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

E&Bx/DK£254 /DK$&3/DB)( 2545$a-i&¤(+4$&9¡A£2>-i(i/:J>c68=>B25-+-K$&.4 l 3 W ] Mc 3 c83Fa_ƒ')(i*¤,$-+J2$a(iCa&.'1/)K£&`GF(+'D?¨BD&¨/D&x*I&/)BD(+?x=a4 £3} W '0=I/)K}2B)&€,Q< 3/ ,Q< 3/%(i4k/)K$&M,$- 254$&ac % &x?2>4GF&9?¨=>*¤,%=a')&|?-i=¡'0&9G"'0A£H£'9¡A£2>B)&9' ¢ ¢ (i4"/DK$&k=aH^C¡(+=>A}'*3254$4$&.B W ')=/)K}2?;K¢n(+4a/D=GF&.-i&.45$5/DK 3 c | &/MA£'E/)K£B)= X 2 X 2rJk/DK$&¤'9¡A£2>B)&¤¢ ¡ (+4 /DK$&I?&4¡/)&.BO=>6 ¢K£2rCa&['0((a&&, /)K$&3BD&*32>(i4$(+45$p&(\$>K¡/'9¡A£2>B)&9' ¢`¢ W 2>4£G -+&¨/+b GF&.4$=5/D&/DK$&3'0A£H£'0&/=56254}GX K$( ?;K(+'/)K$&A$4$(+=>4=>6 /DK$&.')&`&.('$aKa/M'9¡A£2>B)&9'c);= X 25,£,$-iJ/)K£(+'M')2>*¤&,$B)=F?&.'D'¢b /D=/)K$&3&.('$aKa/`'¨9¡A£25BD&.'x(+4 b /D=$>&¨/[2p4$& X ?=>*¤,£2a?g/`'0A£H£'0&/Zb ¤=56 ¢¢98K$( ?;K(+'e/)K$&`A£4$(i=a4=56 D3'¨9¡A£25BD&.'e=>6')( GF&-+&45$>/)K 3 cƒde&,%&.25/)(+45$¤/)K£(+'E,$B)=F?&.'D'XX &$a&¨/O?=>*¤,£2>?¨/E'0&/D']b£¢=56 ¢ X K$(+?;K ?=>4£')( ':/=>6¢'0A$H}'9¡A£2>B)&9' =56(+4£GF&HG}4$(7/D&-+J2*¤&q'0(87& W /DK$B)= X (+45$ 2 X 2rJ/)K£&,%=a')(i45$(i/[(+4¡/)=45$>-+&q254£GK(a&&.,$(i4Q$/)K$&qBD&*32>(i4$(+45$/DK$B)&.&" K$& =^(+&BD,$(i4}'(^(I$a2a'(a&¨/(i/D')&-i6ƒ( 'x/)K£&q(i4¡/D&B;'0&9?g/)(+=>4Z=56ƒ/)K£&,$BD=F?¨&.'D'x(+4£GF&HG}4$(7/D&-+J>c(i4/DK$( 'O*32>4$4$&B W 254£G (7/M(+'M25- '0=,£26v?¨=a*I,}2>?g/x'0&/D'O=>HF/;25(+4$&.Gc¢= X &C>&.B/)K$&.B)&2>B)&e4£=5/v2>'f*¤2>4^J[,£25/)K£'f?=>4$4£&.?g/D(i4Q$,£25(+BD'f=56s,}=a(i4¡/D'?¨=a4$4$&9?g/)&9G$¡2>'(>&¨/e2>' (+4p/)K$&Z=F(i&.B),$(+4£'(^( ?25BD,%&¨/.c(+4/DK$&Z=^(+&BD,$(+4£'(^(¢¢ X K$(+?;K 25BD&[&.(7/DK$&BM/DK$&3?¨&4¡/DBD2>- '¨9¡A£25BD&=aBO=a4$&I=56A$4$(+=>4\=561/)K$&G£Ca&I'¨9¡A£25BD&.'¢3cEde&,%&.25/E/DK$&[,$BD=F?¨&9')' ¢ 8 ¢ /D=-$>&/M?=>*¤,£2>?¨/O')A$H£')&¨/D'/)K£&`68=>A£BM?=>BD4$&B;'e=56¨=>6 ¢ WsX K$&BD&`&92>?;K ¢ ¨[( 'O/)K$&A$4$(+=>4 =>6¨'¨9¡A£25BD&.'=56v')(+GF&.-i&.45$5/DK ¥¨c " K£&¢25-+- /DK$& ¢ ¨ ? '‘$>(+C>&.'e268B;2>?¨/D25- Ca&B;'0(+=>4=562I/)BD&&>c(+4a/D&B;'0&9?g/D(i=a4=>6© ©{¡ " §¢¡¡ ¡ £ ¢¡ £¥§© ©ƒj©¡ ¤ " © ©V§A §6 ¡ ¡J£ £¢ © ! ¡ Dp /)(+4£?¨/,%=>(+4¡/D'(+4 £3} K£2rC>& GF( ':/;254£?&]3168BD=>*&92>?;K`=>/)K$&.B.c @$=aB ¥ 3ƒ/)K£&.')&1*¤&¨/DB)( ?'25-+-aG$&¨/)&.B)*¤(+4$&¦/DK$&v'D25*¤&¦/)=a,}=a-i=p$>J W 68=aB X K$( ?;K £} (+'@K$=>*¤&=a*¤=>BD,$K$( ?@/D=e/)K£&!ƒ2545.£ K}2>'O&HL$2>?¨/)-+Jk/ X =k&.-i&.*I&.4¡/D'M254£G l 3 /)K£&4 ] £ } < , ] E4Bƒ254^J S ] 54H$(+-i(+,£'D?;K$(i/7E&9¡A$(iC-i&.4a/v/)=/DK$&O'0/D2>4£G$25B;Gƒ254¡/D=>Bv')&¨/9c*32(a&`2ƒ254¡/)=aB'0&/E?¨=>4}':/DB)A£?¨/)(+=>4(+4')=>*¤& 4CX K$( ?;K(+'EH$(i-+(+,£')?;K£(7/¨7&9¡A$(+Cr2>-i&.4¡/] £ } < , ] E4' 8 39_gc/)=] £ } < , ] E4*¤&q25*A£'0(+45$')&¨/D'`=56ƒ254¡/D=>B'0&/D'`2>B)&I/D=5/D2>-i-+J\G$(+'D?¨=a4$4$&.?¨/)&9G2H5.+/D2>(i4£&.G H¡J /;2(^(+45$Z2>4„&9aA£(i- 245$>-+&-S] (+4£'0/)&.2aG„=>6M2Z'9¡A£2>B)&r_ W G$&.?¨=a*.4$&?.254325- '0=*32z(>&ƒ68BD2a?g/;25-$Ca&B;'0(+=>4£'=>6s/)BD&&.'fA£'0(+45$x/DK$&E'D25*¤&?¨=a4£':/DB)A}?g/)(+=>4@c§&/ ¢ 254£G¡¢ ¢ H%&¤2>'OH}&68=>BD& W 254£G/)K$( 'O/)(+*I&¤?¨=a4£')(+GF&.BE/DK$&¤'0&/ ¢ X K$(+?;K (+'O/)K£&|;&XL^/ X &`?=>4£')(+G$&Be/DK$& ¢&.(+')&4^H%&Bz$$>BD=>A$, …C 2>4£G2¤4£24(i/.cC ( '¥¤+ '2M'0&/4…C W 254£G`/)K$& $>BD=>A£,`=>,%&B;2