K u =K g K sinαtanφ. Definition 3.3 If the cue C is a pseudo-union cue then K u = 0. If the cue C is a pseudo-asymptote cue then K = 0. If the cue C is a pseudo-geodesic cue then K g = 0.

12 79 Nil Kofoğlu Fom 7) and Definition 3.3: Theoem 3.4 If the cue C has any two of the following popeties it also has the thid: it is a pseudo-union cue, it is a pseudo-asymptote cue, it is a pseudo-geodesic cue poided that m ae not the components of a null ecto field. If φ = 0 o α = 0 o K = 0, we obtain K u = K g. Hence: Theoem 3.5 The necessay and sufficient condition fo a pseudo-union cue to be pseudo-geodesic is one of the following it is a pseudo-asymptotic cue, the conguence consist of the nomals, the diection of the tangent ecto field to C coincides with that of the ecto field l k. Refeences [] V. Hlaaty, Les Coubes de la Vaiete Wn, Memo., Sci. Math., Pais, 934. [] A. Noden and S. Yafao, Theoy of non geodesic ecto fields in two dimensional affinely connected spaces, Iz. Vuzo, Matem., 974), [3] A. Noden, Affinely connected spaces, GRMFL Moscow, 976. [4] T. K. Pan, On a genealization of the fist cuatue of a cue in a hypesuface of a Riemannian space, Can. J. Math., 6 954), no., [5] S. C. Rastogi, On pseudo-union cues in a hypesuface of a Riemannian space, Istanbul Uniesity Science Faculty the Jounal of Mathematics, Physics and Astonomy, 34 97), -5.

13 Pseudo-union cues 793 [6] B. Tsaea and G. Zlatano, On the geomety of the nets in the n- dimensional space of Weyl, Jounal of Geomety, ), no., [7] S. A. Uysal and A. Ozdege, On the Chebyshe nets in a hypesuface of a Weyl space, Jounal of Geomety, 5 994), no. -, [8] G. Zlatano, Nets in the n-dimensional space of Weyl, CR Acad. Bulgae Sci., 4 988), 9-3. Receied: July 8, 07; Published: August 6, 07