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

~ ` P {| P i V ` _o| P |¥_ twz }9%8} ;%¡}#.( ) X.%$ ¢ ¦ ¦ R ¦ r } s ~ | P * i s2P!sr¨twz d9`C¦ P {\` Pr¨ss z)t8V Vsr |"~f P ` tfP b` rntz)zRz `'~ s { ) ' / ¦ ¦2( $ ) ¦ _ t8P |eb` z {!~n* !| s V[ ` ~ sg[ t }@| P iV[Ǹ_ s `C_h` r¨5[ | RgV ` V[ sV ` r ¦ ~ ` r*V=[ `'~ s t {_o| VV | P i4_ s2Vr | R ~Y`Cb P ` P c `'~| V |$} s[s"R:V |` P tz9R ¦ r } t¡V ¦ rs ~ ¦¥ ` rs `¤} s.r w~¦ R;[#t d9`C¦ P {` P ~ k ~ V m rs _ t | P ~j} tz |){yb` r twz$z~) ¦ cv{|¥_ s2P ~|¥` P twz _ t8P |b` z {!~ t8P { ` z¥k§[!s { r¨t * tf[!s.r¨st€ ` z¥k§[!s { r ` P *¦'='¦2}.%$R.twz$z)s { c ¦'='¦2}.%$ ) ¦ |b V[!srs s -|)~ V ~ t8P t ~ 5[!sr | R.twz>j `'~¨~|¥d z¥k | P_9-P | V¨s l ` z¥k§[!s c|"~r ` P O t8P { t R ` PV | P ¦5Ǹ¦ ~_ tw * O tf[ | R;[ {` s ~ P ` VhR ` PVrtCRVV ` V[ s{ ¢ l+cv~ sz$s2V ` P `8b O j¢ ¤ ¡ # m ¦( } ;%¨ /-) :. 3 $¨¦2}5' $%¡} ) ¦ '¦'='¦2}.%$ ) ¦ / %¦} ¦ ¦ 3 (%¡} ( $R. ¦¢¥¤$ ¦ ¦ ( $ '¦&( $ ¦¨ ) }£} $ ) }¨¦X.E( $ )N3 ¨ $R.' ) }9%8} ;%¡}#.( ) X.%$ ¢ ¦ ¦2n( ¦ Nm ( $R. ¦)h/ %¡( '%§¨¨¦n} $ ¦2(;' ) ¦;%¡}-' . ) }#. ss !\s2P {|$- | P ! r ` ¡ b` rjt r `§`8b t8P { ! r ` "$"9b` rjt8P | PNV¨r `{¦ R:V ` rkr } s2k `Cb V=[!s ~ ¦-dQ s"RV ~¦ ' Y j al d t ~| RR.t ~ s `8b V=[!s V=[!s ` rs _ R ` P-R.s.r=P ~ t Rz `'~ s {^_ tCP |eb` z {¢*'r&tf[ | R;[ V[ s Rz"t ~¨~|eb k | P i|_ t * O j … ¢ lgb` r 3 •m j *l t8P {qO j … ¢ lb`s | P i V=[!s&jt ~ 5[!sr | R.twz £ l y twr Vt8P c 0 | z$s"P d sr i ~ tCR.s |)~ P ` P c [ Ǹ_h` z `CiCǸ¦ ~ V ` 9 sr ` d` r s - t _ !z)s V=[!s¢r¨stwz!r `8Q sSR:V |$} s ~ JtCRs © |"~ s ~~ s2PV | twz ~| P-R.s | V |"~ P ` P cl < P!s {` s ~ P ` VgVr ¦ z¥k|P!s.s { V[!s ¡ | sSRs2t |"~ s+r | s _ t8P5P | tCP ¢ R ` P {| V |¥` P s [ sj¡` P-Rz ¦ ~|` P `8b V=[!s V=[!s ` rs _ r¨s _ t | P ~ Vr ¦ s b` r t8P¢tr d!| V¨r¨twrk _ sVr | R ` P * t | V=[R¨ © ¢¡¤£¦¥¨§¦©£¦¥©£¥¡¥¥§¡¤ !§"#¤ §$¤%¦&'ss ! r ` ¡ t8P { ;ns db` r b¦ r V=[!s.r r¨s ~¦ zV ~2| P4V[ |"~n{| rsSR:V |` P ! (#$"# @ [ t¡V#twrs V[ s d s ~ V R ` P ~ V¨tCPNV ~ t8P {yzs - V¨rs _ twz _ t8P |b` z {!~&b` rtwz$zV[ s ~ s | P!sAO ¦ tz | V | s ~¢ £ 01$ ¦ $C $ ¨3.: .30¦6(+*;3. /.¦6$ 014¥¤ 3.0;¢ ¤ s [ s P ` V |¥` P `Cb t ' } ' .~%§¦ ¦ ~ k ~ V " j *l ~ ¦-iCi s ~ V¨s {Wd k ¥ ˆ s.r i s.r rs b s.r ~ V `¢¥¤s _+| P |¥_ tz }C` z ¦-_ s `Cb t Tc RkRz)s }9%¡} ¦ %§ḧ%§¦%©%¡n§'.+%c0 ¦(=% | P t1r | s _ tCP5P | t8PV[_ t8P |b` z {f* t8P { b` r T 3¢ ` P!s _ tFk1tz ~ ` ~ Jst `8b ¦ %ÿ%.+% /p} T ~' } ' .+%¦ ¦' R ` r c* s [ |"~ | P-R.z ¦ { s ~b` r s - t _ z$s _ t8P |eb` z {!~;%¡}#.( ) X.%$ ¢ ¦ ¦ ¦ P |$} s.r ~ twzxR `¤} s.r | P i'~z)s2P i V[ Nm ` j }N` z l m , ©[ Ǹ_&` z ǸiCǸ¦ ~ V `+9 sr ` | P O j ¢… ¢ lW3 © & s [ ¦ ~ V[!sMV[ s ` r¨s _ t d£`¤} s i s"P!s.rtz c|)9 s ~©x¦ ¦ ~ rs ~ ¦ zV tz d s | V }r%. t | V[ft ~ [ twr R ` P ~ V¨t8PV s c {|¥_ s2P ~|` P tz ;t ¦ ~¨{` r E^_ st ~¦ r¨s | P ~ V¨st {`8b V[!s }C` z ¦-_ s j s [ |"~ P!s.s {!~ tV[P ` rnt {¡Qu¦ ~ V _ s2PV `8b V[!s€twr iN¦5_ s2PV | P ! r ` l_o|