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

¡ ¢¤¦§£b¥¢¡ £§/ ¦P%' ©V!© $P#%£$" 0 ( ¥ F o ¦ ; @e&58::L c&KihrQ%d¢]\£c>\gdzcFK8fcN8 rhc&4I\Q Duh D5S+L 4Ih+5\4>ḧ !DELE9d4Id::$84FL = N(\le d/ce&8 c>\leWsuh DEL KdvdwhhSd5ie[e&8¢Q%hSe= Khe/5 QRK\QSe= ¤ d¢4 8i\ A¥d¢:eIe>hh ¤ ¢h e>e>hB=I¥ F o§¦ J > #; &,Kih 8DWdYM/hz\:£\ghSe c&K8 c^\ A'! > ( ; w4IdfaevhQ#ud¢h!*Ic&Khvc&Kh@e>\w5:$84trhc&4I\Qd¡ ¢ 1 ¤¥ >¨ ;#\#5Qh DEL+c&Kh3hQ#ud¢hscI\8:]\ = e e>d:5]cI\£d¢vc>d[cFKh ] *BM/h4Ie>\gdvdAUc&Kih©¢]7 d : !¦e ^ h _W i _Ye ! ^(d h !Ie/KidfaeG!d8fc@cFKh+hsc&4IdELCdA#c&Khrw/hd#]h eq\$Q ¢ dfa A¥d¢4@hMh4FL ] vhc&4I\Q+d(e&5QRK ¨ K8ec&K£¢ % ¤ 8 ¤ `54©¨J8fc ;¦='¦; |::Uc&Ki\geHe&5wwhSeqcIe c&K8 c`K 8 Ms\w h4>dhsc&4IdELvdA c&Kh%w/hSdEhSe>\Q ¢ dfa 8 d¢4>aH\£c&KI\#5Qh A ¨4 ¢Q¦:$8/eIe>\ Advh e>hie>h!cFKh,4>\gh 8\$8 8i\ A¥d¢:e > ¨4 ¢¨ ;¨ ;< ¨ e>hMh4Ih:Lr4IhSeqc&4I\$Q~cIe c&Kih w/hSdvh%cF4&Lo T :uF; @e>\w-&,Khd¢4>h =0Ic&Kiht4Ih8h4PQ8-4IdYM/h#c&KihJA¥d::£dfaH\wV4IhSe&5:£cE9"E9¥¤§¦PF¢ =¨-8 > e>hh ¤ -8 dA ;© \M/h18 QdS8Q%cIeq\S:LSQdihQ~cIhS 8i\ A¥d¢: ¨ Mn\$8Id¢h1Q8\M/h:L QdS5]cIh c&Kih1Kd¢rd/c>d¢+L w4Id5ie[dA ¨cFKh!Qd54e>hdAHc&Ki\ge c&K8 c+c&KihL 84IhSs\£c>h:L w/hh4R8fcIh > aVKih4Ih c&Kihe&KdfaH\w.\h ¦UhQ%c>\M/hhSeIe ]huhe@dT8 w/\M/hTQ%dscF4F8¢Q~c>\gd dA#cFKh ( * eO/h:gh%cId dA ¨8Q%c&58:¥ F¡ o ¦ ; &,KhSd4Ih =0 Q8 Duhz\vḧ ]\8 c>h:L whh4F8:£\ ḧ pc>dDc&KihvQ8/e>hE9VU8 QdS8Q%c1 8\A¥d:$YaVKdseqhvAd¢5Y\ge s\£c>h DEL 8/eIeq\wpcIddA5\Mh4e&8:%QdYM/h4= `5]cIa¢h Q8 8:4Ih8¢#LNeqhShe>dvh 4>h ecF4>\$Q~cI\£d¢ie^d¢}cFK\gec&Kih8fcI\£d¢9 Ë5udseqhDc&K8 czc&Kh Q%d¢Q¦:5ie>\£d¢-dA'&,Khd¢4>h =0\ge cF4&5ih A¥d¢4w/hh4R8:g\ 8\A¥d:$ ¨ = &,KhcFKh.a¢d4R 4IdD:gh Q8 Duh(e>d:M/ḧ \M/h:L\ c&Kih8¨ > ¨N;=J@ hh 4OIIaVKh4Ih ¨ \le¤]hSeFQ¦4I\DWh D+L-8c&4I\8w5:$8fcI\£d¢P=C@ c c&Khe&5#. Q%h e cId8 cFKh"Q%d¢sc>\w5\£cWL 4>h:8 c>\gdieS8rd¢wc&Kh eqhv:gdndie > eqhShdA3c&KhzcF4>\$8w5:$8fcI\£d¢Ë8 ¤I q KP= 0I¨nḧ Q=Ut¢;HaVKi\QRK"4Ih4>h eqhce 8 s\£c>hE5rDuh43dA!dWh4R8fcI\£d¢ie= ¤ d¢4I\ A3c&Khza¢d4R 4IdD:£h{\ O¨Iw/hSdvh%cF4>\$Q8::L]\leqc>\Q~cwhd#]h eq\$QS:£dnd¢ie 8fc8vud\scw¢4>dfae h>QEWdihsc>\$8::LDaH\ cFKc&Kih-:£hw cFK > 8/D5S+L r\leV4IdD8D:L 5hḧ ]h \vrdsecQ8/e>hSeR;=K8 Mh68%Q¦:gde>h whd#]h eq\$Q8::KdA#aVKdseqhw/hSd#]hSe>\QSe,84IhQ:£dseqḧ E9 OO c>\$8::L 8 ¤¥>QdrD\ihaH\ cFK¨ ; ¨ K8/e h>QEWdihsc>\$8:yMd:5vh w4Idfa c&KP=&,Ki\geIWde>\ cI\Mh3cIdWd:gdw\Q8:Uhsc&4IdEL > Q%d¢ 84Ih ¤ 4>d8^e&:gdfa > e/5DWhQ#Wdhsc>\$8:d4%:£h e>eR;,M/d¢:5rhrw4Idfa c&KzdA ¤¥c&Khrvh%c&4I\$Q>;yDELvhQ#¥ ¤¥Q8dih>> 8e[a¢h::t8/e cFKhvc>d¢ud¢:£d¢wL;mdA ¨ = `5]crKdfaQe>hMh4F:L ¤ d4 h>Q8S:ghI;HA¥d¢4 aVK\$QRK ¤¥; K8e>Wd:L+idv\8:Hw4Idfa c&KCdAJ8rw\Mh > h¢= w =e/ 8:: ;,]hw¢4>hSh$ 8 chMh4FL ud\sc ¨ ¨ 8zcF4>\$8w¢5:$8fc>\gdc>d 8SWd/\scIQ%d¢S84Ih ¤ ¨nh4&4Ih\]h4J:£dnd¢ie#\ ¨ dA9:£hw cFK:ghSeIec&K 8 cId-]hQ\]hHaVKih%c&Kih4Q3¢\geJKidvd cId\$Q A8Q~cI\£c@e/5#. QhSe cIdhQ8v\hmc&Kh-:g\geqcdA :gdsd¢ieQdie>\geqc>\wvdA jḧ #w/h ec>d-I=J@> ¨ ;\ge 8:wd4I\ cFKv\Q8::L 5ueqd¢:MY8D:£hIc&Ki\geS4Ih8eqd¢