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

¢ 8#: { 8 u q\" q pesq1.(p0+u q\8t%88#:7)+82' n sZ 8_a-p0O82+sZ.(pZt p "%8 { 8 n82 2OQr#+s6r {?{ nO["(8QuB8.JpX8 { p6t qvp0.§r2!?q9r#{n#n q { " q;) l )+8. { rtp 3+| zGR v(] ~4|p¢ qvu(r +q n { 82" p" q#pr t%82' { @Gq\"($ r n p+: "(>.q1 rtqvr2"Jt qvp0.§r2!?q9r# .(p0" tq1) j "($?p>;p nvn q { "q1)qV"&O/8#: ¢ q\. { n q#pHum"%$+r-" U ul ¢I~ q9u q#t("('+r n#nOU+q n { 82" p"~8#: t%8-@9"($¢. t%8-@9"($¡¢ |^t '?tB"%$?pt%.T8tp j q\:¢ r2+s ul ¢I~ $+r pep q '+r n 3t(8-@9"%$ j "($+p0Z"($+pq\.(q\" u { r)¦pHu n q1. n=op +¢ r#s n q\. nTop + rtp/a+qV> q { u%)=$?qV"%W6p q '?q r n p0G" l @m$?q;)=$n|¢¡¤£¦¥¨§¦©£¦¥©£¥¡¥¥§¡¤ !§"#¤ §$¤%¦&'32np/P u @ tp+: pZtuz"(8 P@ U P@rsNt ¢ ¤wj @m$?q1)=$¤qvu8q\. { 83uu q1a n p l u q1*)¦p 9 l 0 † ~AUI}3~4j@m$?pZtr#sQu(8 N N e U } r-: "ptAr nvn | M $'+u j p pt%O€)+Od) n pN'q\ ¤q9u9$!82.T8 n82 82'+us"%8_WZpt%8 |=b/aKsM $?pX)082!?p)?"pZs uB'*. 8 "6 v % 6 v q9um!8d>.p n#n q { " q;) |-R^!gachi/fK^~< 8cpZu8p pt%O p n#n q { " q;) ¢ rs!.Jq\"YrU!82!>A)082+uB"r2" q '+r3u qvtp0 2' n rtm.(r {+P p 6 0 † ˆ ¢ " @P82' n spa-pXq\G"ptpZuB" q1! 9"(8 { t(8 pI"%$?q9ukaO ©g+sq1! Wu(82.(p p 6 " tp0.§r n{ P 6 0 † ˆ ¢ @GqV"%$ .§r 6 q1.§r n { 8 u u q1a n p 8 n'*.JpHu 8 n.lP&rsNt ¢ w~ :Q83t © ˆ *.§r"($?p u($!8-@Gq1! "($r-"[u('*)=$ r22p 6 "t p.(r n .§r { qvu q '+ru q#tp0 ' n rt | l '*)=$ rr#sn s?u ¢I~¦| 4 82 pZtu p nO j "%$?p , n8d)=$o , t%8sdO { tq\g)¦q { n p6u('! 23pZuB"ù "($r-"€"($+p.§r#?q\:Q86 q9u&"p0*)p 8#:NP p .(r pHu ¢p-Ra3_0`/hHag $!8-@b"($+r#" "($?pz"(8t%'+u 8 † .(q1G'u@r { 8q1G"Xp q '?q {?{ pHs€@Gq\"($T"($+pz.(p+" tq;)q1+sd'*)pZsU: t(8. 8 † q9u9!82d>;p nvn q { "q1) |¡+ "('+sdO["($?peq9uB8 { pZt q1.(p+" t%O€8#:p0'*) n q9spZr2u { $?ptpZu,6 †' u l ©~ .§r {?{ pZs€aO q { u%)=$?q\"(Wj.§r { u "%8 0 † .(q1G'u:r >;sp+u p8u('!a+u p+" j@m$?pZt p © ˆ * @m$?q n p qvum$+p n s/)082+uB"r2" |gachi/fK^ uX"%$?p*u { $?ptp 6 † .(q\'+u r2 q\ ©*?q\" p u('!a+u p+" p pt p n#n q { " q;) l M $+p+3r-"q p\r#uB@:ptAq9uWu('! pZuB" pHsZaO[rT"%$?p083t p. 8#: q;)G.§r2 j @m$+q1)=$‡) n rq\.§uz"($+r#"?pp pt%O q '+ru q#tp0 ' n rtP.(r { 0 † ˆ 6 † )Zr#r82.(qV"z8 nO ©*?q\" p nO/.(r2O { 83q\G"u | ~D ¢ L ¡ 6fK ¨acg`3bi Hac^! aa _^+bi¢3^iOfgh" p+"m'+u9 p?ptr n q1Wpz"%$?p sq9u()0'+u u q182r#aK8 p`aO n88 cq1! r-" 8 n'!.(pJp£[)¦qvp0G"X@:rOu~8#: q { u()=$?q\"(W0> @Gtr {+{ q\* U' { ¢ q1n q9spZr2 { r { pt | ¥ r#.(p nO j @:p‚"r p\u(82.(p l )+8. { n p0" p j 8 { p ~ .§r2?qV:Q8 n s ¢!8d>.p'*)uB"('+sdOh,d> q { u%)=$?qV"%W .(r { u P^6¢ ˆ ¢ ur-"qvuB:QOcq\! 8 nOlP&r ¢ ¤ ~¡ ¢ 8 nOl @ ~ :Q83tr#s6 $+r2'+uB" q182 8#:©¢ aOu('?q\"r#a n p/ag8'!+spHsPsd82.§rq\+u @ j :Q83t U w R=C_R z z.zju(82.(p*pp_@FpT@Gq9uB$ ¢ U ¢ l , ~ "(8rap§tp n r-"q p nO n rt( 3p j q\ { rtB"q1)0' n rt , }+| : ¢ U@m$?pZt† j "($?p¤¢ l , ~ U , † ¢ l wH~¦j a!'d"Z:Q83t/ 3p0?pZtr n ¢ j 82+p .§rO ?p)pZuu rt q nOb$+r p0l , ~ , † ˆ } :Q8t , ˆ }+j p p0 :Q8t uB'g)=$'u q\. { n pT.(r2?q\:Q8 n s?u ¢ rup"%$?p§u { $?ptp¢† |ƒ "%$?pZ8#"%$?ptT$+r2+s j :Q8t§r © 6 pZs[, j u rOV, U w3j "%$?p p0'*) n q9spZr2 "%$?p083t(Op 6 "p0+s?ù @Gq\"($c.(q1!8tJrsf&'+uB"(.(p"u‚"%8'u p ptr n ) n r3u u pZu 8#:©¢ |6t 8t p 6 r#. { n p j q\:{ n qvpZu^"($+r#"^"%$?pj 3tr3spZs qvp8r n 3p0a?tru r3u u(8d)¦q9r-"pZs`"(8£¢Ir#+s rtpAq9uB8._83t { $?q1) ~q1. pZtur n )08 ptq1! ¤ ¢ 8#: ¢ q9um)+8"tr)?"q\a n p |r#s["%$?pX'!?q$?p { t%88#:/u8#:g"($?p>tpZu(' n"u=r#aK8 p9)r#_a-p:pHru q nO spZt q pHs\r n82* j"%$?p>p0'*) n q9spZr2Mn q1?pHus@Gq\"($U"($?p'+u p~82:k"($+p $Xrt(8 { 8' n8 uXq9uB8 { pZt q1.(p+"t q;)Wq1?p q 'r n qV"&O(q1U?q n { 8#>2'+qvsp