Metric Structures for Riemannian and Non-riemannian Spaces M ...

nz ¢¡¤£¦¥¨§¦©£¦¥©£¥¡¥¥§¡¤ !§"#¤ §$¤%¦&'c~ $!8-@ "($+r#" p pt%O©u p)+82s )082'!G"r2a n p_.(p0" tq\WHr#a n pT t%82' { ¢ r3sd.(qV"u r/: t pZpltq1)er)?" q182r82 ¦2|q9uB8.Jp0"FaG _0f =b/aJ l ¨ t%OuB8 ~9< 8 pHuP"($?pZt p p 6 qvuB"Wr -0 ¨0¥W)-"#£;£ u p)+82sr)082'!G"(>¤n p '!?q pZtur n u { r2)p ¦ r#a.(p+"t q;)u_82 ¤ l )082. { rtp R+|R3x l z ~ ~¦| !82"($?pZt`@>rO "(8 { 'd"(qV"§q9u‚"%$+r-"*r l uB' _>wH~ ¢§q9uEq1 rt q9r#G" '*+spt "($?pP t%82' { 82: { pZt(.`'d"r#" q182+uI8#: ¤„@Gq\"($©*?q\" pFu(' {?{ 8t(" |lC~ l~r< p ©*?p ¦§ > '*?q ptu r n q\"&OPq1 "($?p )r-"p0 8t%Oc8#: u { r2)pZu€@Gq\"($2r : tpp { t(8 { pZtl£¢tq1)Wr2)?"q\8[8#:Er n8d)Zr nvnOT)+8. { r2)?" j u p)082+s€)082'!G"r#a n pj 3t(8' { ¢(| $!8-@ "($+pq9uB8.Jp0" K (] ¤ _%/i'hZi_0g`iZg_0ach SiaJKfkHgbieh^+`ach b p+"_'+u n88 r#" "($+pf pX.T8sd' n qu { r)¦p q1©r*u(.§r nvn ?pq1 2$aK8t%$!88s ¦§82:XuB8.Jp u { r)¦p … p U r#aK8l u('[)¦qvp0G" nO u(.§r nvn ~ ¦ )r2Ua-petr#"($?pZts+r#"('?tr nvnOJtp { t pHu p"pZs*r3u ¤¤¦ ¢pj @m$?ptp6{ n rq1?pHs q\‚"($?p:u q1. { n pHu&" )r3u pFq\ R+|RGC|/ s?pZr nvnO j @Fps@:r2G"k"(8~$+r pXu(82.(p^r-"('+tr np ptu r n a?tr2*)=$?pHs )+8 ptq\* i ¤ ˆ @m$?q;)=$`$rug u(8 l …'~ >;tr2.(q ©g)Zr-"q\8 i r#"gB'*?ql p p a-p+"(" pZt j q { u%)=$?qV"%W ~ .§r { u …¨ˆ … e :Q8t r nvn …©RB… e UJ |/, 'd" sd8 !82"~ $ ptu($?q r n u(8©uB'* 2 pHu&"pZs "%$+r-" ¨ t%OuB8Pu { r)¦pHu )+8.(p€: t(8. uB'?q\"r2a n pU.(pZr#>lapHuX82 "%$?pQu { r2)p©2 8#:>q\ ©*?q\" pQu(O.T.Jp0" tq1)T.§r#" tq1)pZu qvp0@:pZs ru tr2+sd82.u('?tnO ~ tr2+sd82. .(p+"t q;)[u { r)¦pQqvu ¦2 |cM 8 .§r p["%$?q9u { t p)¦q9u p j )+82u q9spt`"($+p)¦qvp0G"{ r2)p2 8#:I.(p0" tq1)Zus8 ¤ U u w R=C_R z z.zw r#+s n p+" ¢ a-per { t%82ar#a?q n q\"&OT.(pZru('?tpu2 @GqV"%$r"($?p :Q8 n#n8-@Gq\* { t%8 { pt(" qvpZu |82n { t%8f p)?" q182 l { '+u($d> :Q8t%@:rts ~ 8#: ¢ "(8`"($?peu { r2)p ¢ 8#:k.(p+"t q;)u"($+p~+r#"('?trwjR%C_R z.z z R w q9u>r , 8tp n .(pZru('?tpj@m$!8 u pYuB' {?{ 83tB">p q '+r n u>r n#n 8#: ¢z:Q83t82vup pt%O U w R%C R z z.z |jM $+p0©r n.T83uB"8r nvnFl :Q8t~"($?q9u ¢ ~ .(p+"t q;)u ¤U 2rtp{+{ t(8 6 q1.§r-" p nO ¦2 > '!+q ptu r n j r2+sc$?p*)¦p/"($+pZ)082. { n p+"q\8 8#: l ¤ R ¤ ~ qvurtq1) "(8 ¦2 :Q8t r n.T8 u&" r nvn ¤ U 2| 4 $?p)cq1! Qp pZt(OG"($+q\! r' { qvuq9uB8.Jp0"f&'+uB"Wr#!82"($?pZtGp 6 pt=)¦q9u pj:Q83t9"($?p tpZr3spt |p 6 q9u&"p0*)p 8#:"($?pHu p u { r2)pZuGr#s { t(8 pz"($r-"m"($+p0OQrt p r n#n q9uB8.(p+" tq;)j"%8 ¦2|~ p0"('?t% "(8‡"($?pr)+8+u&"t('g)?" q182 8#: ¦§ ru‚"($?p/"(8-@:pt`8#:9"($+p § >;p 6 " p+u q182ur#s ©*s/82"($?pZt l uB" t%82! 3pt ~ uB" t%'*)?"%'?tpZu982U"($?pHu p l apHu q9spHuP"($?pX.(p+"t q;) ~ @Gq\"($r"($+plPtr#u q\" q p r#'!"(82.T83t { $+qvu(. t%82' {E| l $r p!8#"m:Q' nvnOQu(8 n pZsU"($+qvu>p 6 pZt%)qvu p | ~"@m$?q;)=$ $+r3uWr€!8d> " tq q9r n q9uB8.(p+" t%Or t%82' { ¢p| : ¢p q9uWr ©g?qV"p 3t(8' {Rj "%$?p0'r¦)082+u qvuB"uj8#:Xq9uB8.(p+" t%OZ) n ruu pHum8#:@u('?q\"r#a nOZ.§rt"pHs6u { r)¦pHu … ) n83u pX"%8 … p r3upZr)=$ … U‡ uB'*)=$r"%$+r-" l "($?p n8d)r n qvu(82.(p0" tq#pHu98#: ~ ¤ pZp { " tr2)[82:q9u(82.(p+"t q;)qV: "($?pZt p8qvu:r\u('?q\"r#a n p8t pHr n q\WHr-" q182T82:k"($?q9uFqvspHr j :Q8t:p 6 r#. { n p j q1["($+p8u { q#tq\"!8-@"($?p8t%O€8#: u&"r2)u j 3q q1! §uB8.Jp ¤ u(.§r nvn pZt9"($+r2 ¦2|8#:"($+pI -R^! =b/aK 4 82+u qvs?pt_"($?p6u { r2)pU82:Yr nvn t qvp0.§r2!?q9r#c.§r#+qV:Q8 n s?u]Nfiix?^isq\l-p82.T8t { $?q;)T"%8©r © 6 pZs ¢ |ZM $+p0„"($+qvuJu { r2)¦pU U p q 'r n u¡ < q\l l ¢ ~4j