Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A WEYL SPACE
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1 Jou of Eee d Ntu Scece Mühed e Fe Be De S 5/4 SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A EYL SPACE N KOFOĞLU M S Güze St Üete, Fe-Edeyt Füte, Mtet Böüü, Beştş-İSTANBUL Geş/Receed:..4 Ku/Accepted: 3..5 ABSTRACT Let e hypeufce of the ey pce +. Let (,,..., ) e tet ecto fed eo to d e the ozed o ecto fed of. Code the ( + ) - et (,,...,, ). By u the pooed cot dffeetto, we ft ot the et of fou coepod to the Feet fou fo octed wth cue C o h the tet ecto fed. e the, dee two t coce the othoo eupe (,,..., ), (,,..., ) e dffeete ecto fed o. Keywod: ey pce, Net of cue, Pooed cot dete. MSC ue/uı: 57R55, 58A99. BİR EYL UZAYININ GÖZÖNÜNE ALINAN HİPER YÜZEYLERİNİN BAZI ÖZELLİKLERİ ÖZET ey uzyıı hpeyüzey ou. (,,..., ), e t teğet etö ı e,, + oze edş o ecto ı ou. Geeeştş oyt tüe uı, öce,,...,, ( ) ( + ) - şeee öz öüe ıı. hpeyüzey C eğ teğet ecto ı ğı o Feet foüee teü ede foüe ede edşt. So, (,,..., ) othoo şeee ydııy yt tııştı. Aht Sözcüe: ey uzyı, Eğe şeee, Pooed oyd tüe. de tıı e-pot: ofou@feed.u.edu.t, te: () / 74 4
2 N. Kofoğu S 5/4. INTRODUCTION A deo fod teo j d yetc coecto equto j T whee fo j λ d to e ey pce f t h yetc cofo etc tfy the coptty codto e y the j (.) T deote cot ecto fed. The yetc teo j ozto of the j (.) d the cot ecto fed T T T tfoed y the ue + λ (.3) whee λ pot fucto o []. Let the coodte of ey hypeufce of deo d t eeop pce of ( +) deo e defed y u d x, epectey d et. (.4) u A qutty A ced tete wth weht of of the fudet teo Ude the eozto of the ue A λ A j of the fo j j. j λ, the qutty A che ccod. (.5) The pooed cot dete d the pooed dete of the tete A e defed foow, epectey [] d A A the tete A. A A [3] whee A T A (.6) T A (.7) uu dete of the tete A d A Let w e y ecto eo to ecto w, the cot dete of. If w wth epect to u pt dete of w e the cott copoet of the 4
3 Soe Popete Coce the Hypeufce whee ω ω + p u p ω (.8) p e the coeffcet of the coecto d e defed y the fo j { j } ( T j + T jδ T j ) d { j } e ced the ecod d Chtoffe yo d e defed foow δ (.9) { } ( ) j Hee, the j,, j j, +. (.) j, the pt dete of j wth epect to A ey pce hoty deoted y ( j j, T ) Let ( T ),. u. ıj, e hypeufce of the ey pce + (, Tc ) d (,,, + ) d u (,,, ) e the coodte of (, Tc ) T, T d (, T ) (, ), epectey. The etc of ( ) j y the eto j whee c x x ( j,,, ;,,, + ) j x the cot dete of j x + d + c e coected (.) x wth epect to u. The pooed cot dete of A wth epect to A, epectey. Thee e eted y the codto c A x A (,,, ; c,,, + ) u d c x e A d c. (.) Let the o ecto fed of ( j, T ) e ozed y the codto. Sce the weht of x { }, the pooed cot dete of x, ete to u, e y [] x x ω (.3) w e the coeffcet of the ecod fudet fo of ( T ),. whee j O the othe hd, t ey to ee tht the pooed cot dete of e y ω x. (.4) By e of (.3), the pooed cot dete of j x foud to e [4] 43
4 N. Kofoğu S 5/4 j x Ω Let, j. (.5) d + ( Tc ) d ( j T ) + ( Tc ) d ( T ), x, e the cott copoet of the ecto fed,, epectey. Deot the copoet of, y j d we he [4,5] ete to ete to x. (.6) The pooed cot dete of the ecto fed d t ecpoc e, epectey, e y [5] T, T. (.7). THE FORMULAS BELONGING TO THE ORTHOGONAL NET The pooed cot dete of the decto of, ( ) c e wtte the fo [6,7] (.) whch,,. (.) e c the ft tet ecto fed d the ft cutue ecto fed d the ft cutue of C. Sce ( ), (.3) pepedcu to. T the pooed cot dete of the decto of, we fd tht α + β (.4) whee 44
5 Soe Popete Coce the Hypeufce,,. (.5) By u (.), (.3) d (.5) fo (.4), we otα. Putt β, (.4) ecoe +. (.6) e c the ecod tet ecto fed d the ecod cutue ecto fed d the ecod cutue of C. Sce pepedcu to, the equty tfed. If we te the pooed cot dete of th equty the decto of, we et + + (.7) + e c the (+) th. tet ecto fed d + the (+) th. cutue ecto + + fed d + the (+) th. cutue of C. Poceed the e wy, we fy ot the dete of the fo (.8) Geey, the expeo (.8) c e wtte p+ p (.9) p p+ p whee + d. e c the th. tet ecto fed d the th. cutue ecto fed d + the th. cutue of C. I th wy, we ot utuy othoo ecto,,..., t pot P of C whch tfy the codto δ (,,,..., ) (.) Now, we h ot oe pott eut y the ft d ecod ode pooed cot dete of the ecto fed of the eupe: 45
6 N. Kofoğu S 5/4 Fo (.7), we he If we te the pooed cot dete of oth de of the oe equty the decto of, we he ( + ) + + ( ) ( ) (.) If we utpe (.) y +, we et + + (.) ( ) If + cott, the eft hd de of (.) zeo d f eft hd de of (.) zeo, + cott. Hece: Cooy. The ecey d uffcet codto tht the ecod ode pooed cot dete of the th. tet ecto fed the decto of e othoo to tht + + e cott. Let u utpe (.) y. The we he ( + + ).. (.3) If +, the ht hd de of (.3) zeo. Tht, the eft hd de of (.3) zeo. If eft hd de of (.3) zeo, +. Fo th t foow tht Cooy. The ecey d uffcet codto tht the ecod ode pooed cot d dete of the th. tet ecto fed e othoo to tef tht oth +. 46
7 Soe Popete Coce the Hypeufce If we utpe (.) y, we ot. (.4) If cott, the ht hd de of (.4) zeo d coeey, f the eft hd de of (.4) zeo, the cott. Hece: Cooy.3 The ecey d uffcet codto tht the ecod ode pooed cot dete of the th. tet ecto fed the decto of th. e cott. If we epce y (.), we et tet ecto fed tht Fo (.4) d (.5), we ot e othoo to the ( ). (.5). (.6) Theefoe: Cooy.4 The copoet of the ecod ode pooed cot dete o the ( ) th. tet ecto fed equ to the ete of the copoet of the ( ) tet ecto fed o the th. tet ecto fed. Sce the ecto fed pepedcu to, pooed cot dete of th equty, we fd + th. hod. If we te the. (.7) A f we te the pooed cot dete of the e equty, we he + +. (.8) 47
8 N. Kofoğu S 5/4 th the hep of (.7), we he (.9) Fo hee: Cooy.5 The pooed cot dete of the coecute tet ecto fed the decto of e othoo. Now, f we utpe (.) y, we ot. (.) If o zeo the the eft hd de of (.) zeo d f the eft hd de of (.) zeo, ethe o. Hece: Cooy.6 The ecey d uffcet codto tht the ecod ode pooed cot dete of the th. tet ecto fed e othoo to ( ) th. tet ecto fed tht ethe o. If we utpe (.) y If + o +, we he. (.), the the eft hd de of (.) zeo. Coeey, f the eft hd de of (.) zeo, the ethe o. + + Theefoe: The ecey d uffcet codto tht the ecod ode pooed cot dete of the th. tet ecto fed the decto of e othoo to ( + ) th. tet ecto fed tht ethe o. + + If epced y + (.), we he +. (.) + + If we cope (.) d (.), we et 48
9 Soe Popete Coce the Hypeufce +. (.3) + e ow tht +. If we te the pooed cot dete of oth + + de equto the decto of, we he + A f we te the pooed cot dete of the t equty, we et (.4) + + U (.3) d (.4), we ot (.5) Fo hee: Cooy.7 The ecey d uffcet codto tht the pooed cot dete of the decto of e othoo to the pooed cot dete of tht ethe the + ecod ode pooed cot dete of e othoo to o the ecod ode + pooed cot dete of e othoo to. + Now, we code ptcu ce. Fo expe; the ce of. The we ot fo (.) tht + ( ) 3 3 (.6) 49
10 N. Kofoğu S 5/4 Mutpy (.6) y 3 3 3, the foow equty foud. (.7) If o 3, the the eft hd de of (.4) zeo. Coeey, f the eft o. hd de of (.7) zeo the ethe Theefoe: Cooy.8 If o 3 tet ecto fed the decto of coee o tue. If we utpe (.6) y 3, the ecod ode pooed cot dete of the ft othoo to the thd tet ecto fed. The, we et + If. ( ). (.8), the the eft hd de of (.8) zeo. Coeey, f the eft hd de of (.8) zeo, the Hece: Cooy.9 the ecey d uffcet codto fo the othooty of the ecod ode pooed cot dete of the ft tet ecto fed the decto of to tef. If we utpe (.6) y, we et. (.9) If cott, the the eft hd de of (.9) zeo. Coeey, f the eft hd cott. de of (.9) zeo, we ot Fo hee: Cooy. The ecey d uffcet codto tht the ecod ode pooed cot dete of the ft tet ecto fed the decto of e othoo to the ecod tet ecto fed tht e cott. 5
11 Soe Popete Coce the Hypeufce Mutpy (.6) y, we he. (.3) If cott, the the eft hd de of (.3) zeo. Coeey, f the eft hd de of (.3) zeo, the cott. Theefoe: Cooy. cott the ecey d uffcet codto tht the ecod ode pooed cot dete of the ft tet ecto fed the decto of e othoo to the o ecto fed. Hece: If the ht hd de of (.8) zeo, the the ht hd de of (.7), (.9) d (.3) e o zeo. Fo th t foow Cooy. If the ecod ode pooed cot dete of the ft tet ecto fed the decto of othoo to tef, the t o othoo to the o ecto fed, the ecod tet ecto fed d the thd tet ecto fed. Mutpy (.6) y If, we et (.3), the the eft hd de of (.3) zeo d the coee of t o tue. Hece: Cooy.3 The ecey d uffcet codto tht the pooed cot dete of the ft tet ecto fed the decto of e othoo to the o ecto fed tht e zeo. If we utpe (.6) y If, we ot. (.3), the the eft hd de of (.3) zeo. The coee o tue. Thu, we he 5
12 N. Kofoğu S 5/4 Cooy.4 The ecey d uffcet codto tht the ft ode pooed cot dete of the ft tet ecto fed the decto of e othoo to the ecod tet ecto fed tht e zeo. If d, the the ht hd de of (.3) d (.3) e zeo. I th ce, the ht hd de of (.8) zeo. Theefoe, we he Cooy.5 If the ecod ode pooed cot dete of the ft tet ecto fed othoo to tef, the the ft ode pooed cot dete of the ft tet ecto fed othoo to the o ecto fed we to the ecod tet ecto fed. e he ee fo (.3) tht f the ft ode pooed cot dete of the ft tet ecto fed othoo to the ecod tet ecto fed. The coee of t o tue. But fo (.7) fo, t ee tht the ecod ode pooed cot dete of the ft tet ecto fed othoo to the thd tet ecto fed. Hece: Cooy.6 If the ft ode pooed cot dete of the ft tet ecto fed the decto of othoo to the ecod tet ecto fed, the the ecod ode pooed cot dete of the ft tet ecto fed the decto of thd tet ecto fed. e ow tht the equto of (.8) hd ee expeed othoo to the ( + ). (.33) If cott, the fo (.9) the ecod ode pooed cot dete of the ft tet ecto fed the decto of othoo to the ecod tet ecto fed. If cott, the fo (.3) the ecod ode pooed cot dete of the ft tet ecto fed the decto of othoo to the o ecto fed. If the codto cott d cott e tfed, the we ot fo (.33) tht cott. Cooy.7 If the ecod ode pooed cot dete of the ft tet ecto fed othoo oth to the o ecto fed d to the ecod tet ecto, the th d the ft tet ecto fed cut ech othe ude cott e. 5
13 Soe Popete Coce the Hypeufce 3. THE DERIVATIVE FORMULAS FOR A GEODESIC TANGENT Defto 3.: Let C e cue the ey hypeufce fed of C. If the pooed cot dete of ced eodec,.e.. d et e the tet ecto the decto of tef zeo, the C Let u code the eodec tet ecto fed of the cue C t the pot P d et u deote t y C. Futheoe, et u deote the tet ecto fed, the pcp o ecto fed d the o ecto fed eo to C y,,, epectey e ow fo (.) tht (3.) e c wte τ τ (3.) fo Doux_Rocou Equto, whee τ d τ e the ft d the ecod cutue of the eodec tet to the cue C, epectey, τ d τ e the o cutue d the eodec too of the cue C, epectey. Fo (3.) d (3.), we et τ τ. (3.3) If we utpe (3.3) y tef, we ot τ + τ. (3.4) Theefoe, we c tte the foow theoe: Theoe: 3.. If y two of the foow popete fo cue hypeufce of ey pce + e tfed, the the thd o hod: ) The ft cutue of the eodec tet he. ) The eodec too of the cue zeo. ) The ft cutue of the cue C zeo Theoe: 3.. If the cue C yptotc e, the the pooed cot dete of the o ecto fed the decto of othoo to the cue. 53
14 N. Kofoğu S 5/4 Poof: 3.. Let C e yptotc e. The the o cutue of C zeo, tht, Fo C, κ. κ τ (3.5) tfed. e ow tht ( ) κ. (3.6) Fo (3.5) d (3.6), we ee tht The poof copeted. O the othe hd, we ow tht ( ), we et othoo to. τ τ fo (3.). Sce τ. (3.7) Th y τ. (3.8) Ao fo (3.3) (3.9) Fo (3.8): Cooy 3. The poduct of the pooed cot dete of the o ecto fed o yptotc e y tef the eodec too of the yptotc e. 4. ON THE HYPERSURFACES MEETING UNDER A CONSTANT ANGLE e code cue C wth the tet ecto fed whch coo to two hypeufce d thee hypeufce eet t cott e the the foow codto tfed:. Let d e the o ecto fed wth epect to thee hypeufce. If ( ) (4.) Fo th t foow tht. (4.) { ( )}. + { } 54
15 Soe Popete Coce the Hypeufce Sce d hypeufce, o tht, eet ude cott e, C e of cutue fo oth fed to the cue wth epect to + d dete of the o ecto fed the decto of Ao to the cue wth epect to + d dete of the o ecto fed the decto of Fo the oe foto d (4.), we ot + (4.3) o co, co, e c expe th : Theoe : 4.. whee e the copoet of the tet ecto the e poduct of the pooed cot y tef. whee e the copoet of the tet ecto fed the e poduct of the pooed cot, If cue C coo to two hypeufce y tef.. (4.4) d uch tht they eet ude cott e o C, the (, ) Tc +. of the ey pce (, ) + Tc ue t fo 5. AN INVARIANT ASSOCIATED ITH AN ORTHOGONAL ENNUPLE IN A EYL HYPERSURFACE Theoe: 5.. The u whee the e poduct of the pooed cot dete of the o ecto fed to ey hypeufce the decto, of the th ecto of othoo eupe y tef, t. 55
16 N. Kofoğu S 5/4 Poof: Let u deote the othoo eupe y (,,, ). The pooed cot dete of the o ecto fed the decto of the ecto of the othoo eupe c e expeed, y (.4), ω x (,,, ). (5.) Fo th we ot ( )( ) ( )( ) ω j ω t x x j t j j ω ω j wth the hep of (.) d (5.). Let u deote th c y,.e. ω jω. If we te the u of the que wth epect to, we fd fo (5.) tht j ω j ω (5.) j ω jω, (5.3) ce fo the ecto fed of othoo eupe. Th how tht t. The poof of the theoe copeted. I thfu to Pof. D. Ley Zee Aü fo he udce. REFERENCES [] Node, A. : Affey Coected Spce, GRMFL, Mocow, (976). [] Node, A., Yfo, S. : Theoy of No-eodec Vecto Fed Two Deo Affey Coected Spce, Iz., Vuzo, Mth., No., 9-34, (974). [3] Hty, V. : Le Coue de Vete, Meo. Sc. Mth., P, (934). [4] Uy, S. A., Özdeğe, A. : O the Chyhe Net Hypeufce of ey Spce, Jou of Geoety, V.5, 7-77, (994). [5] Te, B., Zto, G. : O the Geoety of the Net the -Deo Spce of ey, Jou of Geoety, Vo. 38, 8-97, (955). [6] Ku, R. N. : Foue Coepod to Feet Foue, Acd. Roy. Béque, 4, 9-34, (955). [7] Aü, L. Z. : Feet Fou Fo Cue Geezed ey Spce, Gt, Vo.5, No., 49-64, (). 56
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