IOSR Joural of Mathematcs (IOSR-JM) e-issn: 78-578 -ISSN:39-765X Volume 0 Issue Ver VII (Mar-Ar 04) PP 43-5 wwwosrjouralsorg O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem- Elses ad Par of Straght es PK Srvastava & VNJha Galgotas College of Egeerg & Techolog Greater Noda Abstract: It s a terestg roblem to fd the fudametal fucto of a two dmesoal Fsler sace whose geodescs costtute a gve faml of curves M Matsumoto [5][6][8] ) obtaed the fudametal fucto of a two dmesoal Fsler sace whose geodescs are secal coc sectos The am of the reset aer s to obta the fudametal metrc fucto of two dmesoal Fsler sace whose geodescs are semelses ad ar of straght les We show that such sace s locall Mkowska I Prelmares et F = (M ( )) be a -dmesoal Fsler sace o a uderlg smooth mafold M wth the fudametal fucto () The fudametal tesor g j () the agular metrc tesor h j () ad the ormaled suortg elemet l () are defed resectvel b g j = h j + l l j h j = () (j) l = () where () = ad ()(j) = j t The geodesc the etremal of the legth tegral s ( ) where t t 0 d = = t0 alog a oreted curve C: = ( from ot P = (t o ) to ot Q= ( t > 0 s gve b the Euler euato () d 0 where I terms of F( ) = ( )/ () s wrtte the form () d d G ( ) h( where we ut (3) F F g j G () = r r J J ad h( = (d s/ ) / (ds/d Now we cosder two dmesoal Fsler sace ad use the otato () ad ( resectvel stead of ( ) ad ( ) The fudametal fucto (; are ostvel homogeeous of degree oe ad Therefore we have (3 a) = + = + Also from homogeet of ad we have (3 b) W ( sa) where W s called Weerstrass varat Coseuetl the two euatos rereseted b () reduce to the sgle euato (4) + ( ) W 0 Kewords: Fsler sace geodescs locall Mkowska 000 Mathematcs Subject Classfcato: 53B40 ) Numbers suare brackets refer to the refereces at the ed of the aer wwwosrjouralsorg 43 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es Whch s called Weerstrass form of geodesc euato Now cosder the assocated fudametal fucto A() d = = defed as follows: d (5) A() = (; ) (; = A ( ) Therefore =A A =A = A A = A = ( /) A = (-/) A = (/) A O usg these values (4) we have (6) A + A + A A =0 = whch s called the Rashevsk form of geodesc euato 3 We observe that =/ gves = ( ) / Hece from () we have aother form of geodesc euato (7) = G G 3 Now -dmesoal Fsler sace F we have the Berwald coecto B G jk G j G j G / G jk j k V () gve b v v r r v V( j ) G j r J V GrJ V j j wwwosrjouralsorg j ; defed b G / ad two kds of covarat dfferetato of Fslera vector feld Called the h- ad v covarat dervatve of have r (8) lrg Further from (8) we have r r (9) j (/ ) hrjg lrgj V resectvel Sce B s -metrcal e () =0 we F Net from (3) we have g j G J r = r that s (0) G l = r r Now we shall retur to the two-dmesoal case sce the matr (h j ) s of rak oe we get ad the vector feld m () satsfg () h j = m m j so we have () g j = l l j + m m j therefore we get easl (3) l l = l m = m l = 0 m m = Thus we obta the orthoormal frame feld (l m ) called the Berwald frame Therefore we have scalar felds h( ; ad k( ; such that (4) (m m ) = h (-l l ) (m m ) = k( l l ) hk = The euatos () ad (4) gve (5) g(=det g j ) = g g - g = (l m l m ) = h ad (6) (m m ) = (-l l ) h g We alread have h = = W ad h = (m ) = h (l ) = h Coseuetl we have l (7) 3 W = h = g Now we tr to fd the eresso for G the Berwald frame [4] the euatos (9) ad (4) gve r r = h rjg h r G j 44 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es m r = r r m jg m G j m r r = r mg mg m r r = r hl G l G m r = r hg Due to homogeet of G r r r we have G G therefore G m = ( ) / h j j r j j Usg G ( ll m m ) ( r ) l m ( ) / h ad (0) leads to (8) G = ( r r )l + ( /h) ( )m If we ut (9) r r = o ad = M (8) ad usg (6) ad (7) we get (0) G M = 0 G M = 0 W W II From geodescs to the Fsler metrc et us cosder a faml of curves {C(ab)} o the () lae R gve b the euato () = f(ab) wth two arameters (a b) Dfferetatg () wth resect to we get () (= ) = f (ab) Solvg () ad () for a b we fd (3) a = () b= () I vew of (3) the dfferetato of () leads to (4) = f (ab)= u() whch s recsel the secod order dfferetal euato of characterg {C(a b)} Now we are cocered wth the Rashevsk form (6) of geodesc euato = A / ad A = W 3 hece from (5) ad (7) we have (5) 3 W = A 3 A = g Thus t s sutable to call A the assocated Weerstrass varat If we ut B=A the the dfferetato of (6) wth resect to gves (6) B + B + B u + Bu = 0 whch s frst order uaslear artal dfferetal euato Its aular euatos are gve b d d d db (7) u Bu Now defg U(ab) ad V() b (8) U(; a b) = e u ( f f) d V( ) = U (; ) we obta (9) H ( ) B()= V ( ) where H s a arbtrar o-ero fucto of two argumets From A = B we get A the form (0) * A( ) A ( ) C( ) D( ) * A B( ) d d ( B( 0 where C ad D are arbtrar but must be chose so that A ma satsf (6) that s wwwosrjouralsorg 45 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es * * * * () C D = Au A A A If a ar (C 0 D 0 ) has bee chose so as to satsf () The (C C 0 ) =(D D 0 ) so that we have locall a fucto E() satsfg E =C C 0 ad E =D D 0 Thus (0) s wrtte as A=A * +C 0 +D 0 +E +E Therefore (5) leads to fudametal fucto ( ; 0 ( ; e( () * A ( ) C ( ) D ( ) 0 0 0 where e s the derved form gve b (3) e(;=e d+e d Thus we see that the Fsler metrc s uuel determed whe the fuctos H ad E of two argumets are chose Further for dfferet choce of the fucto H we obta Fsler saces whch are rojectve to each other because each oe has the same geodescs {C(ab)} III Faml of sem-elses et us cosder the faml of sem-elses {C(ab)} gve b the euato (3) + a = b b>0 o the semlae R From (3) we have (3) + a = 0 = Coseuetl the fuctos () () ad u() of the recedg sectos are gve b (33) a ( ) b ( ) (34) ' u( ) From (34) we get ' ( ' ) (35) ' ' whch characteres the faml {C(ab)} Now we fd the fucto U(;ab) ad V(;) defed b (8) Dfferetatg (34) we get u ad b U(;ab)= e d V ( ) b b Thus (9) mles that B()= H() = H() O accout of the arbtraress of H we ma wrte B as B=H() ad A * of (0) s wrtte the form (36) A * ()= H( )( d) or (36) A * ()= ( F( F( H t t We have take the lmt of tegrato from to stead of 0 to because t s the deomator of F( wwwosrjouralsorg 46 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es Now f we ut F (=H F (=H t t t t the from (0) ad () we have (37) C D = H( ) F( F t ( ) F ( F t 3 ( ) F ( tf ( t t Therefore we have: Theorem Ever assocated fudametal fucto A() of a Fsler sace R ( ; havg the sem-elses (3) as the geodescs s gve b A()= A * ()+C()+D() where A * s defed b (36) H s a arbtrar fucto of () gve b (33) ad the fucto (CD) must be chose so as to satsf (37) Eamle I artcular we frst ut H()=() for a real umber The F(= hece from (36) t ad (37) we have (38) * A ( t or (38) A * = ad (39) C D = ( ) ( ) ( )( ) If we choose C = ( ) we have ad D = ( ) (30) A()= ( )( ) Therefore t follows from (5) that the fudametal fucto (3) (; = - - - - where was omtted ( )( ) Case I If = the (38) ad (39) gves A * = ( log - +) C D = Choosg C = Coseuetl we obta fudametal fucto D = we have A()= log wwwosrjouralsorg 47 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es (3) ( = log Therefore = = From (3b) we have W so usg these 3 values (4) where we use ' ' ad ' The t leads to (35) mmedatel Case II If = we have smlarl A * = ( log ) C D = Choosg C= ad D= we have A()= log Therefore we obta the metrc (33) (; = log Now we shall retur to the geeral case wth the Fsler metrc (3) If we refer to the ew co-ordate sstem ( ) the we have (= ad the metrc (3) ca be wrtte the form (3) ( ; Sce does ot deed o ad ths s a smle metrc called a locall Mkowska metrc ad ( ) s a adated co-ordate sstem to the structure Further ts ma scalar I s costat Sce (3) s of the form () or (v) of Theorem 353 of [] we have drectl as follows: I () I > 4 ( ) ( ) < 0 +=( ) I 4 () ( ) ( )> 0 I I 4 +=( ) Therefore we have: Proosto The Fsler sace ( R ( ) wth a metrc (3) s locall Mkowska ad has the sgature ad the costat ma scalar I as follows: ( 3) () : = I = ( )( ) () ( 3) : = I = ( )( ) ( 3) Remark = 4+ The grah of I s show fgure ( )( ) 8 6 4 wwwosrjouralsorg 48 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es Sce a Fsler sace of dmeso two s Remaa f ad ol f I = 0 Therefore The Fsler sace R ( uder cosderato s Remaa f ad ol f =3/ IV Faml of ar of straght les et us cosder the faml of ar of straght les {C(ab)} gve b the euato (4) ( a) b = 0 b>0 o the semlae R From (4) we have (4) (-a)= b = 0 = Coseuetl the fuctos () () ad u() of the recedg sectos are gve b (43) a ( ) b ( ) (44) ' 0 u( ) From (44) we get (45) '' 0 whch characteres the faml {C(ab)} Now we fd the fucto U(;ab) ad V(;) defed b (8) Dfferetatg (44) we get u 0 ad U(;ab)= e 0d V ( ) Thus (9) mles that B()= H() Therefore A * of (0) s wrtte the form (46) A ()= * H( )( d) or (46) A * ()= ( F( F( H t t We have take the lmt of tegrato from to stead of 0 to because t s the deomator of F( Now f we ut F (=H F (=H the from (0) ad () we have t t t t (47) C D = F ( F ( Therefore we have: Theorem Ever assocated fudametal fucto A() of a Fsler sace R ( ; havg the ar of straght les (4) as the geodescs s gve b A()= A * ()+C()+D() where A * s defed b (46) H s a arbtrar fucto of () gve b (43) ad the fucto (CD) must be chose so as to satsf (47) Eamle I artcular we frst ut H()=() for a real umber The F(= hece from (46) ad t (47) we have (48) * A ( t or (48) A * = ( ) ( ) ( )( ) ad (49) C D = 0 wwwosrjouralsorg 49 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es If we choose C = ( ) ad D = we have ( ) (40) A()= ( )( ) Therefore t follows from (5) that the fudametal fucto (4) (; = - - / where was omtted ( )( ) Case I If = / the (48) ad (49) gves A * = ( log - +) C D = 0 Choosg D = C = we have A()= log Coseuetl we obta fudametal fucto (4) ( = log Therefore =0 =0 From (3 b) we have W so usg these values (4) 3 where we use ' ' ad ' The t leads to (45) mmedatel Case II If = we have smlarl A * = ( log ) C D = 0 Choosg C= D= we have A()= log Therefore we obta the metrc (43) (; = log Now we shall retur to the geeral case wth the Fsler metrc (4) Sce t s of ute smlar form to (3) we get also the result smlar to roosto as follows: Proosto The Fsler sace ( R ( ) wth a metrc (3) s locall Mkowska ad has the sgature ad the costat ma scalar I as follows: (4 3) () : = I = ( )( ) (4 3) () : = I = ( )( ) (4 3) Remark = 4+ The grah of I s show fgure ( )( ) ( ) 8 6 4 / Sce a Fsler sace of dmeso two s Remaa f ad ol f I = 0 Therefore The Fsler sace R ( uder cosderato s Remaa f ad ol f =3/4 wwwosrjouralsorg 50 Page
O A Two Dmesoal Fsler Sace Whose Geodescs Are Sem-Elses Ad Par Of Straght es Refereces [] P Atoell RS Igarde ad M Matsumoto The Theor of Sras ad Fsler Saces wth Alcatos Phscs ad Bolog Kluwer Acad Publshers Dordrecht/Bosto/odo (993) [] W Blaschke Vorlesuge über Dfferetalgeometre I (Drtte Auflage) Srger 930 (Chelsea 967) [3] M Matsumoto Foudatos of Fsler Geometr ad Secal Fsler Saces Kasesha Press Sakawa Otsu Jaa (986) [4] M Matsumoto Projectvel flat Fsler saces of dmeso two ad a eamle of rojectve chage Proc 5th Natl Sem Fsler ad agrage saces Soc St Mat Romaa Uv Brasov (988) 33-39 [5] M Matsumoto The verse roblem of varato calculus two-dmesoal Fsler sace J Math Koto Uv 9(989) 489-496 [6] M Matsumoto Geodescs of two-dmesoal Fsler saces Mathl Comut Modellg 0 No 4/5 (994) -3 [7] M Matsumoto Ever ath sace of dmeso two s rojectve to a Fsler sace Oe sstems & formato damcs 3-3(995) -3 [8] M Matsumoto Two-dmesoal Fsler saces whose geodescs costtute a faml of secal coc sectos J Math Koto Uv 35-3(995) 357-376 wwwosrjouralsorg 5 Page