It J otem Mat Sceces Vo 7 0 o 9 99-98 O Hyersurface of Seca Fser Saces Admttg Metrc Lke Tesor Fed H Wosoug Deartmet of Matematcs Isamc Azad Uversty Babo Brac Ira md_vosog@yaoocom Abstract I te reset work te yersurfaces of some seca Fser saces suc as -reducbe Sem -reducbe ke P-reducbe P ke S ke ad 3 δ δ + v Recurret Fser saces admttg te tesor fed δ wc satsfes te codto ave bee studed δ δ ke Matematcs Subect assfcato: 5360 Keywords ad Prases :Fser yersurface-reducbe Sem -reducbe v S ke P-reducbe P ke ad recurret T-tesor 3 INTRODUTION osder a o-remaa yersurface F of F ( caracterzed by te equato x x ( u Te fudameta tesor of F s gve by ( were g g B B B x u
90 H Wosoug were u are Gaussa coordate o F ad Greek dces ru from to - Te comoets of aguar metrc tesor are gve by g were s te ormazed eemet of suort By sme cacuato we ca get ( g were g g B B B B B B We use te foowg otatos o Fser yersurface [6] (3 (a 0 (b δ (c (d METRI LIKE FINSLER FIELD IN F Te metrc ke tesor fed s defed by [5]ad studed by ([4] [5] as ( + wc satsfes te codto By a sme cacuato wt te e of reato ( w become k B ad te reato ( te ( + were 0 deote cotracto wt we ave B ad B0 B v Terefore B0B0 Teorem( I a yersurface of F te metrc ke tesor fed s of te form (
O yersurface of seca Fser saces 9 3 THE EISTENE OF OVARIENT TENSOR IN SPEIAL FINSLER HYPERSURFAES I ts secto we study yersufaces of seca Fser saces ke -reducbe Sem -reducbe -ke P-reducbe P -ke S3 -ke ad S 4 -ke wc are admttg metrc ke tesor fed Defto(3 A Fser sace F ( s sad to be -reducbe f te (v-torso tesor s gve as [] (3 ( + k + were g otractg (3 by B ad usg were B B B B B we get (3 ( + + x were te matrx of roecto factor B s of te rak - u Now cotractg (3 by ad usg (3 betwee we ave δ δ [ δ ( + + ] [ ( δ + + ( + + ( + δ δ δ δ δ ] ( δ + δ + δ + + δ δ
9 H Wosoug [ δ + δ + δ + ( δ + ( δ δ δ ] [ δ + δ + δ + ( + ( δ δ ] ( δ + δ + δ terefore (33 δ δ Hece we ave te foowg Teorem (3 I a yersurface of -reducbe Fser sace te covarat tesor fed satsfes ( s of te form (33 δ Defto (3 A Fser sace F ( s to be sem -reducbe f te torso tesor f wt o-zero egt of te torso vector s of te form [3] (34 ( + + k /( + otracto of above reato by g we get + q / (35 were g otractg (35 by B ( k + + /( + q / we get ad +q were (36 ( + + /( + q / B B Terefore we get Teorem (3 A yersurface reducbe otractg (36 wt F of sem -reducbe Fser sace δ ad usg (3 we ave F s sem -
O yersurface of seca Fser saces 93 ( + + δ δ + q δ / [ ( δ + + ( + + ( + δ δ δ δ δ + ( δ + δ ] + q ( δ + δ / ( δ + δ + δ + δ + δ + q δ / [ δ + δ + δ + δ ( δ + δ ( δ ] + q δ / [ δ + δ + δ + δ ( + δ ( ] + q δ / ( δ + δ + δ + q δ / terefore (37 δ δ Terefore we ave Teorem (33 I a yersurface of sem -reducbe Fser sace te covarat tesor fed δ satsfes ( s of te form (37 Aga te yersurface F of a -ke Fser sace s gve by [7] as (38 were stads for ad s o-zero otractg (38 by δ we get δ δ (
94 H Wosoug [ ( ] δ + δ Wc accordg to (3a Terefore (39 δ δ Terefore we ave δ Teorem (34 I a yersurface of a -ke Fser sace te covarat tesor fed δ wc satsfes ( s of te form (39 Defto (33A Fser sace s caed P-reducbe f te torso tesor as (30 P ( Pk + P + k P /( were P P 0 otracto of (30 wt g we get (3 P ( P + Pk + P /( otractg (3 wt B we get (3 P ( P + P + P /( Terefore we ave Teorem (35 A yersurface F of P-reducbe Fser sace otractg (3 by δ we get P δ ( P + P + P δ [ P ( δ + δ + P ( δ + δ + P + ( Pδ + δ P + δ P were we ave made use of (3 Terefore (33 P δ Pδ Hece we get ( δ δ s wrtte F s P-reducbe ]
O yersurface of seca Fser saces 95 Teorem (36 I a yersuface of P-reducbe Fser sace te covarat tesor fed wc satsfes ( s of te form (33 δ Defto (34 A Fser sace F ( f s P -ke f t s caracterzed by (34 P K Were K K ( x y s a covarat vector fed otracto of te above reato wt g t w be caracterzed as foows (35 P K otractg (35 by B λ we get (36 P K λ Terefore we ave Teorem (37 A yersurface λ λ F of P -ke s P -ke otractg ( 36 by δ we get P λ δ ( K λ λ δ K λ ( δ + δ λ ( δ + K λδ λδ Terefore (37 P λ δ Pλδ Were we ave used reato (3 Terefore we ave Teorem(38 I a yersrface of P -ke te covarat tesor fed ( s of te form (37 Defto (35 A Fser sace S s wrtte te form [] δ K λδ λδ + K λ δ δ λ δ δ F ( f 3 δ wc satsfes s caed S3 -ke f te curvature tesor (38 L S S( k k were te scaar curvature S ( S g g s a fucto of osto aoe otracto of (38 wt g we get S3 -ke te form (39 L S ( S k Aga cotractg (39 by B λ we get (30 L S S( λ Terefore we ave Teorem (39 A yersuface of a S3 -ke s a S 3 -ke λ λ
96 H Wosoug otractg (30 by δ we get L S L S + λ δ λ ( δ δ L S[ λ ( δ + δ λ ( δ + L S L S( λ δ ( λ δ λ δ + λ δ λ δ λ ( δ δ λ δ + δ λ ( δ δ L S ( ( λ δ _ λ δ + λ ( δ λ L S( λ δ λ δ terefore we get (3 L Sλ Sλδ Terefore we ave Teorem(30 I a yersurface of S3 -ke Fser sace te covarat tesor fed δ satsfes ( s of te form (3 Defto (36 A Fser sace F s caed -recurret f torso tesor satsfes te equato [] (3 k Were k s some vector fed ad stads for -covarat dfferetato otractg (3 by B λ we get (33 λ k λ Terefore we get Teorem (3 A yersurface of a -recurret Fser sace s a -recurret ] δ Aga cotracto (33 wt δ we get λ δ k λ δ k λ ( δ + δ k λδ Hece (34 λ δ k λδ Terefore we ave
O yersurface of seca Fser saces 97 Teorem (3 I a yersurface of -recurret Fser sace te covarat tesor fed satsfes ( s of te form (34 δ ν Defto (37 A Fser sace F s caed -recurret f te torso tesor satsfes te reato [] (35 a Were a s some covarat vector fed ad s ν covarat dfferetato otracto of (35 wt we obta B λ (36 a λ λ Terefore we ave Teorem (33 A yersurface of Aga cotractg (36 by δ we get a λ δ λ δ a λ ( δ + δ a + ( λδ λ δ ν -recurret Fser sace s a ν -recurret a λδ Terefore we obta (37 λ δ a λδ Tereforewe ave ν Teorem (34 I a yersurface of -recurret Fser sacete covarat tesor fed δ satsfes ( s of te form (37 A T-tesor s troduced smutaeousy but deedety by Kawaguc ad Matsumoto 97 I fact te ν covarat dfferetato of defes te T-tesor T te form T L + + + + k otracto of te above reato wt (38 otractg (38 by (39 k k g we obta T-tesor as foows T g T L + k + + + k B λ we get T λ L + λ λ + λ + λ + λ Hece we ave Teorem (35 A yersurface of Fser sace wt T-tesor s wt T-tesor
98 H Wosoug otractg te above equato (39 by δ we obta T L + + + + ( + λ δ ( λ λ λ λ δ δ λ L + δ λ + λ δλ δ + δλ + δλ + L λ δ + λ δ + λ δ + λ δ + λ δ Tδλ + λ δ λ δ we get Terefore f 0 (330 T λ δ Tδλ Hece we ave te foowg Teorem (36 I a yersurface of Fser sace wt T-tesor te covarat tesor fed satsfes ( s of te form (330 rovded tat 0 δ λ δ REFERENES [] KtayamaM : Fser saces admttg a arae vector fed Baka Joura of geometry ad ts acatos 3(998 9-36 [] Matsumoto M : O -sotroc ad recurret Fser saces J MatKyoto Uv -(97-9 [3] Matsumoto M : Foudatos of Fser geometry ad seca Fser saces Kasesa ress Sakawa Otsu Jaa 986 [4] Narasmamurty SK Avees ST KumarP : Seca Fser saces admttg metrc ke tesor fed : It Joura of Mat Aayss Vo 3 9 No 5- [5] Prasad BN Padey TN ad Dwved PK : Metrc ke tesor fed a Fser sace Ida J ure ad a Mat 7(9 (996 869-873 [6] Rud H : Te dffereta geometry of Fser sace Srger Verag Ber 959 [7] Sg UP Guta BN : Hyersurfaces of ke Fser saces Pubcatos De sttute Matematque Nouvee sere tome 38(5 985 77-8 Receved: October 0