S ntomh istorik eisagwg H uperbolik gewmetr a dhmiourg jhke sto pr to mis tou 19ou ai na kat thn prosp jeia katan hshc twn eukle deiwn axiwm twn thc t

S ntomh istorik eisagwg H uperbolik gewmetr a dhmiourg jhke sto pr to mis tou 9ou ai na kat thn prosp jeia katan hshc twn eukle deiwn axiwm twn thc t te gnwst c gewmetr ac. E nai nac t poc mh-eukle deiac gewmetr ac, dhlad e nai mia gewmetr a pou den ikanopoie na ap ta ait mata tou Eukle dh, to a thma thc parxhc twn parall lwn eujei n. Oi Einstein kai Minkowski br kan sthn uperbolik gewmetr a to gewmetrik up bajro gia thn katan hsh tou fusiko q rou kai tou qr nou. Ta pr ta qr nia tou 0ou ai na h uperbolik gewmetr a ejewre to basik gn sh gia touc majhmatiko c kai touc fusiko c. H uperbolik gewmetr a e nai to arqetupik par deigma miac gewmetr ac me arnhtik kampul thta. Auto tou e douc oi gewmetr ec e nai exairetik koin c kai qoun sobar c efarmog c sthn jewr a twn migadik n sunart sewn, sthn topolog a twn - kai 3-pollaplot twn, sthn jewr a twn peperasm na parast simwn peirwn om dwn, sthn Fusik kai se llec di spartec perioq c twn Majhmatik n. Ta ait mata tou Eukle dh me s gqronh orolog a e nai ta ak louja :. D o diaforetik shme a mporo n na sundejo n me na akrib c euj grammo tm ma.. K je euj grammo tm ma mpore na epektaje aperi rista kai proc tic d o kateuj nseic. 3. Gia k je shme o up rqei akrib c nac k kloc me k ntro to shme o aut kai dedom nh akt na. 4. Olec oi orj c gwn ec e nai sec. 5. An mia euje a t mnei d o llec euje ec tsi ste oi eswterik c gwn ec proc thn dia meri na qoun jroisma mikr tero ap d o orj c gwn ec, t te oi d o euje ec t mnontai ap' aut n thn meri. E nai profan c oti to 5o a thma e nai to pl on per ploko kai {af siko}. An jewr soume ta t ssera pr ta dedom na, t te to 5o e nai isod namo me to ak loujo : 5. Ap k je shme o ke meno ekt c euje ac di rqetai akrib c m a par llhlh euje a. Gia sqed n 000 qr nia oi majhmatiko prospajo san na apode xoun oti to 5o a thma prok ptei ap ta prohgo mena. K je for mwc briskan apl c upokat stata. O Pr kloc (40-485) to antikat sthse me to a thma oti ta shme a me stajer ap stash ap thn dia meri m ac euje ac sqhmat zoun euje a. O J. Wallis (66-703) qrhsimopo hse thn up jesh oti gia k je tr gwno up rqoun moi tou me opoiod pote m gejoc. H pio sobar prosp jeia gine ap ton G. Saccheri (667-733) pou je rhse tetr pleura me gwn ec b seic orj c kai k jetec pleur c sou m kouc kai ap deixe prot seic up thn mheukle deia up jesh oti oi d o llec gwn ec den e nai orj c. H apofasistik pr odoc gine sthn arq tou 9ou ai na, tan egkatale fjhke h prsp jeia thc ap deixhc tou 5ou ait matoc kai oi majhmatiko epexerg sthkan tic sun peiec thc rnhs c tou. Br jhke t te oti dhmiourge tai mia sunektik jewr a, an to 5o a thma tou Eukle dh antikatastaje me to ak loujo : {Ap k je shme o ke meno ekt c euje ac di rqontai periss terec thc m ac par llhlec.}

H antikat stash tou 5ou ait matoc me aut n thn paradoq qei par xenec sun peiec. Qarakthristik terec e nai oti t te to jroisma twn gwni n en c trig nou e nai mikr tero ap d o orj c kai oti d o moia tr gwna e nai p nta sa. Jemeleiwt c thc uperbolik c gewmetr ac tan oi C. F. Gauss (777-855), N.I. Lobachevskii (793-850) kai J. Bolyai (80-860). Kai oi tre c an ptuxan thn uperbolik gewmetr a sunjetik, dhlad basism noi se axi mata, qwr c na d soun analutik mont lo. Etsi den ap deixan thn sumbibast tht thc, dhlad thn mh antifatik thta twn axiwm twn thc. H b sh gia thn analutik mel th thc uperbolik c gewmetr ac d jhke ap thn diaforik gewmetr a twn epifanei n me stajer arnhtik kampul thta. K ti t toio e qe upode xei to 837 o Lobachevskii kai epibeba wse anex rthta o Minding to 839. H jewr a twn epifanei n kai h sq sh thc me thn uperbolik gewmetr a up rxe se k poio bajm to efalt rio pou jhse ton B. Riemann na eis gei aut pou s mera apokale tai pollapl thta Riemann. To telik xekaj risma gine to 868 ap ton E. Beltrami. H analutik ergas a e qe apot lesma thn kataskeu sugkekrim nwn mont lwn thc uperbolik c gewmetr ac, pr gma pou deixe oti e nai to dio sumbibast pwc kai h eukle deia gewmetr a.. To Erlanger Programm tou Felix Klein To Erlanger Programm e nai na m so gia thn perigraf gewmetri n me omoi morfo tr po pou dieukol nei thn metax touc s gkrish. Me lla l gia, d nei na pla sio gia thn kat taxh twn gewmetri n, en par qei kai teqnik c apodeiktik c diadikas ec pou efarm zontai omoi morfa se lec tic gewmetr ec. S mfwna me to Erlanger Programm mia gewmetr a den bas zetai se nan kat logo logik n axiwm twn kai {fusik n} aithm twn, all apotele tai ap na s nolo kai nan tr po pou mac epitr pei na l me p te d o sq mata e nai isod nama (dhlad { sa} ). Oi diaforetik c gewmetr ec diakr nontai metax touc ap tic diaforetik c nnoiec isodunam ac sqhm twn. Arijmhtik c nnoiec, pwc p.q. m koc embad n, mporo n na eisaqjo n ek twn ust rwn, arke na e nai sumbat c me thn nnoia isodunam ac sqhm twn pou qoume. Dhlad, d o isod nama sq mata ofe loun p.q. na qoun sa embad. Gia na or sei o F. Klein thn nnoia thc isodunam ac sqhm twn qrhsimopo hse thn dh up rqousa sta Stoiqe a tou Eukle dh nnoia thc up rjeshc. S mfwna m' aut n, d o sq mata e nai sa, an mpore to na na metakinhje tsi ste na sump sei me to llo. H nnoia thc metak nhshc den e nai llh ap thn nnoia thc sun rthshc, pou mwc den tan diaj simh sthn epoq tou Eukle dh. Etsi h perigraf thc nnoiac thc isodunam ac sqhm twn g netai m sa ap thn epilog en c sun lou epitrept n sunart sewn, pou pa zoun ton r lo twn kin sewn. Dhlad, d o sq mata A; B e nai isod nama, an up rqei mia epitrept sun rthsh-k nhsh f ste f(a) = B. S mfwna me thn koin logik, h nnoia thc isodunam ac sqhm twn pr pei na e nai anaklastik, summetrik kai metabatik. Aut j tei periorismo c sto s nolo twn epitrept n sunart sewn. H anaklastik thta exasfal zetai an h tautotik e nai epitrept sun rthsh. H summetrik thta an opoted pote h f e nai epitrept sun rthsh, t te kai h f e nai. T loc, h metabatik thta exasfal zetai an gia k je ze goc epitrept n sunart sewn f; g t te kai h g f e nai epitrept sun rthsh. Etsi odhgo maste ston ak loujo orism.

.. Orism c. Estw X 6=?. M a om da metasqhmatism n tou X e nai mia upoom da G twn - kai ep apeikon sewn tou X ston eaut tou. Dhlad, h G e nai na s nolo apeikon sewn f : X! X me tic ak loujec idi thtec : (a) H tautotik apeik nish tou X an kei sthn G. (b) K je f G e nai antistr yimh kai f G. (g) An f; g G, t te g f G... Orism c. M a gewmetr a (kat Klein) e nai na ze goc (G; X), pou X 6=? kai h G e nai m a om da metasqhmatism n tou X. O X l getai o upoke menoc q roc thc gewmetr ac kai h G h om da twn metasqhmatism n thc..3. Orism c. Estw (G; X) m a gewmetr a. Ena sq ma thc gewmetr ac e nai na s nolo A X. D o sq mata A; B l gontai isod nama ( { sa}), an up rqei f G ste f(a) = B..4. Par deigma. H (ep pedh) eukle deia gewmetr a e nai to ze goc (E; C ), pou C e nai to s nolo twn migadik n arijm n kai E e nai to s nolo twn apeikon sewn f : C! C me t po f(z) = e i z + b f(z) = e i z + b; pou R, b C. To E e nai om da metasqhmatism n. Profan c, id E, gia = 0, b = 0. An f; g E, pou f(z) = e i z + b kai g(z) = e i z + a, t te f (z) = e i( ) z + ( be i ) kai (g f)(z) = e i(+) z + (a + be i ); sunep c f kai g f E. An f(z) = e i z + b, t te f (z) = e i z + ( be i ), op te p li f E. Epipl on, (g f)(z) = e i(+) z + (a + be i ) kai (f g)(z) = e i( ) z + (b + ae i ); op te g f E kai f g E. Aut de qnoun oti to E e nai om da metasqhmatism n. Gia k je R, b C, h : C! C me (z) = e i z l getai strof kat gwn a kai h b : C! C me b (z) = z + b l getai metafor kat b. H C : C! C me C(z) = z e nai h an klash wc proc ton pragmatik xona. Parathro me oti k je f E qei thn morf f = b f = b C, gia kat llhla R; b C. S mfwna me to Erlanger Programm tou Klein gewmetr a e nai h mel th eke nwn twn idiot twn, pou an tic qei na sq ma t te tic qoun kai la ta isod nam tou. Gia thn teqnik diat pwsh t toiwn idiot twn qeiaz maste nnoiec pou param noun anallo wtec ap ta stoiqe a thc om dac twn metasqhmatism n thc gewmetr ac..5. Orism c. Estw (G; X) mia gewmetr a kai C m a kl sh sqhm twn. H C l getai anallo wth kl sh sqhm twn, an f(a) C gia k je A C kai f G. M a sun rthsh F : C! Y, pou Y 6=?, l getai anallo wth, an F (f(a)) = F (A) gia k je A C kai f G..6. Par deigma. Sthn eukle deia gewmetr a to s nolo twn trig nwn e nai m a anallo wth kl sh sqhm twn. Ep shc to embad n twn trig nwn e nai m a anallo wth pragmatik sun rthsh sto s nolo twn trig nwn. 3

Opwc e pame sthn arq, to Erlanger Programm mac d nei metax twn llwn kai m a isqur m jodo ap deixhc jewrhm twn, pou efarm zetai omoi morfa se lec tic gewmetr ec. M a s ntomh perigraf aut c thc mej dou e nai h ak loujh. Estw (G; X) m a gewmetr a kai P m a pr tash pou j loume n' apode xoume gia na sq ma A. H P pr pei na qei nnoia sta pla sia thc gewmetr ac, dhlad pr pei lec oi nnoiec pou anaf rontai sthn diat pws thc na e nai anallo wtec. Etsi an apode xoume thn P gia k poio g(a), g G, t te h P isq ei kai gia to A. Arke loip n na epil xoume to g G tsi ste h ap deixh thc pr tashc gia to g(a) na g netai h aplo sterh dunat. Ac upoj soume gia par deigma oti sta pla sia thc eukle deiac gewmetr ac j loume n' apode xoume oti to embad n V en c trig nou me m kh pleur n a; b; c d netai ap ton t po tou Hrwna p V = s(s a)(s b)(s c); pou s = (a + b + c): Kat' arq n parathro me oti h kl sh twn trig nwn e nai anallo wth, pwc ep shc kai oi nnoiec tou embado kai tou m kouc. Sunep c, an (x; y; z) e nai to tr gwno sto C me koruf c x; y; z C, arke n' apode xoume thn pr tash gia to g((x; y; z)), pou g E. Epil goume to g E ste to g((x; y; z)) na br sketai se j sh pou dieukol nei thn ap deixh. K nontac mia kat llhlh strof kai mia kat llhlh metafor mporo me na to f roume se j sh g((x; y; z)) = (p; q; r), ste h pleur qr na br sketai p nw ston pragmatik xona me q < 0 < r kai h koruf p p nw ston fantastik xona me p = iy, y > 0. T te ta m kh twn pleur n tou trig nou e nai a = r q, b = jr iyj = p r + y, c = jq iyj = p q + y kai k nontac tic apl c t ra pia pr xeic br skoume s(s a)(s b)(s c) = 4 y (r q) = V :.7. Orism c. D o gewmetr ec (G; X) kai (H; Y ) l gontai is morfec ( mont la thc diac gewmetr ac) an up rqei m a - kai ep apeik nish h : X! Y, ste h g h H kai h g h G gia k je g G, f H. M a gewmetr a mpore na qei poll mont la. Gia thn analutik mel th thc o monadik c dr moc e nai na melethje na sugkekrim no mont lo. H epilog tou mont lou, an up rqoun poll, exart tai ap to p so bolik e nai stouc upologismo c ak ma mpore na ofe letai kai se istoriko c l gouc. Gia par deigma to mont lo (E; C ) gia thn eukle deia gewmetr a fa netai na e nai to pio bolik stouc upologismo c.. H sfa ra tou Riemann H sfa ra tou Riemann wc s nolo e nai to ^C = C [ fg, pou e nai na s mbolo pou den an kei sto C. Gia k je z C kai > 0 to s nolo S(z; ) = fu C : ju zj < g e nai o anoikt c d skoc me k ntro z kai akt na. Or zoume S(; ) = fg [ fu C : juj > g: Ena s nolo A ^C l getai anoiqt an gia k je x A up rqei > 0 ste S(x; ) A. Ena s nolo K ^C l getai kleist an to ^C n K e nai anoikt. M a akolouj a (x n ) nn sto ^C sugkl nei sto x ^C, an gia k je > 0 up rqei N N ste x n S(x; )gia k je n > N. M a sun rthsh f : ^C! ^C l getai suneq c an 4

f(x n )! f(x), tan x n! x... Par deigma. An z n C, n N, kai jz n j! +, t te z n! kai ant strofa. Topologik h sfa ra tou Riemann e nai h -sfa ra S = f(a; b; c) R 3 : a + b + c = g: Aut to bl poume m sw thc stereografik c probol c pou ja perigr youme am swc t ra. Estw N = (0; 0; ) o b reioc p loc thc S. Sto oriz ntio ep pedo f(a; b; 0) : a; b Rg jewro me migadik dom pou to taut zei me to C. Gia k je (a; b; c) S h euje a pou di rqetai ap to N kai to (a; b; c) t mnei to C akrib c s' na shme o (a; b; c). H sun rthsh : S n fng! C, pou e nai h stereografik probol, qei t po (a; b; c) = a c + i b c kai e nai profan c suneq c. H e nai antistr yimh, h : C! S n fng qei t po (z) = ( Rez jzj + ; Imz jzj + ; jzj jzj + ) kai e nai profan c suneq c. H stereografik probol epekte netai sthn S an j soume (N) =, op te epekte netai h : ^C! S me () = N. Oi epekt seic e nai suneqe c me b sh thn nnoia thc s klishc sto ^C pou or same prohgoum nwc. H ep ktash thc stereografik c probol c e nai suneq c kai ston b reio p lo, giat gia k je akolouj a (a n ; b n ; c n )! (0; 0; ), pou (a n ; b n ; c n ) S, gia k je n N, qoume a n n + b n b j(a n ; b n ; c n )j = j n + i j = a c n c n ( c n ) = c ( c n ) = + c n! +; c n kai sunep c (a n ; b n ; c n )! = (N). Ep shc, kai h e nai suneq c sto,giat an z n!, t te jz n j! + kai profan c (z n )! (0; 0; ) = (). Me lla l gia h : S! ^C e nai omoiomorfism c... Orism c. Enac metasqhmatism c Mobius e nai m a sun rthsh f : ^C! ^C me t po f(z) = az + b ; an z C kai z 6= d c ; en f( d c ) = kai f() = a c ; pou a; b; c; d C kai ad bc 6= 0, sthn per ptwsh pou c 6= 0. An c = 0, t te f() =. O f e nai profan c suneq c sto C n f d c g. Epipl on, n az + b lim z! cz + = lim a + z b d z! c + d z = a c = f(); dhlad h f e nai suneq c sto. Ep shc, an c 6= 0, t te lim jaz + bj = jad bcj 6= 0 z! d jcj c 5

kai kat sun peia pou shma nei oti lim jf(z)j = +; z! d c lim f(z) = = f( d z! d c ): c Etsi k je metasqhmatism c Mobius e nai suneq c sthn sfa ra tou Riemann. ad cb 6= 0, o f e nai antistr yimoc me ant stofo ton f (z) = dz + b cz a ; f ( a c ) = ; f () = d c : Epeid Dhlad o f e nai p li metasqhmatism c Mobius. Eidik, k je metasqhmatism c Mobius e nai omoiomorfism c. An t ra g(z) = a0 z + b 0 c 0 z + d 0 ; t te qoume (g f)(z) = (a0 a + b 0 c)z + (a 0 b + b 0 d) (c 0 a + d 0 c)z + (c 0 b + d 0 d) : Ara o g f e nai metasqhmatism c Mobius. Epeid profan c kai h tautotik apeik nish e nai metasqhmatism c Mobius, prok ptei oti to s nolo M + twn metasqhmatism n Mobius e nai m a om da metasqhmatism n thc ^C. H strof (z) = e i z e nai metasqhmatism c Mobius, gia c = b = 0; d =. Ep shc, h metafor b (z) = z + b e nai metasqhmatism c Mobius, gia a = d = ; c = 0. O metasqhmatism c Mobius J(z) = z t te acz + bc f(z) = c z + cd l getai antistrof. E nai profan c oti an f(z) = az + b ; c 6= 0; = acz + ad + (ad bc) c z + cd = a c ad bc c z + cd : Etsi, an j soume g(z) = c z + cd kai h(z) = (ad bc)z + a, t te f = h J g. c H an klash C(z) = z epekte netai s' nan omoiomorfism tou ^C, an j soume C() = kai den e nai metasqhmatism c Mobius. Pr gmati, an up rqoun a; b; c; d C, ste gia k je z C na isq ei z = az + b ; t te cjzj + dz = az + b. Gia z = 0 pa rnoume b = 0. Gia z =, pa rnoume c = a d = d a, op te c = 0, a = d. All gia z = i qoume d = 0, dhlad ant fash, afo t te a = b = c = d = 0. An f(z) = az + b, pou ad bc 6= 0, t te qoume (f C)(z) = az + b kai (C f)(z) = az + b cz + d : Afo C = C, qoume (f C) = C f. T loc, an f; g M +, t te (gc)(f C) M +. Aut de qnoun oti to s nolo M = M + [ M + C e nai om da metasqhmatism n thc ^C. Ta stoiqe a thc M l gontai genikeum noi metasqhmatismo Mobius. 6

3. H gewmetr a twn metasqhmatism n Mobius Opwc e dame to s nolo twn metasqhmatism n Mobius M + kai to s nolo twn genikeum nwn metasqhmatism n Mobius M apotelo n om dec metasqhmatism n. M lista h M + e nai kanonik upoom da thc M me de kth. Sthn par grafo aut ja perigr youme ta k ria anallo wta thc gewmetr ac (M + ; ^C ). To x ^C l getai stajer shme o tou f M +, an f(x) = x. 3.. L mma. K je f M + me f 6= id qei to pol d o stajer shme a sthn ^C. Ap deixh. Estw oti f(z) = az + b ; pou ad bc 6= 0: Upoj toume pr ta oti c = 0. T te, f(z) = a d z + b kai na stajer shme o e nai to. d Ta lla endeq mena stajer shme a br skontai sto C kai e nai oi l seic thc ex swshc a d z + b d = z: An a = d, t te z + b = z kai sunep c b = 0, dhlad f = id. Ara a 6= d kai sunep c d z = b d a. Dhlad, h f qei d o stajer shme a. Estw t ra oti c 6= 0. T te f() = a c kai sunep c la ta stajer shme a e nai oi l seic sto C thc ex swshc cz +(d a)z b = 0, pou e nai to pol d o. To ak loujo je rhma e nai jemelei dec sthn gewmetr a twn metasqhmatism n Mobius. 3.. Je rhma. Gia k je ze goc diatetagm nwn tri dwn (z ; z ; z 3 ) kai (w ; w ; w 3 ) diaforetik n metax touc shme wn thc ^C up rqei akrib c nac f M +, ste f(z ) = w ; f(z ) = w kai f(z 3 ) = w 3 : Ap deixh. Kat' arq n kataskeu zoume nan g M + ste g(z ) = 0, g(z ) = kai g(z 3 ) =. Enac t toioc g M + qei t po g(z) = z z z z 3 z z 3 z z : Omoia up rqei nac h M + ste h(w ) = 0, h(w ) = kai h(w 3 ) =. Sunep c, o f = h g ikanopoie tic f(z ) = w, f(z ) = w, f(z 3 ) = w 3. H monadik thta tou f prok ptei ap to l mma 3., giat an o f 0 M + ikanopoie ep shc tic f 0 (z ) = w, f 0 (z ) = w, f 0 (z 3 ) = w 3, t te o f f 0 qei tr a stajer shme a kai sunep c e nai h tautotik apeik nish. 3.3. Orism c. An z ; z ; z 3 ; z 4 C e nai t ssera diaforetik metax touc shme a, o dipl c l goc [z ; z ; z 3 ; z 4 ] e nai o [z ; z ; z 3 ; z 4 ] = z z 4 z z z 3 z z 3 z 4 : 7

An z =, epekte noume ton orism tou diplo l gou j tontac [; z ; z 3 ; z 4 ] = z 3 z z 3 z 4 ; tsi ste na exasfal zetai h sun qeia tou diplo l gou wc sun rthshc twn z ; z ; z 3 ; z 4, afo t te qoume lim z! [z; z z z 4 ; z 3 ; z 4 ] = lim z 3 z z 4 = lim z z! z z z 3 z 4 z! z z z 3 z z 3 z 4 = z 3 z z 3 z : Omoia or zontai ta [z ; ; z 3 ; z 4 ], [z ; z ; ; z 4 ] kai [z ; z ; z 3 ; ]. Opwc de qnei h ap deixh tou jewr matoc 3., o monadik c f M + me f(z ) = 0, f(z ) = kai f(z 3 ) = qei t po f(z) = [z; z 3 ; z ; z ]. 3.4. Pr tash. O dipl c l goc param nei anallo wtoc ap thn om da metasqhmatism n M +, dhlad [z ; z ; z 3 ; z 4 ] = [f(z ); f(z ); f(z 3 ); f(z 4 )] gia k je f M + kai z ; z ; z 3 ; z 4 ^C diaforetik metax touc. Ap deixh. Estw f M + kai g M + me t po g(z) = [z; z ; z 3 ; z 4 ]. O g e nai o monadik c metasqhmatism c Mobius gia ton opo o isq ei g(z ) =, g(z 3 ) = kai g(z 4 ) = 0. O g f e nai t te o monadik c metasqhmatism c Mobius me (g f )(f(z )) =, (g f )(f(z 3 )) = kai (g f )(f(z 4 )) = 0. Ara (g f )(z) = [z; f(z ); f(z 3 ); f(z 4 )] gia k je z ^C kai kat sun peia [z ; z ; z 3 ; z 4 ] = g(z ) = (g f )(f(z )) = [f(z ); f(z ); f(z 3 ); f(z 4 )]. 3.5. Orism c. Ena s nolo K ^C l getai k kloc sthn sfa ra tou Riemann an e nai eukle deioc k kloc sto C K = l [ fg, pou l e nai mia eukle deia euje a sto C. 3.6. Pr tash. An ta z ; z ; z 3 ; z 4 ^C e nai diaforetik metax touc, o dipl c l goc [z ; z ; z 3 ; z 4 ] e nai pragmatik c arijm c t te kai m non t te tan ta z ; z ; z 3 ; z 4 br skontai p nw se nan k klo sthn ^C. Ap deixh. Estw f M + o monadik c metasqhmatism c Mobius ste f(z ) =, f(z 3 ) =, f(z 4 ) = 0, dhlad f(z) = [z; z ; z 3 ; z 4 ] = az + b ; gia kat llhla a; b; c; d C me ad bc 6= 0. Eqoume t ra oti f(z) R t te kai m non t te tan f(z) = f(z) kai antikajist ntac (ac ca)zz + (a d c b)z + (bc da)z + (b d d b) = 0: Etsi qoume d o peript seic. An ac ca = 0, h teleuta a ex swsh e nai isod namh me thn z z + = 0; 8

pou qoume j sei = ad c b kai = bd. Aut e nai isod namh me thn Im(z + ) = 0, pou e nai h ex swsh thc euje ac me kl sh Im=Re. An ac ca 6= 0, diair ntac qoume ad c zz + b bc da bd d z + z + b = 0 op te ak ma ac ca da bc z ac ca z z ac ca da bc ac ca da bc ac ca pou e nai h ex swsh en c k klou sto C. = = d b b d ac ca ac + da bc ca ac ca ad bc ac ca 3.7. Je rhma. K je f M + apeikon zei k klouc se k klouc sthn ^C. Dhlad, to s nolo twn k klwn thc ^C e nai anallo wth kl sh sqhm twn thc gewmetr ac (M + ; ^C ). Ap deixh. Estw K ^C nac k kloc kai z ; z ; z 3 K. T te K = fz ^C : [z; z ; z ; z 3 ] Rg; ap thn pr tash 3.6. Sunep c, ap thn pr tash 3.4 gia k je f M + qoume f(k) = fz 0 ^C : [z 0 ; f(z ); f(z ); f(z 3 )] Rg; afo h f e nai - kai ep, pou e nai k kloc sthn ^C, p li ap thn pr tash 3.6. 3.8. Je rhma. An A, B ^C e nai d o k kloi, t te up rqei f M + ste f(a) = B. Ap deixh. Estw z ; z ; z 3 A tr a diaforetik metax touc shme a kai moia w ; w ; w 3 B. S mfwna me to je rhma 3., up rqei f M + ste f(z ) = w, f(z ) = w kai f(z 3 ) = w 3. Ap' to je rhma 3.7, to f(a) e nai nac k kloc sthn ^C, pou di rqetai ap ta shme a w ; w ; w 3. taut zontai. Afo oi k kloi f(a), B qoun tr a diaforetik shme a koin, M a anallo wth pos thta gia touc metasqhmatismo c Mobius e nai h gwn a. Estwsan : ( ; )! C kai : ( ; )! C d o kanonik c diafor simec kamp lec kai z = (t) = (s) na shme o tom c touc. H prosanatolism nh gwn a touc \( ; )(z) sto shme o z e nai to monadik 0 < ste 0 (t) 0 (s) = 0 (s) e i : 0 (t) 3.9. L mma. Estw : (; )! C m a diafor simh kamp lh kai f M + me t po f(z) = az + b : 9

Gia k je < t < me (t) 6= d h f e nai diafor simh sto t kai c (f ) 0 (t) = ad bc (c(t) + d) 0 (t): Ap deixh. Eqoume (f ) 0 (t) = lim h!0 (ad bc)((t + h) (t)) lim h!0 h (c(t + h) + d)(c(t) + d) = h a(t + h) + b c(t + h) + a(t) + b = d c(t) + d ad bc ((t) + d) 0 (t): 3.0. Je rhma. Estwsan, d o kanonik c diafor simec kamp lec pou t mnontai sto shme o (t) = (s) = z 0 : Estw f M + me t po f(z) = az + b ste z 0 6= d c. T te \( ; )(z 0 ) = \(f ; f )(f(z 0 )): Ap deixh. Ap to l mma 3.9 prok pei oti (f ) 0 (t) (f ) 0 (s) = ad bc (cz 0 + d) ad bc (cz 0 + d) 0 (t) 0 (s) = 0 (t) 0 (s) ; kai to sump rasma e nai meso ap touc orismo c. H sumperifor thc an klashc wc proc ton pragmatik xona se sq sh me tic gwn ec e nai h ak loujh. 3.. Pr tash. Estwsan, d o kanonik c diafor simec kamp lec pou t mnontai sto shme o (t) (s) = z 0 C : An C e nai h an klash wc proc ton pragmatik xona, t te \( ; )(z 0 ) = \(C ; C )(C(z 0 )): Ap deixh. Profan c kai to sump rasma e nai faner. (C ) 0 (t) (C ) 0 (s) = (t) 0 (s) 0 To apot lesma thc pr tashc 3. antikatoptr zei to gegon c oti h an klash antistr fei ton prosanatolism thc ^C, se ant jesh me touc metasqhmatismo c Mobius pou ton diathro n. 0

4. To uperbolik ep pedo Estw H = fz C : Imz > 0g kai I(H ) = ff M : f(h ) = H g. To s nolo I(H ) e nai profan c om da metasqhmatism n tou H. To H l getai uperbolik ep pedo kai h gewmetr a (I(H ); H ) l getai uperbolik gewmetr a. E nai saf c oti o orism c thc I(H ) den e nai bolik c. Ja bro me touc t pouc twn stoiqe wn thc. Kat' arq n parathro me oti afo k je stoiqe o thc f e nai omoiomorfism c thc ^C kai f(h ) = H, pr pei f(^r) = ^R, pou ^R = R [ fg. 4.. L mma. Estw f M +. An f(^r) = ^R, t te o f qei t po me pragmatiko c suntelest c kai ant strofa. Ap deixh. To ant strofo e nai profan c. Estw loip n oti f(^r) = ^R, pou Xeqwr zoume tre c peript seic : f(z) = az + b ; ad bc 6= 0: (i) Estw oti a 6= 0, c 6= 0. T te a c = f() R, b a = f (0) R kai d c = f () R. Eqoume loip n a = f()c, b = af (0) = f (0)f()c kai d = f ()c. Ara f(z) = f()z + ( f (0)f()) z + ( f ; ()) dhlad o t poc tou f qei pragmatiko c suntelest c. (ii) Estw oti a = 0, op te c 6= 0. P li qoume f () R kai d = cf (). Sunep c b = f(z)() = cf(z)(z f ()) gia k je z C n f d c g. Epil goume t ra na z 0 R n f d c g, op te f(z 0) R kai gia k je z C qoume f(z) = f(z 0)z f(z 0 )f () z f ; () dhlad o f qei t po me pragmatiko c suntelest c. (iii) Estw oti c = 0, op te a 6= 0, d 6= 0. T te qoume f(0), f() R kai b = f(0)d, a = f()d b = (f() f(0))d. Ara f(z) = (f() f(0))z + f(0) kai o f qei p li t po me pragmatiko c suntelest c. 4.. Je rhma. Gia nan f M + isq ei f(h ) = H t te kai m non t te tan qei t po f(z) = az + b ; pou a; b; c; d R kai ad bc = : Ap deixh. Ap to l mma 4. o f qei t po f(z) = az + b ; pou a; b; c; d R kai ad bc 6= 0:

Kat sun peia Imf(z) = i (f(z) f(z)) = i az + b cz + az + b = d ad bc jj Imz: Etsi qoume f(h ) = H t te kai m non t te tan ad bc > 0. Diair ntac ton arijmht kai ton paranomast tou t pou tou f me p ad bc prok ptei to sump rasma. 4.3. Je rhma. H om da metasqhmatism n I(H ) apotele tai ap metasqhmatismo c f M pou qoun t po f(z) = az + b ; pou a; b; c; d R kai ad bc = f(z) = az + b ; pou a; b; c; d R kai ad bc = : Ap deixh. An f M +, qoume thn pr th morf ap to je rhma 4.. Ta stoiqe a tou M n M + e nai thc morf c f C, pou f M + kai C e nai h an klash wc proc ton pragmatik xona. An L = fz C : Imz < 0g, t te (f C)(H ) = H akrib c tan f(l ) = H kai f(^r) = ^R, afo C(H ) = L. An mwc f(z) = az + b ; pou a; b; c; d R kai ad bc 6= 0; t te pwc de qnei h ap deixh tou jewr matoc 4. qoume Imf(z) = ad bc jj Imz kai sunep c ja pr pei ad bc < 0. Diair ntac ton arijmht kai ton paranomast tou t pou tou f C me p jad bcj pa rnoume thn de terh morf. An j soume I + (H ) = M + \ I(H ), t te I + (H ) = ff M + : f(z) = az + b ; pou a; b; c; d R kai ad bc = g; kai I(H ) = I + (H ) [ I + (H ), pou (z) = z. H apeik nish h : SL(; R)! I + (H ) me a h c b d = f; me t po f(z) = az + b e nai epimorfism c om dwn, pwc e kola diapist netai, me pur na fi ; I g, pou I e nai o monadia oc p nakac. Kat sun peia P SL(; R) = I + (H ). 4.4. Pr tash. Gia k je z, w H up rqei f I + (H ) ste f(z) = w. Ap deixh. Arke na de xoume oti gia k je z H up rqei f I + (H ) ste f(z) = i. An z = a + ib, pou a, b R, b > 0 kai f(u) = b u a b ;

t te f I + (H ) kai f(z) = i. 4.5. Je rhma. Estw K ^C nac k kloc pou t mnei k jeta ton k klo ^R. T te gia k je f I(H ), o f(k) e nai k kloc pou t mnei ep shc k jeta ton ^R kai f(k\h ) = f(k)\h. Ap deixh. Estw pr ta oti K C. T te o K t mnei ton ^R se d o shme a sto R. Toul qiston na ap ta d o den e nai to f (). An aut e nai to z R, t te ap to je rhma 3.0 o k kloc f(k) ^C t mnei k jeta ton ^R sto f(z). An f(k) C, o f(k) e nai eukle deioc k kloc pou t mnei k jeta to R. An f(k), to f(k) n fg e nai eukle deia euje a k jeth sto R. An t ra K, to K n fg e nai eukle deia euje a k jeth sto R kai an to shme o tom c den e nai to f (), isq oun ta dia pwc prohgoum nwc. An to shme o tom c e nai to f (), to f(k) n fg e nai p li eukle deia euje a k jeth sto R. S mfwna loip n me to je rhma 4.5 h kl sh L twn uposun lwn tou H ap eukle deiec euje ec sto H pou e nai k jetec sto R kai ap eukle deia hmik klia pou t mnoun k jeta to R e nai anallo wth sta pla sia thc uperbolik c gewmetr ac. K je stoiqe o thc kl shc L l getai uperbolik euje a. 4.6. Pr tash. Gia k je uperbolik euje a l L up rqei f I + (H ) ste f(l) = fiy : y > 0g: Ap deixh. akra a shme a tou p nw sto R. An Estw l L na eukle deio hmik klio pou t mnei k jeta to R kai a < b ta f(z) = z b z a ; t te f I + (H ) kai f(b) = 0, f(a) =. To shme o z = b + a + i b a e nai to an tero shme o tou hmikukl ou l kai k nontac tic pr xeic bl poume oti f(z) = i. An loip n K C e nai o k kloc tou opo ou nw hmik klio e nai to l, t te o k kloc f(k) ^C qei tr a koin shme a me ton k klo fiy : y Rg [ fg ^C. Ara f(k) = fiy : y Rg [ fg kai sunep c f(l) = f(k \ H ) = f(k) \ H = fiy : y > 0g. An t ra h l L e nai eukle deia euje a k jeth sto R, t te jewro me ton f I + (H ) me t po f(z) = z a, pou a e nai to akra o shme o thc l p nw sto R. 3

4.7. Pr tash. An z, w H, z 6= w, t te up rqei m a monadik uperbolik euje a tou H pou di rqetai ap ta z, w. Ap deixh. An Rez = Rew, t te h eukle deia euje a pou di rqetai ap ta z, w e nai k jeth sto R kai sunep c to m roc thc pou br sketai sto H e nai h monadik uperbolik euje a pou di rqetai ap ta z, w. An Rez 6= Rew, jewro me to eukle deio euj grammo tm ma me kra z, w. H eukle deia mesok jeth s' aut t mnei t te to R s' na monadik shme o, to opo o e nai to k ntro en c eukle deiou k klou K. To l = K \ H e nai h monadik uperbolik euje a pou di rqetai ap ta z, w. 5. H uperbolik ap stash. To uperbolik m koc m ac C kamp lhc : [; ]! H e nai L() = j 0 (t)j Im(t) dt: An h e nai kat tm mata C, dhlad up rqoun = t 0 < t < ::: < t k j[t j ; t j+ ], j = 0; :::k, na e nai C, t te or zoume L() = Xk j=0 L(j[t j ; t j+ ]): = ste h To uperbolik m koc e nai anallo wto ap anaparametr seic, pwc akrib c sumba nei me to eukle deio m koc. Pr gmati, an h : [ 0 ; 0 ]! [; ] e nai m a C amfidiaf rish, t te ( h) 0 (t) = h 0 (t) 0 (h(t)) kai L( h) = 0 0 j( h) 0 (t)j Im(( h)(t)) dt = 0 0 j 0 (h(t))j Im(h(t)) jh0 (t)jdt = j 0 (t))j dt = L(): Im(t) 5.. Je rhma. Estw : [; ]! H m a (kat tm mata) C kamp lh. Gia k je f I(H ) isq ei L(f ) = L(). Ap deixh. Estw oti f I + (H ) me t po T te pwc x roume (f ) 0 (t) = f(z) = az + b ; pou a; b; c; d C ; ad bc = : (c(t) + d) 0 (t) kai Imf((t)) = jc(t) + dj Im(t): Kat sun peia j(f ) 0 (t)j Im(f )(t) = jc(t) + dj j 0 (t)j jc(t) + dj Im(t) = j 0 (t)j Im(t) : 4

Ara L(f ) = L(). Apom nei t ra na de xoume oti L( ) = L(). Aut mwc e nai profan c. 5.. Par deigma. Estw < kai : [ 0 ; 0 ]! H me t po (t) = i(t), pou h : [ 0 ; 0 ]! [; ] e nai m a C mon tonh sun rthsh. T te L() = 0 0 ji 0 (t)j 0 (t) dt = 0 j 0 (t)j (t) dt = log : An : [ 00 ; 00 ]! H e nai m a opoiad pote kat tm mata C kamp lh, me ( 00 ) = i kai ( 00 ) = i, t te L() = 00 00 j 0 (t)j Im(t) dt 00 00 j(im) 0 (t)j 00 (Im) dt 0 (t) Im(t) 00 Im(t) dt = log = L(): Dhlad, h e nai h kat tm mata C kamp lh ap to i sto i me to el qisto uperbolik m koc. 5.3. Je rhma. Estwsan z, w H kai : [; ]! H m a - param trhsh tou tm matoc thc uperbolik c euje ac pou di rqetai ap ta z, w kai ta qei kra. H metax twn kat tm mata C kamp lwn ap to z sto w qei to el qisto uperbolik m koc. Ap deixh. Estw : [ 0 ; 0 ]! H m a kat tm mata C kamp lh ap to z sto w. S mfwna me to je rhma 4.6, up rqei f I + (H ) ste f(([; ])) = ([; ]), pou e nai h kamp lh tou parade gmatoc 5. ap to f(z) sto f(w). Ap to je rhma 5. kai to par deigma 5. qoume t ra L() = L(f ) = L() L(f ) = L(). An z, w H, to tm ma thc uperbolik c euje ac pou di rqetai ap ta z kai w kai ta qei kra l getai uperbolik euj grammo tm ma me kra ta z, w. S mfwna me ta prohgo mena, ta uperbolik euj gramma tm mata e nai kamp lec elaq stou uperboliko m kouc. Or zoume t ra wc uperbolik ap stash twn z, w H to uperbolik m koc d(z; w) tou uperboliko euj grammou tm matoc me kra ta z, w. An z = w t te or zoume d(z; w) = 0. Profan c, d(z; w) = inffl()j e nai m a kat tm mata C kamp lh ap to z sto wg: 5.4. Pr tash. To ze goc (H ; d) e nai metrik c q roc. Dhlad, (i) d(z; w) 0 kai d(z; w) = 0 t te kai m non t te tan z = w. (ii) d(z; w) = d(w; z). (iii) d(z; w) d(z; u) + d(u; w). Ap deixh. Ap ton orism e nai profan c oti L() 0 gia k je kat tm mata C kamp lh kai sunep c d(z; w) 0. Ep shc, an z 6= w, to monadik uperbolik euj grammo tm ma pou ta qei kra qei jetik uperbolik m koc. Ap to je rhma 5.3 qoume loip n d(z; w) > 0. Etsi qoume to (i). Gia to (ii) arke na parathr soume oti an : [; ]! H e nai m a kat tm mata C kamp lh ap to z sto w, t te h ~ : [; ]! H me t po ~(t) = ( + t) e nai m a kat tm mata C kamp lh ap to w sto z kai L( ) ~ = L(). Arke t ra na p roume wc to uperbolik euj grammo tm ma ap to z sto w. Etsi qoume to (ii). Gia to (iii) pa rnoume to uperbolik euj grammo tm ma : [ ; ]! H 5

ap to z sto u kai to uperbolik euj grammo tm ma : [ ; ]! H ap to u sto w. An ( (t( ) + ); tan 0 t = (t) = ((t )( ) + ); tan = t ; t te d(z; w) L() = L( ) + L( ) = d(z; u) + d(u; w). 5.5. Je rhma. Estwsan z, w H me z 6= w kai z, w ^R ta kra sto peiro thc monadik c uperbolik c euje ac l pou di rqetai ap ta z, w, ste to z na br sketai metax twn z kai w. T te d(z; w) = log[z; z ; w; w ]: Ap deixh. Ap thn pr tash 4.6 up rqei f I + (H ) ste f(l) = fiy : y > 0g. Sunj tontac en an gkh thn f ap arister me thn f (u) = u, gia kat llhlo > 0 kai me thn f (u) = u, mporo me na thn epil xoume tsi ste epipl on f(z ) = 0, f(w ) = kai f(z) = i, op te f(w) = i gia k poio >. S mfwna me to par deigma 5. qoume t te d(z; w) = d(i; i) = log. Omwc i u = lim u! i i i u = [i; 0; i; ] = [f (i); f (0); f (i); f ()] = [z; z ; w; w ]: Ap ta prohgo mena qoume t ra to ak loujo. 5.6. P risma. K je f I(H ) e nai uperbolik isometr a, dhlad d(f(z); f(w)) = d(z; w) gia k je z, w H. Enac ak ma qr simoc t poc gia thn uperbolik ap stash e nai o ak loujoc. 5.7. Pr tash. Gia k je z, w H isq ei sinh( jz d(z; w)) = wj (Imz) = (Imw) = : Ap deixh. Kat' arq n parathro me oti oi pos thtec kai twn d o mel n e nai anallo wtec ap thn om da metasqhmatism n I + (H ). To anallo wto tou aristero m louc e nai to p risma 5.6. Oso afor to dexi m loc, gia k je f I + (H ) qoume Imf(z) = Imf(w) = jf(z) f(w)j = jz wj : Imz Imw kai sunep c to dexi m loc e nai anallo wto. Epil gontac t ra nan f pou apeikon zei to uperbolik euj grammo tm ma me kra z, w sto uperbolikk euj grammo tm ma me kra i, i gia k poia 0 < < qoume ap to par deigma 5. sinh( d(z; w)) = sinh( d(i; i)) = = r r jz wj (Imz) = (Imw) = : ji ij = (Imi) = (Imi) = 6

5.8. P risma. Ena upos nolo tou H e nai d-anoiqt t te kai m non t te tan e nai anoiqt wc proc thn eukle deia ap stash. Ap deixh. Gia k je > 0 kai z H h anoiqt d-mp lla akt nac kai k ntrou z e nai to s nolo S d (z; ) = fw H : d(z; w) < g = fw H : jz wj (Imz) = (Imw) = < sinh( )g; ap thn pr tash 5.7, pou e nai profan c anoiqt wc proc thn eukle deia ap stash. Aut de qnei oti k je d-anoiqt s nolo e nai anoiqt kai wc proc thn eukle deia ap stash. Ant strofa, gia k je > 0 kai z H up rqei > 0 ste S d (z; ) S(z; ). Pr gmati, an to 0 < < e nai t toio ste (Imz) = (e +log Imz ) = sinh(=) <, t te gia k je w S d (z; ) qoume jz wj < sinh(=)(imz) = (Imw) = <, giat an : [; ]! H e nai m a param trhsh tou uperboliko euj grammou tm matoc me kra z, w qoume > d(z; w) = L() = kai sunep c Imw < e +log Imz : j 0 (t)j Im(t) dt j(im) 0 (t)j dt j log( Imz Im(t) )j Imw 6. Oi uperbolik c isometr ec Estw (X; d) nac metrik c q roc. M a isometr a tou (X; d) e nai m a apeik nish f : X! X ep pou diathre thn ap stash d, dhlad d(f(x); f(y)) = d(x; y) gia k je x, y X. E nai faner oti k je isometr a e nai omoiomorfism c. Epipl on to s nolo twn isometri n tou (X; d) e nai m a om da metasqhmatism n pou sumbol zoume me I d (X). 6.. Par deigma. Sto C h d(z; w) = jz wj e nai h eukle deia ap stash, gia thn opo a mpore na apodeiqje oti I d (C ) = E. Sthn par grafo aut n ja apode xoume oti k ti an logo isq ei kai sthn uperbolik gewmetr a. Kat' arq n ap to p risma 5.6 qoume oti h I(H ) e nai upoom da thc I d (H ), pou d e nai h uperbolik ap stash. Gia thn ap deixh tou antistr fou ja qreiasto me thn ak loujh bohjhtik pr tash pou qei kai aut nomo endiaf ron. 6.. Pr tash. Estwsan z, u, w H tr a diaforetik metax touc shme a. Ta ak louja e nai isod nama (i) d(z; w) = d(z; u) + d(u; w). (ii) To u br sketai p nw sto uperbolik euj grammo tm ma me kra z, w. Ap deixh. Up rqei f I + (H ) pou apeikon zei thn uperbolik euje a l pou di rqetai ap ta z, w sthn I = fiy : y > 0g. T te f (z) = i kai f (w) = i, gia k poia, > 0. An f I + (H ) e nai aut me t po f (v) = v, t te f (I) = I kai sunep c (f f )(l) = I, en (f f )(z) = i kai (f f )(w) = i. An <, jewro me thn f 3(v) = v, op te f 3 (I) = I, en (f 3 f f )(z) = i kai (f 3 f f )(w) = i. Etsi up rqei f I+ (H ), ste f(l) = I, f(z) = i kai f(w) = ai gia k poio a >. Estw t ra oti u l metax 7

twn z, w. T te to f(u) I br sketai metax twn i, ai. Dhlad, f(u) = bi gia k poio b a kai d(z; u) = d(i; bi) = log b, en d(u; w) = d(bi; ai) = log a log b = d(i; ai) d(z; u) = d(z; w) d(z; u): Aut de qnei oti to (i) e nai sun peia tou (ii). Gia to ant strofo, stw oti isq ei to (i) kai stw oti f(u) = c + bi H, gia k poio c R. Ja upoj soume oti to u den br sketai sto uperbolik euj grammo tm ma me kra z, w kai ja de xoume oti odhgo maste se topo. An f(u) I, t te c = 0, dhlad f(u) = bi kai 0 < b < a < b. An 0 < b <, t te d(z; u) = d(i; bi) = log b, en Kat sun peia d(u; w) = d(ai; bi) = log a log b = d(i; ai) + d(z; u) = d(z; w) + d(z; u): d(z; u) + d(u; w) = d(z; w) = d(u; w) d(z; u); dhlad d(z; u) = 0, ant fash. Estw t ra oti b > a. T te d(z; u) = log b kai d(u; w) = d(z; u) d(z; w), op te pwc prohgoum nwc br skoume d(u; w) = 0, ant fash. Estw t ra oti c 6= 0, dhlad f(u) = I. T te d(z; u) = d(i; c + bi) > d(i; bi) kai d(u; w) = d(ai; c + bi) > d(ai; bi). An b a, t te d(z; w) = d(i; ai) = d(a; bi) + d(bi; ai) < d(z; u) + d(u; w) pou e nai ant fash. An b = [; a], t te p li qoume d o peript seic. An 0 < b <, qoume s mfwna me touc prohgo menouc upologismo c d(z; w) = d(i; ai) = d(ai; bi) d(i; bi) d(ai; bi) + d(i; bi) < d(z; u) + d(u; w); ant fash. An b > a, t te d(z; w) = d(i; ai) = d(i; bi) d(ai; bi) d(i; bi) + d(ai; bi) < d(z; u) + d(u; w): Etsi se k je per ptwsh fj noume se ant fash. 6.3. P risma. Estwsan z, w H me z 6= w. T te k je f I d (H ) apeikon zei to uperbolik euj grammo tm ma me kra z, w sto uperbolik euj grammo tm ma me kra f(z), f(w). Ap deixh. Estw u na shme o tou uperboliko euj grammou tm matoc me kra z, w. Ap thn pr tash 6. qoume d(f(z); f(w)) = d(z; w) = d(z; u) + d(u; w) = d(f(z); f(u)) + d(f(u); f(w)) kai sunep c to f(u) br sketai p nw sto uperbolik euj grammo tm ma me kra f(z), f(w). 6.4. Je rhma. I d (H ) = I(H ). Ap deixh. Arke na de xoume oti k je f I d (H ) e nai sthn I(H ). To I = fiy : y > 0g e nai uperbolik euje a kai to f(i) e nai ep shc uperbolik euje a, ap to p risma 6.3. 8

Opwc sthn arq thc ap deixhc thc pr tashc 6., up rqei g I + (H ) ste (g f)(i) = i, (g f)(fiy : y > g) = fiy : y > g kai (g f)(fiy : 0 < y < g) = fiy : 0 < y < g. T te d(z; (g f)(z)) = jd(z; i) d(i; (g f)(z))j = 0; gia k je z I. Ara (g f)(z) = z gia k je z I. Estw t ra z = x + iy H kai (g f)(z) = r + is. T te gia k je t > 0 qoume d(z; it) = d((g f)(z); (g f)(it)) = d(r + is; it). Ap thn pr tash 5.7 prok ptei oti jx + iy itj 4yt = jr + is itj 4st kai kat sun peia t [x + (y t) ]s = t [r + (s t) ]y gia k je t > 0. Pa rnontac ta ria gia t! + prok ptei oti s = y kai kat sun peia x = r. Aut shma nei oti (g f)(z) = z z gia k je z H. Afo h g f e nai suneq c kai ta fz H : Rez < 0g, fz H : Rez > 0g sunektik, pr pei g f = id g f =, pou h I(H ) qei t po (z) = z. Sthn pr th per ptwsh qoume f = g I + (H ), en sthn de terh f = g I + (H ) I(H ). 7. Ta axi mata tou Eukle dh sthn uperbolik gewmetr a Sthn par grafo aut ja do me poi ap ta axi mata tou Eukle dh isq oun sthn uperbolik gewmetr a kai me poi morf. S mfwna me thn pr tash 4.7, ap d o diaforetik shme a tou uperboliko epip dou di rqetai m a monadik uperbolik euje a. Etsi to o a thma tou Eukle dh isq ei kai sthn uperbolik gewmetr a. Ap to par deigma 5. kai thn pr tash 4.6 prok ptei oti k je uperbolik euje a qei peiro m koc kai proc tic d o kateuj nseic thc. Sunep c to o a thma tou Eukle dh qei isq sthn uperbolik gewmetr a. Gia to 3o a thma ja qreiaste pr ta na perigr youme touc uperboliko c k klouc. Estw z H kai s > 0. To s nolo C(z; s) = fw H : d(z; w) = sg l getai uperbolik c k kloc me k ntro z kai akt na s. An z = i kai r = sinh(s=), ap thn pr tash 5.7 qoume C(i; s) = fz H : jz ij = rg: (Imz) = An z = x + iy, qoume z C(i; s) t te kai m non t te tan x + (y ) isod nama x + [y (r + )] = 4r (r + ): = 4r y Dhlad, o uperbolik c k kloc C(i; s) e nai nac eukle deioc k kloc me k ntro i(r + ) kai akt na r(r + ) =. Afo gia k je z H up rqei f I + (H ) ste f(z) = i kai o f apeikon zei eukle deiouc k klouc se eukle deiouc k klouc, en e nai kai uperbolik isometr a, prok ptei oti k je uperbolik c k kloc e nai eukle deioc k kloc wc s nolo, all me llo k ntro kai llh akt na. Aut shma nei oti to 3o a thma tou eukle dh isq ei kai sthn uperbolik gewmetr a. 9

Jewr ntac thn dia nnoia gwn ac gia thn uperbolik gewmetr a pwc kai sthn eukle deia, lec oi orj c gwn ec sto uperbolik ep pedo e nai sec, afo oi uperbolik c isometr ec diathro n tic gwn ec. Etsi isq ei kai to 4o a thma tou Eukle dh. D o uperbolik c euje ec l, l l gontai par llhlec an l \ l =?. Estw l m a uperbolik euje a kai z H, z = l. Diakr noume d o peript seic. Estw oti h l e nai eukle deia hmieuje a k jeth sto R. T te h eukle deia hmieuje a pou e nai k jeth sto R kai di rqetai ap to z e nai uperbolik euje a par llhlh proc thn l. Estw x R me y < x < Rez, pou y e nai to asumptwtik kro thc l ep tou R. Up rqei nac monadik c eukle deioc k kloc C pou di rqetai ap ta x, z kai e nai k jetoc sto R. To C \ H e nai uperbolik euje a pou di rqetai ap to z kai e nai par llhlh proc thn l. Aut de qnei oti up rqoun uperarijm simec sto pl joc par llhlec proc thn l pou di rqontai ap to z. Estw oti h l e nai to nw hmik klio eukle deiou k klou k jetou sto R. Estw x 0 R to k ntro auto tou eukle deiou k klou. Estw K o eukle deioc k kloc me k ntro x 0 pou di rqetai ap to z. To K \ H e nai uperbolik euje a par llhlh thc l. An x R e nai na opoiod pote shme o metax twn eukleide wn k klwn K, l, t te up rqei nac monadik c eukle deioc k kloc C pou di rqetai ap ta x, z. To C \ H e nai uperbolik euje a par llhlh thc l. Etsi kai se aut n thn per ptwsh up rqoun uperarijm simec sto pl joc uperbolik c euje ec par llhlec thc l pou di rqontai ap to z. Ta prohgo mena de qnoun oti to 5o a thma tou Eukle dh den isq ei sthn uperbolik gewmetr a, all isq ei to ak loujo. 7.. Je rhma. Gia k je uperbolik euje a l kai z H me z = l up rqoun uperarijm simec sto pl joc uperbolik c euje ec par llhlec thc l pou di rqontai ap to z. M a idiaiter thta pou qei to uperbolik ep pedo se sq sh me to eukle deio e nai to ide dec s noro, pou e nai ex orismo to s nolo @H = ^R. Ta shme a tou ide douc sun rou l gontai shme a sto peiro. K je uperbolik euje a qei akrib c d o shme a sto peiro. Ant strofa, d o diaforetik shme a sto peiro or zoun akrib c m a uperbolik euje a thc opo ac e nai ta shme a sto peiro. Sto uperbolik ep pedo qoume d o peript seic parall lwn eujei n. D o par llhlec uperbolik c euje ec l, l e te qoun na koin shme o sto peiro e te den qoun kan na. An den qoun kan na l gontai uperpar llhlec. 0

8. To uperbolik embad n kai o t poc twn Gauss-Bonnet To uperbolik embad n en c sun lou X H e nai to (X) = X y dxdy; an to olokl rwma up rqei. Jum zoume oti h parxh tou oloklhr matoc exart tai ap to X. Etsi to uperbolik embad n den up rqei gia la ta upos nola tou uperboliko epip dou. 8.. Pr tash. To uperbolik embad n e nai anallo wto ap tic uperbolik c isometr ec. Dhlad, gia k je X H, tou opo ou to uperbolik embad n up rqei kai k je f I(H ) isq ei (X) = (f(x)). Ap deixh. H ap deixh e nai sun peia tou t pou allag c metablht c kat thn olokl rwsh. Estw pr ta oti f =, pou (z) = z. An z = x+iy, t te (x; y) = ( x; y) kai sunep c 0 D(x; y) = : 0 Ara ((X)) = (X) Estw t ra oti f I + (H ) me t po T te pwc x roume kai Ara kai sunep c y X dxdy = j det D(x; y)jdxdy = (X): y f(z) = az + b ; pou a; b; c; d R; ad bc = : Imf(z) = jj Imz f(x; y) = ( acx + acy + bd + bcx + adx y (cx + d) + c y ; (cx + d) + c y ): Df(x; y) = 0 B @ (cx + d) c y cy(cx + d) [(cx + d) + c y ] [(cx + d) + c y ] cy(cx + d) (cx + d) c y [(cx + d) + c y ] [(cx + d) + c y ] det Df(x; y) = ; pou z = x + iy: jj4 Etsi ap ton t po allag c metablht c kat thn olokl rwsh qoume (f(x)) = f(x) y dxdy = X y [(cx + y) + y ] C A [(cx + y) + y dxdy = (X): ]

Ena uperbolik n-gwno, n 3, e nai na kleist upos nolo P tou H [ @H, pou fr ssetai ap n uperbolik euj gramma tm mata, pou l gontai pleur c. Ta shme a tom c twn pleur n l gontai koruf c. Epitr poume k poiec ap tic koruf c na br skontai sto @H = ^R. T toiec koruf c apokalo ntai ide deic koruf c kai b baia den an koun sto P \ H. P nta mwc qoume intp H. An to P den qei kamm a ide dh koruf, e nai kleist kai fragm no, dhlad sumpag c. 8.. Pr tash. Estw na uperbolik tr gwno me m a m non ide dh koruf. An 0 ; e nai oi eswterik c gwn ec stic d o llec koruf c, t te () = : Ap deixh. Qrhsimopoi ntac na kat llhlo stoiqe o thc I + (H ) mporo me na metasqhmat soume to tr gwno, ste h ide dhc koruf tou na e nai h, op te oi d o pleur c pou t mnontai s' aut n e nai tm mata eukleide wn eujei n k jetwn sto ^R. Metasqhmat zontac sthn sun qeia to tr gwno me stoiqe a thc I + (H ) thc morf c f(z) = z + b, b R kai g(z) = z, > 0, to f rnoume se j sh ste h tr th pleur na peri qetai sto eukle deio hmik klio me k ntro to 0 R kai akt na. To uperbolik embad n kai oi gwn ec param noun anallo wta ap touc metasqhmatismo c auto c, ap thn pr tash 8. kai to je rhma 3.0. Eqoume t ra () = y dxdy = cos cos( ) J tontac x = cos, 0, br skoume () = + p x cos y dy dx = cos( ) sin d = : sin p x dx:

8.3. Je rhma. Estw H na sumpag c uperbolik tr gwno me eswterik c gwn ec,,. To uperbolik embad n tou e nai () = ( + + ): Ap deixh. Metasqhmat zontac to tr gwno me kat llhla stoiqe a thc I + (H ) pa rnoume na isod namo tr gwno tou opo ou kamm a pleur den e nai m roc eukle deiac euje ac k jethc sto ^R. Opwc p nta to embad n kai oi gwn ec den all zoun. Estwsan A, B, oi koruf c me ant stoiqec eswterik c gwn ec,,. Proekte nontac thn pleur AB proc thn kate junsh tou B, h uperbolik euje a, m roc thc opo ac e nai h AB, qei na shme o B 0 sto peiro. To uperbolik tr gwno me koruf c A, B 0, qei m non m a ide dh koruf, thn B 0 kai to dio isq ei gia to uperbolik tr gwno me koruf c B, B 0,. An e nai h eswterik gwn a tou trig nou sthn koruf, qoume () = ( ) ( ) = [ + + )] [ ( + )] = ( + + ): 8.4. P risma. To jroisma twn eswterik n gwni n en c uperboliko trig nou e nai mikr tero ap kai h diafor e nai to uperbolik embad n tou trig nou. 9. To mont lo tou d skou tou Poincare Sthn par grafo aut n ja perigr youme na enallaktik mont lo thc uperbolik c gewmetr ac ston monadia o d sko D = fz C : jzj < g. O M + me t po (z) = iz + z + i l getai metasqhmatism c tou Cayley kai (H ) = D, giat an z = x + iy C, qoume j(z)j < t te kai m non t te tan dhlad y > 0. O ant strofoc qei t po x + (y ) x + (y + ) < ; (z) = iz + : z i 3

To s nolo I(D ) = fg M : g(d ) = D g = f f : f I(H )g e nai om da metasqhmatism n tou D kai to ze goc (I(D ); D ) e nai m a gewmetr a is morfh me thn (I(H ); H ). Me lla l gia, h (I(D ); D ) e nai na de tero mont lo thc uperbolik c gewmetr ac. J toume I + (D ) = f f : f I + (H )g. T te I(D ) = I + (D ) [ I + (D ), afo = sthn ^C. K je stoiqe o thc I + (D ) e nai thc morf c g = f, pou f(z) = az + b ; me a; b; c; d R; ad bc = : K nontac tic pr xeic br skoume oti o t poc tou g e nai J tontac t ra pa rnoume ton t po g(z) = (f( (z))) = Omoia an g I + (D ), t te [a + d + i(b c)]z + [b + c + i(a d)] [b + c i(a d)]z + [a + d i(b c)] : A = [a + d + i(b c)] kai B = [b + c + i(a d)] g(z) = Az + B Bz + A ; pou jaj jbj = : g(z) = Az B Bz A ; pou jaj jbj = : An : [; ]! D e nai m a kat tm mata C kamp lh, or zoume wc uperbolik m koc thc to L() = L( ). Afo ap ton kan na thc alus dac qoume ( ) 0 (z) = (z i) ; ( ) 0 (t) = Ap' thn llh meri, gia k je z D qoume Im (z) = i iz + z i ((t) i) 0 (t): iz + = jzj z i jz ij : Etsi to uperbolik m koc sto mont lo tou d skou d netai ap ton t po L() = j( ) 0 (t)j Im( )(t) dt = j(t) ij j 0 (t)j 4 j(t)j j(t) ij dt = j 0 (t)j j(t)j dt:

Opwc sto H or zoume thn uperbolik ap stash ston D wc (z; w) = inffl()j e nai kat tm mata C kamp lh ap to z sto wg: To ze goc (D ; ) g netai tsi metrik c q roc kai h : (H ; d)! (D ; ) isometr a metrik n q rwn. Ep shc I (D ) = I(D ). Opwc qoume apode xei, oi uperbolik c euje ec sto H qoun an mesa stic kat tm mata C kamp lec to el qisto uperbolik m koc. O wc metasqhmatism c Mobius apeikon zei tic uperbolik c euje ec sto H se tm mata k klwn thc ^C m sa ston D, pou e nai k jetoi ston monadia o k klo S = @D = fz C : jzj = g. To @H = ^R apeikon zetai ston S, pou e nai to ide dec s noro tou D. Oi uperbolik c euje ec ston D qoun b baia to el qisto uperbolik m koc. 9.. Par deigma. Oi uperbolik c euje ec ston D pou di rqontai ap to 0 e nai oi eukle deiec di metroi tou D, giat (i) = 0 kai oi k kloi thc ^C pou di rqontai ap to 0 kai e nai k jetoi ston S e nai eukle deiec euje ec (me to ). Estw z D kai : [0; ]! D h param trhsh tou uperboliko euj grammou tm matoc ap to 0 sto z me t po (t) = tjzj. T te (0; z) = jzj 0 + jzj t dt = log : jzj An epil soume wc proc jzj br skoume kai jzj = tanh( (0; z)). To gegon c oti oi uperbolik c euje ec tou D pou di rqontai ap to 0 e nai oi eukle deiec di metroi tou D, se sunduasm me to gegon c oti h I(D ) qei lec tic idi thtec pou qei h I(H ), bohj ei k poiec for c na sugkr noume uperbolik c me eukle deiec apost seic. O an logoc t poc thc pr tashc 5.7 e nai o ak loujoc. 9.. Pr tash. Gia k je z, w D isq ei sinh ( (z; w)) = jz wj ( jzj )( jwj ) : Ap deixh. H diadikas a thc ap deixhc e nai moia me thc pr tashc 5.7. Kat' arq n parathro me oti gia k je g I + (D ) isq ei (g(z) g(w)) = g 0 (z)g 0 (w)(z w) gia k je z, w D. Pr gmati, an o t poc thc g e nai g(z) = Az + B Bz + A ; jaj jbj = ; 5

t te kai kat sun peia (g(z) g(w)) = g 0 (z) = ( Bz + A) (z w) ( Bz + A) ( Bw + A) = g0 (z)g 0 (w)(z w) : Ta d o m lh thc is thtac pou j loume na apode xoume e nai anallo wta ap thn om da metasqhmatism n I + (D ). To arister m loc profan c e nai. Oso afor to dexi m loc, gia k je g I + (H ) qoume kai sunep c jg(z)j = Az + B Az + B Bz + A Bz + = A jg0 (z)j( jzj ) jg(z) g(w)j ( jg(z)j )( jg(w)j ) = jg 0 (z)jjg 0 (w)jjz wj jg 0 (z)j( jzj )jg 0 (w)j( jwj ) = jz wj ( jzj )( jwj ) : Epil goume t ra g I + (D ), ste g(z) = 0, pou up rqei ap thn pr tash 4.4, op te jz wj ( jzj )( jwj ) = jg(w)j jg(w)j = tanh ( (g(z); g(w))) tanh ( = tanh ( (z; w)) tanh ( (z; w)) = sinh ( (z; w)): (g(z); g(w))) Opwc to uperbolik m koc, tsi kai to uperbolik embad n metaf retai ap to mont lo tou H sto mont lo tou D m sw tou metasqhmatismo tou Cayley. Or zoume gia k je X D to uperbolik tou embad n wc to (X) = ( (X)), tan to dexi m loc up rqei. Ap ton t po allag c metablht c kat thn olokl rwsh qoume (X) = pou z = x + iy. Omwc (X) op te antikajist ntac br skoume (X) = X y dxdy = X det D (z) = j( ) 0 (z)j = jzj jz ij 4 (Im (z)) j det D (z)jdxdy; jz ij 4 dxdy = X An k noume allag se polik c suntetagm nec br skoume (X) = X 4r ( r ) drd: 6 4 jz ij 4 ; 4 ( x y ) dxdy:

Ena par deigma boliko upologismo ston D e nai o t poc tou Lobachevskii gia thn uperbolik ap stash en c shme ou ap m a uperbolik euje a m sw thc gwn ac parallhlismo. Estw z D kai l m a uperbolik euje a me z = l. Up rqoun akrib c d o uperbolik c euje ec pou di rqontai ap to z, pou e nai par llhlec thc l kai qoun ap na koin shme o sto peiro me thn l. H gwn a pou sqhmat zei h m a ap tic d o me thn k jeth uperbolik euje a ap to z proc thn l l getai gwn a parallhlismo. An m a uperbolik euje a di rqetai ap to z kai sqhmat zei me thn k jeth ap to z proc thn l gwn a megal terh ap, t te e nai uperpar llhlh proc thn l. An w e nai to shme o tom c thc k jethc uperbolik c euje ac ap to z proc thn l me thn l, h ap stash tou z ap thn l e nai (z; l) = inff(z; z 0 ) : z 0 lg = (z; w): 9.3. Je rhma. Estw l m a uperbolik euje a kai z na shme o ekt c aut c. An e nai h gwn a parallhlismo, t te e (z;l) = tan( ): Ap deixh. Efarm zontac na kat llhlo stoiqe o thc I(D ), mporo me na apeikon soume to z sto 0, op te arke na apode xoume ton t po gia to z = 0, afo ta stoiqe a thc I(D ) diathro n tic gwn ec. Ap to par deigma 9. qoume t ra + jwj (0; l) = (0; w) = log jwj kai Kat sun peia jwj = sin tan = cos cos : e (z;l) = jwj cos + sin = + jwj cos sin + = tan( ): 7

0. Uperbolik trigwnometr a Opwc sthn eukle deia gewmetr a, tsi kai sthn uperbolik up rqoun k poiec sq seic metax twn eswterik n gwni n en c sumpago c uperboliko trig nou kai twn pleur n tou. Oi upologismo twn uperbolik n apost sewn pou qrei zetai na k noume e nai pio aplo sto mont lo tou d skou. Estw na sumpag c uperbolik trigwno me koruf c ta shme a A, B,, ant stoiqec eswterik c gwn ec,, kai uperbolik m kh pleur n a = (B; ), b = (A; ), c = (A; B). Up rqei na g I + (D ) pou apeikon zei to A sto 0 kai to B se k poio r R \ D. To g e nai na stoiqe o thc I + (D ) pou apeikon zei thn uperbolik euje a tm ma thc opo ac e nai h pleur c sthn uperbolik euje a R \ D. Efarm zontac sthn an gkh kai thn an klash (z) = z, mporo me na apeikon soume to shme o A sto 0 kai to B se k poio r > 0. Etsi k je uperbolik tr gwno ston D e nai isod namo me na uperbolik tr gwno tou opo ou h koruf A = 0 kai sunep c oi pleur c c, b e nai tm mata eukleide wn diam trwn tou D kai B = r R \ D, r > 0. Estw oti = se i, gia k poio 0 < s <. Ap to par deigma 9. qoume r = tanh( c) kai s = tanh( b). Ap to eukle deio pujag reio je rhma gia to eukle deio tr gwno me koruf c A, B, qoume jb j = r + s rs cos = tanh ( c) + tanh ( b) tanh( c) tanh( c) cos : Ap thn pr tash 9. qoume ep shc jb j = ( r )( s ) sinh ( a) = cosh ( c) cosh ( b) sinh ( a): Kat sun peia ta dexi m lh twn parap nw isot twn e na sa. bl poume oti h is thta aut e nai isod namh me thn is thta K nontac tic pr xeic Eqoume t ra to ak loujo. cosh a = cosh b cosh c sinh b sinh c cos : 8