EISAGWGH STHN

PROBOLIKH GEWMETRIA

Iwnnhc P. Zhc

E-mail: i.zois@exeter.oxon.org

10 Dekembrou 2006

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Perieqmena

1 Prlogoc 8

2 Eisagwg 9

3 Proboliko Qroi 12

3.1 Stoiqea Grammikc 'Algebrac . . . . . . . . . . . . . . . . . . 12

3.2 Orismc Proboliko Qrou . . . . . . . . . . . . . . . . . . . 15

3.3 Upqwroi tou Proboliko Qrou . . . . . . . . . . . . . . . . 17

3.4 Basikc Idithtec Probolikn Qrwn . . . . . . . . . . . . . . 18

3.5 Omogenec kai Anomogenec Suntetagmnec . . . . . . . . . . . 19

3.6 Paradegmata Probolikn Qrwn . . . . . . . . . . . . . . . . 21

3.7 Parrthma: Stereografik Probol . . . . . . . . . . . . . . 27

4 Proboliko Metasqhmatismo 29

4.1 Basiko Orismo . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Shmea se Genik Jsh . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Proboliko Metasqhmatismo kai Sqdio . . . . . . . . . . . . 34

5 Dusmc 38

5.1 Dukc Probolikc Qroc kai Arq Dusmo . . . . . . . . . 38

5.2 Mhdenistc Dianusmatiko Qrou kai Dusmc . . . . . . . . . 41

5.3 Efarmogc Dusmo: Jerhma Desargues, Jerhma Pppou . 43

6 Kamplec b' bajmo (Quadrics) 466.1 Digrammikc Morfc . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Paradegmata Kampuln b' bajmo . . . . . . . . . . . . . . . 49

6.3 Genikeumnec Kwnikc . . . . . . . . . . . . . . . . . . . . . . . 52

6.4 Mia Efarmog ap thn Algebrik Gewmetra . . . . . . . . . . 54

7 Grafdec Tetragwnikn (Pencils of Quadrics) 567.1 Diagwniopohsh Zegouc Pinkwn . . . . . . . . . . . . . . . . 56

7.2 Grafdec sto Migadik Probolik Eppedo . . . . . . . . . . . 58

8 Grammiko Qroi Tetragwnikn 62

8.1 To Orjognio Sumplrwma . . . . . . . . . . . . . . . . . . . 62

8.2 Paradegmata Tetragwnikn Kampuln kai Epifanein . . . . 64

9 To Exwterik Ginmeno 67

9.1 Anaparstash epipdwn tou qrou msw tou exwteriko gino-

mnou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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9.2 Deterh exwterik dnamh enc dianusmatiko qrou kai to

sfhnoeidc ginmeno . . . . . . . . . . . . . . . . . . . . . . . . 69

10 H Exwterik 'Algebra 74

10.1 Anterec exwterikc dunmeic enc dianusmatiko qrou . . . 74

10.2 Paradegmata . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

11 H Tetragwnik Klein 8011.1 H sunjkh aposnjeshc . . . . . . . . . . . . . . . . . . . . . 80

11.2 H gewmetrik ermhnea thc tetragwnikc Klein . . . . . . . . . 82

12 Grammiko Upqwroi thc Tetragwnikc Klein 8412.1 Gewmetrik ermhnea twn upoqrwn thc Tetragwnikc Klein . . 8412.2 Sqlia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

13 Prosggish Klein thc Gewmetrac kai Omoparallhlik Gew-metra 89

13.1 O rloc twn omdwn metasqhmatismn . . . . . . . . . . . . . 89

13.2 H Omda GLn twn Genikn Grammikn Metasqhmatismn . . . 9013.3 Stoiqea Jewrac Omdwn . . . . . . . . . . . . . . . . . . . . 92

13.4 H Omda twn Probolikn Metasqhmatismn PGLn . . . . . . 9313.5 H omda twn probolikn metasqhmatismn thc eujeac kai

Omoparallhliko Metasqhmatismo . . . . . . . . . . . . . . . 95

14 Optik kai Eidik Jewra Sqetikthtac 100

14.1 Optik Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . 10014.2 Eidik Jewra Sqetikthtac . . . . . . . . . . . . . . . . . . . 102

15 Upoomdec thc Probolikc Omdac tou Migadiko Epipdou

PGL2(C) 104

16 To Axwma thc Parallhlac 108

16.1 To Eukledeio Athma . . . . . . . . . . . . . . . . . . . . . . . 108

16.2 Ta Aximata thc Eukledeiac Gewmetrac . . . . . . . . . . . . 109

16.3 H probolik gewmetra kai ta aximata tou Eukledh . . . . . 111

16.4 Eujeec tou nw migadiko hmiepipdou . . . . . . . . . . . . . 112

16.5 Gwnec sto nw migadik hmieppedo . . . . . . . . . . . . . . . 113

16.6 Suzuga Trignwn . . . . . . . . . . . . . . . . . . . . . . . . 114

17 Parrthma 116

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Sntomo biografik shmewma tou suggrafa:

O Iwnnhc Panagitou Zhc gennjhke to 1968 sth Broia. Met to

prac twn proptuqiakn tou spoudn sto Tmma Fusikc tou Panepisth-

mou Ajhnn, ap' pou apofothse me 'Arista, sunqise tic metaptuqiakc

tou spoudc sthn Aggla: Mster (Part III of the Mathematical Tripos)sto Panepistmio tou Kimpritz (Kollgio Emmanoul) kai Didaktorik sto

Panepistmio thc Oxfrdhc (Kollgio 'Exeter). H trimelc epitrop thc di-

daktorikc tou diatribc apartiztan ap touc kajhghtc Simon K. Donald-son FRS (Fields Medal 1986, Crafoord Prize 1996), Roger Penrose FRS kaiSheung-Tsoun Tsou. 'Htan uptrofoc, metax llwn, thc Eurwpakc 'Enw-shc, thc Brettanikc Kubrnhshc, tou IKU ('Idruma Kratikn Upotrofin),

tou Koinwfeloc Idrmatoc A.S. Wnshc kai tou Koinwfeloc Idrmatoc

A.G. Lebnthc. 'Eqei ergasje sto pareljn wc ereunhtc kai panepisthmia-

kc dskaloc sta Panepistmia thc Oxfrdhc kai tou Kimpritz thc Agglac,

sto Institut des Hautes Etudes Scientifiques (IHES) (wc mloc thc ereunhti-kc omdac tou kajhght Alain ConnesFields Medal 1982, Crafoord Prize

2002) kai sthn Ecole Normale Superieure (ENS) sto Parsi thc Gallackai allo. Ta ereunhtik tou endiafronta empptoun sta gnwstik anti-

kemena thc Mh-metajetikc Gewmetrac, thc Topologac kai Gewmetrac twn

Pollaplottwn (eidiktera twn Pollaplottwn Qamhln Diastsewn) kai

efarmogc autn sth Jewrhtik Fusik (Kosmologa, Jewrec Enopohshc,

Jewra Qordn/Membrann). 'Eqei dhmosiesei mqri stigmc perpou ekosi

rjra se diejn ereunhtik episthmonik periodik (se hlektronik kai ntuph

morf).

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in girum imus nocte et consumimur igni

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Ant Bibliografac:

Oi didaktikc autc shmeiseic proljan ap to mjhma tou 3ou examnou

(2ou touc) Probolik Gewmetra pou o suggrafac ddaxe kat to akadh-

mak toc 2005-2006 sthn Antath Sqol Topografac thc Gewgrafikc

Uphresac Strato (GUS), sqol pou akolouje to prgramma spoudn thc

sqolc Topogrfwn Mhqanikn tou Ejniko Metsobou Poluteqneou thc

Ajnac (EMP).

H lh tou majmatoc bassthke en polloc sto mjhma tou 3ou trimnou

tou prtou touc Grammik Gewmetra to opoo o suggrafac eqe didxei

sto pareljn stouc proptuqiakoc foithtc twn Majhmatikn Tmhmtwn twn

Panepisthmwn thc Oxfrdhc kai tou Kimpritz thc Agglac. (Shmeinoume

pwc sthn pio gkurh prsfath diejn lsta me ta 200 korufaa panepist-

mia tou ksmou twn efhmerdwn Times Londnou kai N. Urkhc kajc kaitou periodiko Time, ta en lgw panepistmia brskontai stic jseic 1 kai 2sthn ep mrouc kattaxh pou afor ta majhmatik kai tic jetikc epistmec).

To prwtogenc ulik twn shmeisewn autn (kai thc lhc kat' epkta-

sh) prorqetai ap tic exairetikc shmeiseic tou kajhght Nigel J. HitchinFRS, smera sto Panepistmio thc Oxfrdhc all palaitera sto Panepi-stmio tou Kimpritz thc Agglac.

Pra ap th metfrash twn shmeisewn sta ellhnik gine kai mia pro-

spjeia na analujon perisstero poll shmea all kai na emploutisjon

me kpoiec epiplon efarmogc. Tautqrona sumperielfjhsan kai arketo

orismo ap llouc kldouc twn majhmatikn (kurwc ap thn 'Algebra all

kai ap thn Topologa, th Gewmetra k..) ste na gnoun oi shmeiseic autc

kat to dunat autodnamec sthn melth touc. H prjesh tan af' enc na

prosferje na mjhma uyhlo proptuqiako epipdou smfwna me ta diejn

prtupa, all tautqrona h lh na enai prosit kai se na akroatrio me

teqnologik katejunsh.

Ekfrzontai jermc euqaristec arqik proc ton kajhght Nigel J. Hi-tchin gia th shmantik sumbol tou sto lo egqerhma, proc ton EmmanoulMegalooikonmou gia thn poltimh bojei tou sthn exeresh kai qrsh el-

lhnikc kdoshc tou Latex to opoo qrhsimopoijhke gia thn daktulogrfhshtwn shmeisewn all kai sthn Elisbet Giannotsou pou eqe thn filologi-

k epimleia twn shmeisewn. Tloc idiaterec euqaristec ekfrzontai kai

proc touc foithtc gia ton enjousiasm touc, thn dijesh gia melth, thn

enjrrunsh, tic parathrseic kai ta poikla sqli touc. H didaskala e-

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nai amfdromh diadikasa kai qwrc th dik touc sumbol enai amfbolo an

to mjhma ja eqe aut th morf. H elpda enai na mhn apogohtejhkan oi

foithtc kai epshc na mhn apogohteujon kai oi tuqn loipo anagnstec.

Ajna, Mioc 2006

I.P.Z.

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1 Prlogoc

Oi didaktikc autc shmeiseic aforon to kommti thc gewmetrac pou

ja mporose genik na qarakthrisje grammik, se antidiastol me th mh-

grammik jewra thc topologac thc diaforikc gewmetrac. Ap th qrsh

tou rou grammik katalabanei ekola kpoioc ti h grammik lgebra pazei

shmantik rlo. Gi' aut xekinme me tic paragrfouc 3-8 pou anafroume

ousiastik stoiqea ap th jewra twn dianusmatikn qrwn (peperasmnhc

distashc), ta opoa mwc apokton gewmetrik upstash msw thc probo-

likc gewmetrac. Oi pargrafoi 9 kai 10 epektenoun aut th grammik l-

gebra orzontac diaforikc morfc kai to exwterik ginmeno metax autn.

To knhtro prorqetai ap thn prospjeia perigrafc eujein ston tris-

distato Eukledeio qro. Oi pargrafoi 11 kai 12 epiloun to prblhma

thc perigrafc twn eujein diatupnontac thn antistoiqa Klein. Oi par-grafoi 13-15 perigrfoun thn prosggish Klein thc gewmetrac wc melthanallowtwn posottwn ktw ap kpoia omda metasqhmatismn, prosg-

gish pou enai gnwst wc prgramma Erlangen, (Erlangen Programm), mazme kpoiec efarmogc ap th fusik. (To prgramma aut rqise ap ton

F. Klein to 1872 tan gine kajhghtc sto panepistmio thc plhc Erlangenpou brsketai kont sth Nurembrgh). Tloc h pargrafoc 16 exetzei to

uperbolik eppedo up to prsma thc klasikc axiwmatikc prosggishc thc

gewmetrac msw twn axiwmtwn tou Eukledh. Sto Parrthma (Pargrafoc

17) brskontai sugkentrwmnoi diforoi qrsimoi pijanc gia ton anagnsth

orismo.

'Oson afor ta proapaitomena gia th melth twn shmeisewn autn,

epishmanetai ti, an kai katabljhke prospjeia oi shmeiseic na enai ma-

jhmatik autnomec, ja tan qrsimo oi anagnstec na qoun sto pareljn

parakoloujsei na mjhma grammikc lgebrac pou na kalptei th basik

lh ap th jewra (pragmatikn kurwc) dianusmatikn qrwn peperasmnhc

distashc. 'Ena klasik sggramma gia th melth twn dianusmatikn qrwn

peperasmnhc distashc enai gia pardeigma to biblo tou P.B. Halmos: -Finite Dimesnional Vector Spaces.

Oi diforec protseic, orismo, jewrmata, anafrontai arqik me ton

arijm thc paragrfou sthn opoa brskontai, akoloujomeno ap to sugke-

krimno arijm thc prtashc msa sthn en lgw pargrafo. Gia pardeigma

h prtash 4.1.2 anafretai sthn prtash 2 thc paragrfou 4.1, en to tloc

miac apdeixhc sumbolzetai me na leuk tetrgwno.

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2 Eisagwg

Enai elogo na anarwthje kpoioc gia poio lgo na meletsei kanec

llh gewmetra ektc ap thn Eukledeia gewmetra stic 3 diastseic, pou

anaparist to fusik mac ksmo msa ston opoo zome kai h opoa peri-

grfetai majhmatik ap ta diansmata efodiasmna me to eswterik kai to

exwterik ginmeno autn. Gia na dsoume mia peistik apnthsh sto para-

pnw erthma, h opoa tautqrona ja mac dsei kai knhtra gia th melth

tou en lgw majmatoc, ja anafroume orismna paradegmata sta opoa h

qrsh thc Eukledeiac gewmetrac den enai h plon katllhlh.

A. Probolc

Oi kallitqnec (gia pardeigma zwgrfoi all qi mno) thc Anagnnhshc

antimetpizan to prblhma thc anaparstashc tou fusiko ksmou se kpoio

pnaka. Pio sugkekrimna jelan na anaparastsoun diforec 2-distatec

yeic (katyeic) 3-distatwn antikeimnwn. O trpoc pou lusan to prblhma

tan na jewrsoun to mti tou kallitqnh wc na shmeo, stw P , sto qro.Kje shmeo tou pragmatiko antikeimnou pou jelan na anaparastsoun,

stw A, orzei mia eujea me to mti sto qro, thn PA. An parembloumemia difanh ojnh metax tou matio kai tou antikeimnou, aut tmnei thn

eujea PA se na shmeo, stw A. Metabllontac to shmeo A tsi stena kalyei lo to antikemeno, tte ta antstoiqa shmea A ep thc ojnhcsuniston thn probol tou fusiko antikeimnou pnw sth dosmnh ojnh.

To shmerin antstoiqo tou kallitqnh thc Anagnnhshc, ektc bbaia

ap touc sgqronouc kallitqnec, enai oi sqediastc kai oi mhqaniko, oi

opooi pollc forc, se antjesh me touc kataskeuastc montlwn make-

tn pou douleoun sto qro twn 3 diastsewn, qreizetai na apeikonsoun

na 3-distato antikemeno sto eppedo ap diaforetikc yeic. Aut akri-

bc epitugqnetai msw thc probolikc gewmetrac. Sthn pragmatikthta, oi

anaparastseic sto eppedo enc 3-distatou antikeimnou ap diaforetik

optik gwna prokptoun pwc ja dome, msw probolikn metasqhmatismn.

B. Eidik Jewra Sqetikthtac

Sthn Eidik Jewra Sqetikthtac, (A. Einstein 1905), basik rlo pazei homda Lorentz twn legmenwn yeudo-orjogniwn metasqhmatismn (me or-zousa sh me +1) ston eppedo qwrqrono 4-diastsewn (strofc pou diath-ron anallowth th metrikMinkowski). Gia na brome mwc thn plrh omdasummetrac prpei na sumperilboume kai tic metatopseic stic 4-diastseic,

opte katalgoume sthn omda Poincare, h opoa apotele antstoiqh thcomdac metasqhmatismn tou Galilaou thc klasikc fusikc. Knontac mia

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strof Wick, dhlad jewrntac to qrno fantastik, h metrik Minkowskiapokt Eukledeia upograf (se 4 diastseic) kai h omda Lorentz gnetaih omda twn eidikn orjogniwn metasqhmatismn (orjognioi metasqhma-

tismo me orzousa sh me +1) stic 4 diastseic. H omda Poincare mwcmpore na jewrhje wc h omda probolikn metasqhmatismn PGL2(C) toumigadiko epipdou C2.

G. Tomografa

Mia diaforetik pleur thc gewmetrac enai h melth qi twn shmewn all

twn eujein twn epipdwn sto qro. Gia pardeigma mpore kanec na rwt-

sei poia enai h gewmetra tou qrou twn eujein tou Eukledeiou 3-distatou

qrou. Mia eujea mpore epshc na orisje ap ta shmea tomc thc me do

eppeda sto qro, opte mpore na kajorisje msw do paramtrwn. Sune-

pc o qroc twn eujein tou Eukledeiou 3-distatou qrou apotele qro

distashc 2, all qi Eukledeio qro. 'Eqei th dik tou gewmetra, pwc ja

dome.

Uprqoun tso majhmatik so kai praktik problmata, sta opoa oi

eujeec enai pio shmantikc ap ta shmea. Gia pardeigma, periptseic pou

aforon sarwtc (scanners) aktnwn Q. H epistmh aut lgetai tomogra-fa. O sarwtc pargei aktnec Q kat mkoc diafrwn dieujnsewn sto

qro. 'Otan ma ap autc dirqetai msa ap to sma enc asjenoc (

msa ap opoiodpote antikemeno), ekteletai mia mtrhsh pou anaparist

th msh puknthta lhc kat mkoc thc dierqmenhc aktnac. Me aut ton

trpo parnoume mia sunrthsh me pedo orismo to qro twn eujein tou

R3. To zhtomeno enai mia sunrthsh thc puknthtac tou smatoc me pedoorismo to R3, dhlad to qro twn shmewn. Gia na gnei aut h metatrop,qreiazmaste th sqsh metax shmewn kai eujein. Gia pardeigma, oi euje-

ec pou dirqontai ap dosmno shmeo, diagrfoun na eppedo sto qro twn

eujein. Aut h gewmetra, pou den enai Eukledeia, lgetai gewmetra Klein(proc timn tou Germano majhmatiko Felix Klein, pou zhse sto detero mi-s tou 19ou wc tic arqc tou 20ou aina), kai thn opoa ja meletsoume se

aut to mjhma. Egkuklopaidik anafroume epshc, pwc o sunduasmc thc

gewmetrac Klein me touc probolikoc metasqhmatismoc thc Eidikc Jewracthc Sqetikthtac (bl. pardeigma 2 parapnw) apotelon th bsh thc pe-

rfhmhc sustrofikc gewmetrac (twistor geometry) tou korufaou diejncsmera majhmatiko kosmolgou Roger Penrose h opoa (sustrofik gewme-tra) apotele (ektc twn llwn) kai mia prospjeia gia thn enopohsh lwn

twn fusikn allhlepidrsewn.

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D. Gewmetrik Optik

O qroc twn eujein tou R3 emfanzetai sth fusik kai me kpoio diaforeti-k trpo, sth melth optikn diatxewn (susthmtwn katptrwn kai fakn).

Se aut thn perptwsh oi fwteinc aktnec diajtoun kai for didoshc, opte

apotelon kateujunmenec eujeec. 'Otan mia fwtein aktna (kateujunmenh

eujea) mpei se mia optik ditaxh, ja uposte seir anaklsewn, diajlsewn,

klp. kai sthn xodo thc optikc ditaxhc ja proume mia llh fwtein aktna

(kateujunmenh eujea). H optik ditaxh sunepc aske na metasqhmatism

ston (majhmatik) qro twn kateujunmenwn eujein tou fusiko qrou. H

melth autn twn metasqhmatismn apotele th bsh thc gewmetrikc pro-

sggishc sthn optik.

Uprqoun kai pollo lloi tomec pou enai qrsimh h probolik gewme-

tra: gia pardeigma, an ant tou smatoc twn pragmatikn arijmn R epil-xoume na llo peperasmno sma, h probolik gewmetra qrhsimopoietai sth

jewra kwdkwn, sthn kruptografa, ston sqediasm logismiko gia progrm-

mata sqedashc se H/U all kai sth jewra arijmn. Genkeush thc pro-

bolikc gewmetrac apotele h melth twn probolikn poikilin (projectivevarieties) h opoa ankei sto qro thc algebrikc gewmetrac. H algebri-k gewmetra qei stentath sqsh me th jewra arijmn kai to prgramma

Atiyah-Langlands gia thn enopohsh twn majhmatikn.

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3 Proboliko Qroi

3.1 Stoiqea Grammikc 'Algebrac

Upenjumzoume orismnec basikc nnoiec ap th grammik lgebra.

'Estw F sma me qarakthristik diforh tou 2 (dhlad isqei ti 1+1 6= 0,pou 0 kai 1 ta oudtera stoiqea thc prsjeshc kai tou pollaplasiasmosto sma antstoiqa) kai V na mh-ken snolo efodiasmno me mia eswterikprxh + pou ja th lme prsjesh, (dhlad h prxh + enai mia apeiknish+ : V V V ) kai me mia exwterik prxh pou ja th lme bajmwtpollaplasiasm, (dhlad h prxh enai mia apeiknish : F V V ).

Orismc 1. To snolo V efodiasmno me tic parapnw prxeic ja lge-tai dianusmatikc qroc me sma F (sunjwc to sma ja enai oi pragmatiko oi migadiko arijmo) an ikanopoiontai oi exc idithtec:

To zegoc (V,+) apotele Abelian omda (dhlad isqoun h prosetairi-stik idithta, h antimetajetik idithta kai uprqei oudtero stoiqeo kajc

kai to summetrik kje stoiqeou tou V ). O bajmwtc pollaplasiasmc ikanopoie tic exc idithtec (upojtoume ti, F kai a, b V ): (a+ b) = a+ b(+ ) a = a+ a() a = ( a), pou sthn arister parnjesh ennoetai o pollaplasia-smc metax twn stoiqewn tou smatoc

1 a = a, pou 1 F to monadiao stoiqeo (oudtero stoiqeo tou pollapla-siasmo) sto sma F.

Pardegma 1: To snolo twn elejerwn dianusmtwn tou epipdou

tou qrou apotelon paradegmata pragmatikn dianusmatikn qrwn me

prxeic thn gnwst prsjesh dianusmtwn (kannac parallhlogrmmou) kai

bajmwt pollaplasiasm ton gnwst pollaplasiasm pragmatiko arijmo

ep dinusma.

Shmewsh 1: Sth sunqeia gia aplopohsh tou sumbolismo ja parale-

poume to smbolo tou bajmwto pollaplasiasmo. Epshc den ja anaf-roume to sma tan enai safc gia poio sma milme (sunjwc ja enai oi

pragmatiko arijmo).

Mporome na prosjtoume diansmata v, w V gia na proume to dinu-sma v + w V kai na pollaplasizoume ta stoiqea tou V me ta stoiqea

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F kai na parnoume to stoiqeo v V . Autc oi do prxeic ikanopoiontic gnwstc idithtec twn dianusmatikn qrwn (prosetairistik, epimeristi-

k kai antimetajetik).

Orismc 2. 'Enac dianusmatikc upqwroc U V enai na uposnoloU tou V , to opoo enai kleist wc proc thn prsjesh kai to bajmwt pol-laplasiasm, dhlad, an gia kje v, w U kai F, isqei ti v + w Ukai v U . Me lla lgia, to U apotele to dio na no dianusmatik qropeperasmnhc distashc me tic diec prxeic kai to dio sma pwc to V .

Orismc 3. En U1 kai U2 enai do diaforetiko dianusmatiko upqwroitou diou dianusmatiko qrou V , dhlad U1 V kai U2 V , tte orzoumeto (euj) jroisma U1+U2 twn upoqrwn U1 kai U2 tou V wc to snolo twndianusmtwn v V ta opoa mporon na grafon sth morf v = u1 + u2 meu1 U1 kai u2 U2 antstoiqa.

Orismc 4. H tom twn dianusmatikn upoqrwn U1U2 tou V orzetaiwc to snolo twn stoiqewn tou V pou periqontai tso ston U1 so kaiston U2.

Shmewsh 2: Upenjumzoume ti tso h tom so kai to (euj) jroisma

dianusmatikn upoqrwn enc dianusmatiko qrou apotelon nouc dianu-

smatikoc upoqrouc (den sumbanei bbaia to dio kai me thn nwsh dianu-

smatikn upoqrwn).

Orismc 5. Mia bsh tou V apoteletai ap na snolo dianusmtwn{v0, v1, ..., vn} tou V ta opoa enai metax touc grammik anexrthta (dhladkanna touc den mpore na grafe wc grammikc sunduasmc twn upolopwn)

kai ta opoa pargoun ton V , dhlad kje stoiqeo v V mpore na grafewc grammikc sunduasmc autn,

v =n

i=0

ivi (1)

pou i F, me i = 0, 1, 2, ..., n. Kje dianusmatikc qroc peperasmnhcdistashc qei toulqiston mia bsh. To pljoc twn dianusmtwn twn b-

sewn enc dianusmatiko qrou enai stajer kai onomzetai distash tou

dianusmatiko qrou. H distash exarttai ap to sma. Ja asqolhjome

me dianusmatikoc qrouc peperasmnhc distashc. Sto parapnw pardeig-

ma, dimFV = n+ 1.

13

An epilxoume mia bsh {vi}(i=0,1,...,n) ston V , tte kje dinusma v Vgrfetai me monadik trpo wc

v =n

i=0

ivi (2)

kai oi suntelestc i F, me i = 0, 1, 2, ..., n, lgontai suntetagmnec toudiansmatoc v V wc proc th sugkekrimnh bsh (oi suntetagmnec exar-tntai ap thn epilog bshc).

Shmewsh 3: Upenjumzoume pwc gia to dianusmatik qro R2 distashc2, ta diansmata (1, 0) kai (0, 1) apotelon th legmenh sunjh bsh tou R2.Anloga gia to dianusmatik qro R3 distashc 3, ta diansmata (1, 0, 0),(0, 1, 0) kai (0, 0, 1) apotelon th legmenh sunjh bsh tou R3. H genkeushenai profanc qi mno gia to dianusmatik qro Rn me opoiadpote pepe-rasmnh distash n N all kai gia kje sma F.

Orismc 6. 'Estw V kai W do dianusmatiko qroi me to dio sma F.Mia apeiknish f : V W ja lgetai grammik an isqei to exc:

f(v1 + v2) = f(v1) + f(v2)

v1, v2 V kai , F.

H basik sqsh isodunamac metax twn dianusmatikn qrwn enai h n-

noia tou isomorfismo:

Orismc 7. Do dianusmatiko qroi V kaiW (me to dio sma) lgontaiismorfoi, gegonc pou ja to sumbolzoume V = W , an uprqei metax toucgrammik apeiknish pou na enai 1-1 kai ep.

To paraktw jerhma epilei to prblhma thc kathgoriopohshc twn dia-

nusmatikn qrwn peperasmnhc distashc (classification problem) kai maclei pwc h basik anallowth posthta stouc dianusmatikoc qrouc enai h

distas touc:

Jerhma 1. Do dianusmatiko qroi (peperasmnhc distashc kai me

to dio sma) enai ismorfoi en kai mno en qoun thn dia distash.

Isqei h paraktw qrsimh prtash:

14

Prtash 1. An U1 kai U2 enai do dianusmatiko upqwroi tou idoudianusmatiko qrou V , tte isqei h exc jemelidhc sqsh metax twndiastsewn tou ajrosmatoc kai thc tomc dianusmatikn upoqrwn:

dim(U1 + U2) = dimU1 + dimU2 dim(U1 U2)

3.2 Orismc Proboliko Qrou

'Estw V pragmatikc dianusmatikc qroc peperasmnhc distashc (dh-lad wc sma jewrome to sma twn pragmatikn arijmn).

Orismc 1. O probolikc qroc P (V ) enc pragmatiko dianusmatikoqrou V peperasmnhc distashc orzetai wc to snolo twn monodistatwndianusmatikn upoqrwn tou V .

'Estw x P (V ) na shmeo tou proboliko qrou. Se aut bseitou parapnw orismo antistoiqe nac dianusmatikc upqwroc Vx V medimVx = 1. 'Enac monodistatoc dianusmatikc upqwroc tou V apoteletaiap la ta bajmwt pollaplsia v, pou R, enc sugkekrimnou mh-mhdeniko diansmatoc v V . To v lgetai antiproswpeutik dinusma toushmeou x P (V ) pou orzetai ap to sugkekrimno monodistato upqwroVx tou V . Kje llo mh-mhdenik (bajmwt) pollaplsio tou v apotelediaforetik antiproswpeutik dinusma tou diou shmeou x P (V ). Sune-pc uprqei mia plhjra antiproswpeutikn dianusmtwn gia kje shmeo

tou proboliko qrou.

Par' ti to V apotele dianusmatik qro, o probolikc qroc P (V ) denapotele dianusmatik qro. Pio sugkekrimna isqei to exc:

Prtash 1. O probolikc qroc P (V ) apotele (pragmatik), (diafor-simh) pollaplthta me distash kat na mikrterh ap th distash tou V ,dhlad dimP (V ) = (dimV ) 1.

Paratrhsh 1: Ap ton orism thc pollaplthtac prokptei ti o pro-

bolikc qroc apotele qro pou mno topik moizei (dhlad enai omoiomor-

fikc) me to dianusmatik qro R(dimV )1.

Apdeixh: Den ja apodexoume ed thn Prtash 1 (o endiafermenoc

anagnsthc mpore na bre thn apdeixh se poll bibla diaforikc gewmetr-

ac, pou meletntai diaforsimec pollaplthtec), all ja thn exhgsoume:

jewrntac ti dimV = n + 1, prgma pou shmanei pwc an epilxoume mia

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bsh, qreiazmaste n+ 1 paramtrouc (suntetagmnec) gia na perigryoumeta diansmata tou V , tte qreiazmaste mia suntetagmnh ligterh gia naperigryoume ta shmea x tou proboliko qrou P (V ) exaitac thc eleuje-rac pou uprqei sthn epilog tou antiproswpeutiko diansmatoc (la ta

bajmwt pollaplsia enc antiproswpeutiko diansmatoc apotelon ep-

shc isodnama antiproswpeutik diansmata), me dedomno ti o probolikc

qroc wc pollaplthta, toulqiston topik, apotele epshc dianusmatik

qro. 'Ara an dimV = n+ 1, tte dimP (V ) = n.

An dimP (V ) = 1 tte o probolikc qroc P (V ) lgetai probolikeujea (profanc tte dimV = 2).

An dimP (V ) = 2 tte o probolikc qroc P (V ) lgetai probolikeppedo (profanc tte dimV = 3).

Shmewsh 1: An jewrsoume to snolo twn shmewn tou 2-distatou

epipdou ( tou 3-distatou fusiko qrou antstoiqa) efodiasmno me na

Kartesian Ssthma Suntetagmnwn, tte mporome na tautsoume ta sh-

mea tou epipdou ( tou qrou antstoiqa) me ton pragmatik dianusmatik

qro R2 twn elejerwn dianusmtwn tou epipdou, ( me ton pragmatikdianusmatik qro R3 twn elejerwn dianusmtwn tou qrou antstoiqa),qrhsimopointac to dinusma jshc tou shmeou: Dhlad, kje shmeo anti-

stoiqe sto elejero dinusma pou qei arq thn arq twn axnwn kai prac

to en lgw shmeo. Me autn ton trpo apoktme mia antistoiqa 1-1 kai

ep metax tou sunlou twn shmewn tou gewmetriko epipdou ( tou q-

rou) me ta stoiqea tou pragmatiko dianusmatiko qrou R2 twn elejerwndianusmtwn tou epipdou ( tou pragmatiko dianusmatiko qrou R3 twnelejerwn dianusmtwn tou qrou antstoiqa).

Gia pardeigma, stw ti tautzoume to gewmetrik eppedo me ton prag-

matik dianusmatik qro R2 twn elejerwn dianusmtwn tou epipdou, jew-rntac Kartesianc suntetagmnec sto eppedo me xona tetmhmnwn ton xx,xona tetagmnwn ton yy, arq to shmeo tomc O me suntetagmnec (0, 0) kaiparnontac wc bsh ta gnwst orjokanonik monadiaa diansmata i = (0, 1)kai j = (1, 0). Tte o probolikc qroc P (R2) smfwna me ton orism jaapoteletai ap to snolo twn monodistatwn dianusmatikn upoqrwn tou

R2. Gewmetrik, aut antistoiqe sto snolo twn eujein pou dirqontai apthn arq twn axnwn O. 'Omwc gia na perigryoume mia eujea sto eppedo,apaitetai mno ma parmetroc, gia pardeigma h gwna pou sqhmatzei h en

lgw eujea me ton xona twn tetmhmnwn, sunepc dimP (R2) = 1. Shmei-noume pwc mia eujea tou epipdou apoteletai ap peira (uperarijmsima)

16

shmea tou epipdou all ston probolik qro olklhrh h eujea antistoiqe

se na kai mno shmeo tou proboliko qrou.

Me bsh thn parapnw tatish tou gewmetriko epipdou ( tou fusi-

ko 3-distatou qrou) me ton 2-distato dianusmatik qro R2 ( me ton3-distato dianusmatik qro R3 antstoiqa), parathrome pwc mno oi eu-jeec pou dirqontai ap thn arq twn axnwn O apotelon dianusmatikocupoqrouc distashc 1 tou dianusmatiko qrou R2 ( isodnama tou dia-nusmatiko qrou R3), diti oi llec eujeec pou uprqoun sto eppedo (sto qro antstoiqa) all den pernon ap thn arq twn axnwn O den apo-telon dianusmatikoc upoqrouc diti den periqoun to mhdenik dinusma

(pou enai to oudtero stoiqeo thc prsjeshc metax twn dianusmtwn) kai

pou tautzetai me thn arq twn axnwn O. To parapnw enai mia shmantikleptomreia diti to men snolo twn dierqmenwn ap thn arq twn axnwn

eujein tou epipdou, kai pou ex orismo apotele ton probolik qro P (R2),ja dome pwc apotele pollaplthta distashc 1 pou enai omoiomorfik me

ton kklo S1, en to snolo genik lwn twn eujein tou epipdou, pwc jadome sthn topologa twn epifanein sto mjhma thc Diaforikc Gewmetr-

ac, apotele (mh-prosanatolsimh, pollapl sunektik, lea) pollaplthta

distashc 2 pou enai omoiomorfik me th dsmh Mobius.

Anafroume tloc pwc o probolikc qroc P (R2) mpore isodnama to-pologik na jewrhje kai wc o qroc phlko pou prokptei ap to snolo

lwn twn eujein tou epipdou an to diairsoume me th sqsh isodunamac

pou orzetai ap thn parallhla metax twn eujein tou epipdou, dhlad o

probolikc qroc P (R2) mpore na tautisje kai me to qro twn dieujnsewntou epipdou. Upenjumzoume pwc sthn topologa h nnoia tou omoiomorfi-

smo enai ma ap tic basikc sqseic isodunamac: do (topologiko) qroi

lgontai omoiomorfiko en uprqei matax touc mia apeiknish pou na enai

1-1, ep, suneqc kai h antstrof thc (pou uprqei afo h arqik enai 1-1

kai ep) enai epshc suneqc.

3.3 Upqwroi tou Proboliko Qrou

Gia na orsoume (grammikoc) upqwrouc enc proboliko qrou P (V )xekinme ap na dianusmatik upqwro U V . Afo to dio to U apo-tele dianusmatik qro, mporome na orsoume ton probolik qro P (U)pou apoteletai ap touc monodistatouc dianusmatikoc upqwrouc tou U .O probolikc qroc P (U) apotele na grammik upqwro tou probolikoqrou P (V ), dhlad an U V , tte P (U) P (V ).

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3.4 Basikc Idithtec Probolikn Qrwn

Kpoiec idithtec twn probolikn qrwn enai smfwnec me th diasjhs

mac, pwc aut pou perigrfei h paraktw prtash (h opoa isqei kai sthn

Eukledeia gewmetra):

Prtash 1. Uprqei monadik probolik eujea pou ennei kje zegoc

ap do tuqaa diaforetik shmea enc proboliko qrou P (V ).

Apdeixh: 'Estw x, y P (V ) me x 6= y do diaforetik shmea tou pro-boliko qrou P (V ). Epilgoume antiproswpeutik diansmata v V gia tox kai w V gia to y. Afo x 6= y tte to w den enai bajmwt pollaplsiotou v, me lla lgia ta diansmata v, w enai grammikc anexrthta kaisunepc pargoun na dianusmatik upqwro U V me dimU = 2. TtedimP (U) = 1 kai ex orismo ta shmea x, y P (U), sunepc h P (U) enai hzhtomenh probolik eujea.

'Estw tra P (U ) ma llh probolik eujea pou pern ap ta shmeax, y P (U ). Tte afo dimP (U ) = 1 petai ti dimU = 2 kai o dianu-smatikc upqwroc U V epshc periqei ta antiproswpeutik diansmatav, w twn shmewn x, y antstoiqa. Afo mwc ta diansmata v, w U , ttekai loi oi grammiko sunduasmo touc epshc ja ankoun ston U (diti toU wc dianusmatikc upqwroc tou V ja enai kleist wc proc tic prxeictou dianusmatiko qrou prsjesh kai bajmwt pollaplasiasm). 'Omwc ta

diansmata v, w pargoun to dianusmatik upqwro U V opte U U .'Omwc afo U U kai dimU = dimU , petai ti U = U , opte h probolikeujea P (U) enai monadik.

Antjeta, h paraktw idithta-kleid twn probolikn qrwn den isqei

sthn Eukledeia gewmetra:

Prtash 2. Se na probolik eppedo kje zeugri probolikn eujein

tmnetai, dhlad den uprqei h nnoia thc parallhlac sthn probolik gew-

metra.

Apdeixh: 'Estw P (V ) to probolik eppedo, dhlad dimP (V ) = 2,opte dimV = 3 kai stw P (U1) kai P (U2) do tuqaec diaforetikc probo-likc eujeec tou P (V ). 'Eqoume dhlad dimP (U1) = dimP (U2) = 1 opteU1, U2 V kai dimU2 = dimU2 = 2. Jewrome to euj jroisma U1 + U2.Profanc U1 U1 + U2 opte dimU1 dim(U1 + U2).

18

'Estw ti isqei isthta, dhlad dimU1 = dim(U1 + U2). Tte mwcU1 = U1 + U2 opte kje dinusma tou U2 ja ankei kai ston U1 (h u1 su-nistsa mhdenzetai), opte U2 U1. Afo dimU1 = dimU2 = 2, aut jashmanei ti U1 = U2 opte kai P (U1) = P (U2), topo, diti upojsame dia-foretikc probolikc eujeec.

Sunepc anagkastik ja prpei dimU1 < dim(U1+U2). Ja qoume loipn2 = dimU1 < dim(U1 + U2). 'Omwc h distash twn dianusmatikn qrwnenai fusikc arijmc, ra ja prpei dim(U1 + U2) 3. 'Omwc U1 + U2 V kai dimV = 3, opte dim(U1 + U2) = 3. Efarmzontac th sqsh pousundei th distash thc tomc me th distash tou eujwc ajrosmatoc twn

dianusmatikn upoqrwn (Prtash 3.1.1), parnoume:

dim(U1 + U2) = dimU1 + dimU2 dim(U1 U2)

An antikatastsoume tic timc dimU1 = dimU2 = 2 kai dim(U1 + U2) =3 sthn parapnw sqsh ja proume ti dim(U1 U2) = 1. 'Ara loipnuprqei monadikc dianusmatikc upqwroc tou V me distash 1, o U1U2, oopooc periqetai ston U1 kai ston U2. Dhlad uprqei monadik shmeo pouankei sthn probolik eujea P (U1) all kai sthn probolik eujea P (U2)kai to shmeo aut enai o probolikc qroc P (U1 U2) pou qei distash 0afo dim(U1 U2) = 1, ar prkeitai gia na kai mno shmeo. Sunepc oiprobolikc eujeec P (U1) kai P (U2) tmnontai.

3.5 Omogenec kai Anomogenec Suntetagmnec

Enai nomzoume anagkao se aut to shmeo na katanosoume kalte-

ra th sqsh metax eujein kai epipdwn ston probolik qro me tic gnw-

stc mac eujeec kai ta eppeda sto fusik 3-distato qro. Ja exetsoume

prta th genik perptwsh enc tuqaou dianusmatiko qrou V distashcdimV = n+ 1 kai sth sunqeia ja dome thn eidik perptwsh pou V = R3(pou tautzetai me to fusik 3-distato qro).

Epilgoume mia bsh {v0, v1, v2, ..., vn} tou V . Kje dinusma v V qeisuntetagmnec (0, 1, 2, ..., n) pou

v =n

i=0

ivi (3)

Ta antiproswpeutik diansmata twn shmewn tou proboliko qrou P (V )enai aut twn opown lec oi suntetagmnec den enai mhdn (V 3 v 6= 0).

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Epeid o probolikc qroc P (V ) me dimP (V ) = n apotele pollaplthta,dhlad qro pou topik moizei me to dianusmatik qro Rn, mporome, tou-lqiston topik, na anaparistome ta shmea tou me na snolo suntetagm-

nwn (pou den enai lec mhdn), qontac up' yin ti an 6= 0, tte kje snolosuntetagmnwn thc morfc (0, 1, 2, ..., n) anaparist to dio shmeopou anaparist kai to snolo twn suntetagmnwn (0, 1, 2, ..., n). Oi sun-tetagmnec autc onomzontai omogenec suntetagmnec. Dhlad oi omogenec

suntetagmnec anapariston shmea tou P (V ) qrhsimopointac mwc ousia-stik tic suntetagmnec twn antiproswpeutikn dianusmtwn tou V .

Jewrome tra ta shmea x P (V ) pou anaparistntai ap tic omogenecsuntetagmnec (0, 1, 2, ..., n) me 0 6= 0. Kje ttoio shmeo kajorzetaisafc kai ap tic suntetagmnec (1, 1/0, 2/0, ..., n/0), dhlad ap n-wcproc to pljoc suntetagmnec (x1, x2, ..., xn) pou jsame

xi :=i0.

Oi suntetagmnec autc onomzontai anomogenec suntetagmnec tou proboli-

ko qrou P (V ).

Ac dome merik paradegmata: stw V = R4. Tte afo dimP (R4) = 3,ta shmea auto ja anaparstantai ap tic anomogenec suntetagmnec ~x =(x1, x2, x3). Mia (probolik) eujea ston P (R4) apoteletai ap to snolotwn mh-mhdenikn dianusmtwn me omogenec suntetagmnec a(0, 1, 2, 3)+b(0, 1, 2, 3) = 0, me a, b R. An 0, 0 6= 0, mporome na qrhsimopoi-soume anomogenec suntetagmnec

xi =i0kai

yi =i0,

pou i = 1, 2, 3, opte h probolik eujea enai to snolo twn mh-mhdenikndianusmtwn tou R4 me anomogenec suntetagmnec (a + b, ax1 + by1, ax2 +by2, ax3 + by3) h opoa se anomogenec suntetagmnec apoteletai ap dian-smata thc morfc

(a

a+ b)~x+ (

b

a+ b)~y = ~x+

b

a+ b(~y ~x).

H parapnw sqsh gia a, b R ddei thn exswsh eujeac metax twn shmewnme diansmata jshc ~x = (x1, x2, x3) kai ~y = (y1, y2, y3).

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'Ena (probolik) eppedo ston P (R4) tmnei ta shmea me 0 6= 0 se nasunhjismno eppedo. Sunepc to snolo twn shmewn X P (R4) me 0 6=0 enai na antgrafo tou R3 pou oi probolikc eujeec kai ta probolikeppeda tmnoun to X se sunhjismnec eujeec kai eppeda. O plrhc 3-distatoc qroc P (R4) apokttai an sumperilboume kai ta shmea me 0 =0. Afo ta antiproswpeutik diansmata (0, 1, 2, 3) suniston nan 3-distato dianusmatik upqwro tou R4, o antstoiqoc probolikc qroc jaqei distash 2. 'Ara P (R4) = R3

P (R3) pou to smbolo

dhlnei thn

xnh nwsh (disoint union) metax twn sunlwn.

3.6 Paradegmata Probolikn Qrwn

Ac dome kpoia paradegmata pragmatikn kai migadikn probolikn q-

rwn stic diforec diastseic.

1. dimRV = 1'Estw ti o pragmatikc dianusmatikc qroc V qei distash 1. Tte oV enai ismorfoc me to R, dhlad V = R. Aut shmanei pwc o antstoi-qoc probolikc qroc ja qei dimP (R) = 0. Gia na dome ti shmanei aut:epame pwc o probolikc qroc orzetai wc to snolo twn monodistatwn

dianusmatikn upoqrwn enc dianusmatiko qrou, sunepc o probolikc

qroc P (R) ja apoteletai ap touc monodistatouc dianusmatikoc upoq-rouc tou R. 'Omwc to R qei distash 1, sunepc qei mno na monodistatodianusmatik upqwro, ton eaut tou. 'Ara to P (R) ' , dhlad o proboli-kc qroc P (R) apoteletai ap na kai monadik shmeo pou to sumbolzoumepwc sthn topologa me . H distash auto tou qrou enai prgmati 0.

Enallaktik, mpore kanec na qrhsimopoisei th gewmetrik eikna: afo

V = R, qrhsimopoiome th gewmetrik anaparstash tou R wc mia eujea,lgou qrin ton xona twn tetmhmnwn xx tou epipdou ( tou qrou), pnwsthn opoa qoume orsei na tuqao shmeo wc arq O. O probolikc qrocorzetai wc to snolo twn eujein pou dirqontai ap thn arq O all pouenai tautqrona kai upqwroi tou en lgw dianusmatiko qrou, dhlad thc

diac thc eujeac. Profanc mno mia eujea uprqei me autc tic idithtec, o

xonac xx o dioc, sunepc o probolikc qroc apoteletai ap na kai mnoshmeo.

2. dimRV = 2Kje pragmatikc dianusmatikc qroc distashc do enai ismorfoc me to

dianusmatik qro R2, dhlad se aut thn perptwsh V = R2 en dimP (R2) =1. Ac prospajsoume na perigryoume autn to qro. Kat' arqn efodizou-

21

me to eppedo me na Kartesian Ssthma Suntetagmnwn me arq to shmeo

O. O probolikc qroc P (R2) apoteletai ap to snolo twn monodistatwndianusmatikn upqwrwn tou dianusmatiko qrou R2. Gewmetrik, o pro-bolikc qroc P (R2) tautzetai tso me to qro twn eujein tou epipdoupou dirqontai ap thn arq twn axnwn O, so kai me to qro twn dieujn-sewn sto eppedo. Wc gnwstn, kje eujea tou epipdou pou dirqetai ap

to shmeo O prosdiorzetai monosmanta ap mia parmetro, th gwna pouaut sqhmatzei, gia pardeigma, me ton xona twn tetmhmnwn xx. An me-trme tic gwnec se aktnia, h gwna ja parnei tic timc sto kleist disthma

[0, pi]. 'Omwc h eujea pou sqhmatzei me ton xona xx gwna 0, pou enai odioc o xonac xx, tautzetai me thn eujea pou sqhmatzei me ton xona xxgwna pi, pou enai kai pli o xonac xx. An sto kleist disthma [0, pi],pou topologik enai na eujgrammo tmma, tautsoume ta kra tou, 0 pi,tte to eujgrammo aut tmma ja klesei kai ja sqhmatisje nac kkloc.

Sunepc topologik o probolikc qroc P (R2) enai omoiomorfikc me tonkklo, gegonc pou to sumbolzoume P (R2) ' S1.

Enallaktik, mpore kanec na dei to parapnw qrhsimopointac sunte-

tagmnec: stw (x, y) oi Kartesianc Suntetagmnec tou epipdou pou taut-zetai me to dianusmatik qro R2 kai pazoun to rlo twn omogenn sunte-tagmnwn tou proboliko qrou P (R2). Upenjumzoume ti dimP (R2) = 1.Epilgoume x 6= 0 kai jewrome tic anomogenec suntetagmnec (1, ) stonP (R2), pou jsame := y/x. Ta shmea me (anomogenec) kartesianc sun-tetagmnec (1, ) anapariston msa sto eppedo thn eujea x = 1, mia eujeaparllhlh me ton xona yy pou tmnei ton xona xx sto shmeo me sun-tetagmnec (1, 0). H eujea x = 1 topologik enai omoiomorfik me to R,sunepc o probolikc qroc P (R2) ja periqei na antgrafo tou R, dhla-d R P (R2). Den teleisame mwc, diti ja prpei msa ston probolikqro na sumperilboume kai ta shmea me omogenec suntetagmnec (0, y). Tashmea aut anapariston ton xona yy tou epipdou pou topologik enaiepshc omoiomorfikc me to R. 'Omwc olklhroc o xonac yy ston probolikqro ja dsei na kai mno shmeo, aut me omogenec suntetagmnec (0, y),diti pwc edame P (R) ' . Sunepc o probolikc qroc P (R2) ja enai:

P (R2) ' R

P (R) ' R{} ' S1.H teleutaa sqsh prokptei diti ap thn topologa gnwrzoume pwc o

kkloc apotele th sumpagopohsh enc shmeou thc eujeac. O aploste-

roc trpoc gia na to dei aut kpoioc pou den enai exoikeiwmnoc me thn

topologa enai na skefje wc exc: Jewrome mia eujea kai pnw se autn

epilgoume aujareta na shmeo O. Jewrome nan kklo S1 tuqaac aktnac

22

kai kntrou stw K pou na efptetai sthn eujea sto shmeo O. 'Estw O

to antidiametrik shmeo tou O ston kklo. Mporome na orsoume mia apei-knish 1-1 kai ep metax twn sunlwn S1 {O} R wc exc: stw tuqaoshmeo S tou kklou (diaforetik ap to O). Froume th qord OS toukklou kai thn proektenoume spou na tmsei thn eujea. H OS ja tmseithn eujea se na kai mno shmeo, stw E. Antstrofa, an E na tuqao sh-meo thc eujeac, tte to eujgrammo tmma OE ja tmsei ton kklo se nakai mno shmeo, stw S. Katalabanei kanec pwc to mno shmeo tou kkloupou den qei eikna sthn eujea kat thn parapnw apeiknish enai to dio

to O diti h eujea pou efptetai ston kklo sto shmeo O enai parllhlhme thn eujea pou efptetai ston kklo sto shmeo O afo ta shmea O kaiO enai antidiametrik shmea. (H eikna tou shmeou O kat thn parapnwapeiknish enai to dio to shmeo O miac kai to O ankei kai ston kklo kaisthn eujea). Sunepc lme pwc o kkloc prokptei ap thn eujea me thn

prosjkh enc ep' peiron shmeou, pou enai to shmeo O, isodnama sthntopologa lme ti o kkloc enai h sumpagopohsh enc shmeou thc eujeac.

To parapnw enai kat' analoga me th gnwst stereografik probol metax

tou epipdou kai thc 2-distathc sfarac, mno pou ed qoume mia distash

ligterh.

Paratrhsh 1: O upoyiasmnoc anagnsthc katanoe ap to parapnw

pardeigma to baj topologik nhma thc algebrikc prtashc 3.4.2 per thc

mh parxhc parallhlac ston probolik qro.

3. dimRV = 3Kje pragmatikc dianusmatikc qroc distashc tra enai ismorfoc me to

dianusmatik qro R3, dhlad se aut thn perptwsh V = R3 en dimP (R3) =2. Ac prospajsoume na perigryoume autn to qro:Efodizoume to fusik 3-distato qro me na Kartesian Ssthma Sunte-

tagmnwn me arq to shmeo O. Me bsh th shmewsh 3.2.1 ta shmea toufusiko 3-distatou qrou mporon na tautiston me to dianusmatik q-

ro R3 twn elejerwn dianusmtwn tou qrou. O probolikc qroc P (R3)apoteletai ap to snolo twn monodistatwn dianusmatikn upqwrwn tou

dianusmatiko qrou R3. Gewmetrik o probolikc qroc P (R3) tautzetaime to qro twn eujein tou fusiko 3-distatou qrou pou dirqontai ap

thn arq twn axnwn O, o opooc epshc tautzetai me to qro twn dieujn-sewn sto qro. Wc gnwstn, kje eujea tou qrou pou dirqetai ap to

shmeo O prosdiorzetai monosmanta ap do paramtrouc, gia pardeigmatic gwnec pou aut sqhmatzei me opoiousdpote do ap touc treic xonec

tou Kartesiano Sustmatoc Suntetagmnwn. An metrme tic gwnec se akt-

nia, h kje gwna ja parnei tic timc sto kleist disthma [0, pi], sunepc

23

o qroc twn dieujnsewn ja proljei ap to qro twn paramtrwn pou d-

detai ap to kartesian ginmeno [0, pi] [0, pi]. O qroc twn paramtrwn[0, pi] [0, pi] topologik anaparist na tetrgwno.

'Omwc oi dieujnseic tou qrou pou antistoiqon sto snoro (permetro)

tou tetragnou tautzontai (kat' analoga me aut pou sunbh sthn perptw-

sh thc distashc 2 tou amswc prohgomenou paradegmatoc sta sunoriak

shmea 0 kai pi tou kleisto diastmatoc [0, pi]). Ed mwc ta prgmata me tictautseic twn dieujnsewn ston 3-distato fusik qro enai pio polploka:

Kat' arqn efodizoume to tetrgwno (qroc twn paramtrwn) me na no

Kartesian Ssthma Suntetagmnwn (x, y) me arq mia koruf tou tetrag-nou. Wc proc aut to no Kartesian Ssthma Suntetagmnwn loipn, stw

ti oi korufc tou tetragnou qoun suntetagmnec (0, 0), (pi, 0), (pi, pi) kai(0, pi). Katpin ja prpei na tautiston ta shmea twn do orizntiwn pleu-rn tou tetragnou metax touc wc exc: (x, 0) (pi x, pi), me x [0, pi].Tautqrona mwc ja prpei na tautiston kai ta shmea twn kjetwn pleu-

rn tou tetragnou metax touc wc exc: (0, y) (pi, pi y), me y [0, pi].

Dhlad tautzoume tic apnanti pleurc tou tetragnou an do knontac

tautqrona kje for gia kje zeugri kai mia topologik strof (topologicaltwist).

Aut h epifneia pou prokptei onomzetai epifneia Boy. Enai mialea pollaplthta me distash 2, sumpagc, pollapl sunektik kai mh-

prosanatolsimh. H epifneia aut den mpore na emfuteuje (embedded) ston3-distato qro R3, mpore mno na embaptisje (immersed) ston 3-distatoqro R3 (oi orismo twn ennoin autn thc topologac/gewmetrac uprqounsto Parrthma, sthn pargrafo 17 sto tloc). 'Opwc ja dome sto mjhma

thc Diaforikc Gewmetrac sto sqetik keflaio pou ja meletsoume thn

topologa twn epifanein, h parapnw perigraf msw tou qrou twn para-

mtrwn mac ddei na eppedo montlo gia thn perigraf thc epifneiac Boypou apotele mh-prosanatolsimh epifneia me k = 1 (mh-prosanatolsimh epi-fneia me na cross cap, staurwt skofo).

Ja exhgsoume sth sunqeia giat qreizetai h topologik strof. Gia na

fane aut kajar akoloujome diaforetik trpo skyhc: qrhsimopoiome

sfairikc suntetagmnec kai qwrc blbh thc genikthtac, afo aut den eph-

rezei thn topologa, gia na brome to qro twn dieujnsewn tou 3-distatou

fusiko qrou, ant na jewrsoume to qro twn eujein tou 3-distatou

qrou pou dirqontai ap thn arq twn axnwn O, jewrome to qro twneujgrammwn tmhmtwn mkouc 2 (wc proc kpoia monda mtrhshc) pou dir-

24

qontai ap thn arq twn axnwn O, pou to shmeo O enai sto mson lwnautn twn eujgrammwn tmhmtwn. Faner, ta kra lwn autn twn eu-

jgrammwn tmhmtwn sqhmatzoun mia monadiaa sfara S2 distashc 2 mekntro to O. O qroc twn dieujnsewn den enai aut h monadiaa sfaraditi kje diejunsh (dhlad kje eujgrammo tmma) suneisfrei do shmea

(ta do kra tou) sto sqhmatism autc thc sfarac en emec jloume na

shmeo ap kje diejunsh. 'Ara ja prpei na diairsoume aut thn 2-distath

sfara dia do, dhlad prpei na proume mno to na hmisfario maz me ton

ishmerin kklo pou ja enai to snor thc. Kai pli mwc den teleisa-

me diti ta shmea tou ishmerino kklou (pou antistoiqon sta eujgramma

tmmata pou pernon ap to kntro O kai brskontai pnw sto eppedo touishmerino) ja prpei ek nou na diairejon dia do diti kje eujgrammo

tmma tou ishmerino dskou suneisfrei do shmea (pou enai ta kra tou)

ston ishmerin kklo. Aut h diaresh mwc tra prpei na gnei tautzon-

tac ta antidiametrik shmea tou ishmerino kklou. Ap autn akribc thn

tatish prokptei h topologik strof. To apotlesma enai kai pli mia

epifneia Boy.

Gia lgouc plhrthtac ja anafroume kai mia parametrik parstash thc

epifneiac Boy sto qro R3, th legmenh paramtrhsh Bryant. Dojntocenc migadiko arijmo z C, me |z| 1, (pou |z| to mtro tou migadikoarijmo z, blpe kai thn epmenh pargrafo me thn stereografik probol),jtoume:

g1 = 32Im[

z(1 z4)z6 +

5z3 1],

g2 = 32Re[

z(1 + z4)

z6 +5z3 1],

g3 = Im(1 + z6

z6 +5z3 1)

1

2,

g = g21 + g22 + g

23,

(pou Im kai Re sumbolzoun to fantastik kai to pragmatik mroc ant-stoiqa twn migadikn arijmn), tsi ste

x =g1g,

y =g2g,

z =g3g,

25

na enai oi Kartesianc Suntetagmnec enc tuqaou shmeou thc epifneiac

Boy sto qro R3. H sugkekrimnh parametrik parstash ddei mia ekpe-frasmnh embptish thc epifneiac Boy ston qro R3, msw thc apeikni-shc ~r : D R3, pou orsame parapnw, pou D enai o monadiaoc dskocD := {z C : |z| 1} tou epipdou Argand. Shmeinoume pwc se sumfwname sa epame pio pnw, aut h apeiknish den apotele emfteush diti apo-

tele mno topik omoiomorfism epeid h apeiknish den enai 1-1.

4. dimCV = 1Sthn perptwsh enc migadiko dianusmatiko qrou distashc 1, prokptei

mesa ti P (C) ' .

5. dimCV = 2Sthn perptwsh enc migadiko dianusmatiko qrou distashc 2, (migadik

eppedo), qrhsimopointac th stereografik probol, prokptei ti P (C2) 'C{} ' S2. (Shmeinoume pwc to migadik eppedo C2 qei pragmatikdistash 4 en h sfara S2 me pragmatik distash 2 wc migadik pollapl-thta qei distash 1).

Sqlio 1: Mia genkeush twn probolikn qrwn enai h exc: Dojntoc

enc dianusmatiko qrou V me kpoio tuqao sma F kai me dimV = n,mporome geniktera na orsoume to qro pou sumbolzetai Grn,k(V ), pouoi n, k enai fusiko arijmo n, k N, me 1 k n, wc to snolo twndianusmatikn upoqrwn distashc k tou V . Oi qroi auto lgontai qroiGrassmann kai apotelon kai pli pollaplthtec distashc k(n k). Pro-fanc an k = 1 parnoume ton probolik qro.

Sqlio 2: Genik gia ton upologism pragmatikn probolikn qrwn

(peperasmnhc distashc) isqei o paraktw anadromikc tpoc (n N pe-perasmnoc fusikc arijmc):

P (Rn+1) ' Rn

P (Rn)

Sqlio 3: Mia ap tic shmantikterec anakalyeic thc (diagnwstikc)

Iatrikc enai anamfbola autc tou axoniko kai tou magnhtiko tomogr-

fou. Oi efeurtec touc timjhkan me to brabeo Nompl Iatrikc to 1979

(G.N. Hounsfield-UK, A.M. Cormack-USA) kai to 2003 (P. Mansfield-UK,P.C. Lauterbur-USA) antstoiqa. 'Opwc anafrei o A.M. Cormack, to sh-mantiktero prblhma pou eqan na antimetwpsoun tan majhmatikc fshc,

h exeresh tou antstrofou metasqhmatismo Radon. Qwrc na anafroumeleptomreiec, o metasqhmatismc autc stic 2 diastseic enai o oloklhrwti-

26

kc metasqhmatismc enc (oloklhrwtiko) purna sto qro twn eujein tou

epipdou. Gia pardeigma, en mia eujea anaparstatai wc x cos +y sin = s,pou s h apstash thc eujeac ap thn arq twn axnwn kai h gwna pousqhmatzei h eujea me ton xona yy, tte

R[f ](, s) =

f(x, y)(x cos + y sin s)dxdy.

Stic 3 diastseic enai nac oloklhrwtikc metasqhmatismc miac sugkekrim-

nhc sunrthshc (oloklhrwtikc purnac) sto qro twn eujein tou qrou,

dhlad to pedo orismo tou oloklhrwtiko metasqhmatismo enai mia epi-

fneia Boy. To prblhma sthn exeresh tou antstrofou metasqhmatismoRadon phgzei akribc ap to gegonc ti h epifneia Boy, lgw thc topo-logac thc, den mpore na emfuteuje ston R3 all mno na embaptisje. Heplush autc thc duskolac ousiastik fere to brabeo Nompl Iatrikc

stouc A.M. Cormack, G.N. Hounsfield.

3.7 Parrthma: Stereografik Probol

Gia eukola ston anagnsth, parajtoume ta basik stoiqea thc ste-

reografikc probolc tou Riemann metax thc 2-distathc sfarac kai touepipedou Argand.

Jewrome Kartesianc Suntetagmnec (x, y) sto eppedo (2 diastseic)me xona tetmhmnwn ton xx, xona tetagmnwn ton yy kai arq to shmeo O.An tautsoume ton xona twn tetmhmnwn me touc pragmatikoc arijmoc kai

ton xona twn tetagmnwn me touc fantastikoc arijmoc, tte kje shmeo

stw A tou epipdou me suntetagmnec A(x, y) mpore na tautiste me to miga-dik arijm = x+ iy, pou i h fantastik monda, me i2 = 1. Antstrofa,se kje migadik arijm C mporome na apeikonsoume na shmeo touepipdou A me tetmhmnh to pragmatik mroc

lo thc sfarac antstoiqa. Mporome na orsoume nan omoiomorfism metax

tou sunlou S2 {B} kai tou epipdou Argand, pou lgetai stereografikprobol tou Riemann wc exc: jewrome ti h sfara efptetai sto eppedosto shmeo N , opte apeikonzoume to shmeo N thc sfarac sto shmeo touepipdou pou aut efptetai. Katpin, stw A tuqao shmeo thc sfarac(diaforetik ap to B pou qei exaireje kai diaforetik kai ap to N giato opoo milsame dh). Antistoiqome to shmeo A thc sfarac se ekenoto monadik shmeo tou epipdou pou h proktash thc eujeac BA tmnei toeppedo. Antstrofa, kje shmeo P tou epipdou antistoiqe sto monadikshmeo thc sfarac pou h eujea PB tmnei th sfara.

Ac dome tic analutikc ekfrseic twn parapnw apeikonsewn: stw ti

qoume na Kartesian Ssthma Suntatagmnwn (x, y, z) ston 3-distatoqro (pou ton tautzoume me to dianusmatik qro R3), me arq to shmeo O.'Estw S2 = {(x, y, z) R3|x2+y2+z2 = 1} h monadiaa sfara 2-diastsewnemfuteumnh ston 3-distato qro (pou lgetai sfara Riemann). Jewrometi h arq twn axnwn O tautzetai me to kntro thc monadiaac sfarac opteo breioc ploc ja qei suntetagmnec B(0, 0, 1). Orzoume thn apeiknishp : S2 {B} C me tpo:

p(x, y, z) :=x+ iy

1 zApodeiknetai ti h p apotele nan omoiomorfism (dhlad enai 1-1, ep,suneqc kai h antstrof thc enai epshc suneqc). H antstrofh apeiknish

p1 : C S2 {B}( R3) qei analutik tpo:

p1() := ( + i

1 + ||2 ,

i(1 + ||2) ,||2 11 + ||2 )

pou o suzugc migadikc tou kai || to mtro tou migadiko arijmo ,dhlad an = x+ iy, tte = x iy kai ||2 = = x2 + y2.

Shmeinoume tloc pwc an qrhsimopoisoume sfairikc suntetagmnec

(, ) gia na perigryoume th monadiaa sfara, me , R pou [0, pi]kai [0, 2pi], tte h analutik kfrash thc stereografikc probolc qeith morf

= eicot(

2)

pou wc gnwstn ei = cos+ isin.

28

4 Proboliko Metasqhmatismo

4.1 Basiko Orismo

Xekinme me kpoia stoiqea ap th grammik lgebra:

Orismc 1. 'Estw V kai W do dianusmatiko qroi (peperasmnhcdistashc) me to dio sma F. Mia apeiknish T : V W ja lgetaigrammik apeiknish ( grammikc metasqhmatismc) an isqei to exc:

T (v1 + v2) = T (v1) + T (v2)

v1, v2 V kai , F.

O purnac thc T orzetai wc to snolo Ker(T ) := {v V |Tv = 0} enh eikna thc T orzetai wc to snolo Im(T ) := {w W |w = Tv}.

Isqei h exc basik prtash ap th jewra twn dianusmatikn qrwn

peperasmnhc distashc:

Prtash 1. 'Estw T : V W mia grammik apeiknish pwc parapnw.Tte to snolo Ker(T ) V apotele dianusmatik upqwro tou V , tosnolo Im(T ) W apotele dianusmatik upqwro touW en oi diastseictouc ikanopoion th sqsh

dim[Ker(T )] + dim[Im(T )] = dimV.

En Ker(T ) = 0, dhlad an o dianusmatikc upqwroc Ker(T ) V pe-riqei mno to mhdenik dinusma, kai an dimV = dimW , tte h grammikapeiknish T enai antistryimh, dhlad h T enai 1-1 kai ep, opte uprqeikai h antstrofh grammik apeiknish T1 : W V pou enai epshc 1-1 kaiep.

'Estw T : V W mia grammik apeiknish kai stw v V kpoio tuqaodinusma. Sumbolzoume me Vv V to monodistato dianusmatik upqwrotou V pou pargetai ap to dinusma v. Jewrome thn eikna T (Vv) Wtou Vv msa stonW msw thc T . Tte an Tv 6= 0, tte to T (Vv) apotele tomonodistato dianusmatik upqwro WTv W tou W pou pargetai ap todinusma Tv W . Sunepc h T apeikonzei touc monodistatouc dianusma-tikoc upqwrouc tou V pou pargontai ap ta diansmata tou V pou denankoun ston purna thc T , stouc antstoiqouc monodistatouc dianusmati-koc upqwrouc tou W , sunepc orzei mia na apeiknish:

: P (V ) P (KerT ) P (W )

29

An h T enai antistryimh, tte KerT = 0 opte h orzetai se olklhroton probolik qro P (V ) kai ddei mia amfeiknish (dhlad mia apeiknishpou enai 1-1 kai ep):

: P (V ) P (W )Smfwna me ta parapnw loipn qoume ton akloujo orism:

Orismc 2. 'Enac probolikc metasqhmatismc ap ton probolik qro

P (V ) ston probolik qro P (W ) enai h apeiknish : P (V ) P (W ) pouorzetai pwc perigryame pio pnw msw enc antistryimou grammiko me-

tasqhmatismo T : V W .

Shmewsh 1: Faner, kje mh-mhdenik pollaplsio T , pou F,enc antistryimou grammiko metasqhmatismo T orzei ton dio probolikmetasqhmatism . Ekola apodeiknetai kai to antstrofo.

Paradegmata:

1. Metasqhmatismo Mobius thc probolikc eujeac P (R2). 'Estw T : R2 R2 o antistryimoc grammikc metasqhmatismc o opooc wc proc th sunjhbsh tou R2 qei tpo: T (0, 1) = (a0 + b1, c0 + d1), pou (0, 1) hanaparstash wc proc th sunjh bsh enc tuqaou diansmatoc tou R2,kai pou a, b, c, d R me ad bc 6= 0. Qrhsimopointac thn anomogensuntetagmnh x := 1

0parnoume ton antstoiqo probolik metasqhmatism

: P (R2) P (R2) me tpo:

(x) =a+ bx

c+ dx

pou onomzetai grammikc klasmatikc metasqhmatismc metasqhmatismc

Mobius. O parapnw metasqhmatismc emfanzetai suqn sta majhmatik(gia pardeigma sth migadik anlush).

2. Proboliko Metasqhmatismo tou proboliko epipdou P (R3). 'EstwT : R3 R3 o antistryimoc grammikc metasqhmatismc o opooc wc procth sunjh bsh tou R3 anaparstatai me th morf tou paraktw 3 3 anti-stryimou pnaka:

T =

a0 a1 a2b0 b1 b2c0 c1 c2

me mh-mhdenik orzousa detT 6= 0. 'Estw (0, 1, 2) kpoio ssthma omo-genn suntetagmnwn tou proboliko epipdou P (R3). Qrhsimopointac tic

30

anomogenec suntetagmnec x1 :=10kai x2 :=

20parnoume ton antstoiqo

probolik metasqhmatism : P (R3) P (R3) me tpo:

(x1, x2) = (b0 + b1x1 + b2x2a0 + a1x1 + a2x2

,c0 + c1x1 + c2x2a0 + a1x1 + a2x2

).

4.2 Shmea se Genik Jsh

O paraktw orismc enai qrsimoc sth melth twn probolikn metasqh-

matismn:

Orismc 1. 'Estw P (V ) pragmatikc probolikc qroc me dimP (V ) = n(ra dimV = n + 1). 'Ena snolo ap kpoia (n + 2)-wc proc to pljocshmea tou P (V ) ja lme ti brskontai se genik jsh an kje uposnoloapotelomeno ap (n+1)-wc proc to pljoc shmea ex autn qoun antipro-swpeutik diansmata pou enai grammik anexrthta.

Paradegmata:

1. Kje trida diaforetikn metax touc shmewn thc probolikc eujeac

P (R2) brskontai se genik jsh. H apdeixh tou isqurismo enai h exc:'Estw ti isqei to antjeto, dhlad stw (x1, x2, x3) mia trida diaforeti-kn metax touc shmewn thc probolikc eujeac pou de brskontai se genik

jsh (pou kje shmeo anaparstatai msw thc antstoiqhc anomogenoc

suntetagmnhc tou). Tte do ex autn, gia pardeigma aut me anomoge-

nec suntetagmnec x1 kai x2, ja qoun antiproswpeutik diansmata v1 kaiv2 antstoiqa pou enai grammik exarthmna, dhlad to na ja enai bajmwtpollaplsio tou llou, stw v1 = v2 gia kpoio R. Tte mwc ta v1 kaiv2 ja brskontai ston dio monodistato dianusmatik upqwro tou R2 opteston probolik qro P (R2) ja prpei na anapariston to dio shmeo, topo,diti ex arqc upojsame ti ta tra shmea enai metax touc la diaforetik.

2. Kje tetrda diaforetikn metax touc shmewn tou proboliko epi-

pdou P (R3) brskontai se genik jsh an kami trida ex autn den enaisuneujeiak shmea, dhlad ta shmea den brskontai sthn dia probolik

eujea tou proboliko epipdou. (Apdeixh anlogh me to parapnw par-

deigma).

H qrhsimthta tou parapnw orismo sunstatai sto paraktw jerhma:

Jerhma 1. 'Estw x0, x1, ..., xn+1 na snolo ap (n+2)-wc proc to pl-joc shmea enc pragmatiko proboliko qrou P (V ) me dimP (V ) = n (kai

31

ra dimV = n+1) pou brskontai se genik jsh. 'Estw epshc y0, y1, ..., yn+1na snolo ap (n + 2)-wc proc to pljoc shmea enc llou pragmatikoproboliko qrou P (W ) me dimP (W ) = n (kai ra dimW = n + 1) poubrskontai epshc se genik jsh. Tte uprqei monadikc probolikc meta-

sqhmatismc : P (V ) P (W ) me(xi) = yi

gia la ta i = 0, 1, ..., n+ 1.

Apdeixh: 'Estw vi V , ta antstoiqa antiproswpeutik diansmatatwn shmewn xi P (V ), pou o dekthc i parnei tic timc i = 0, 1, ..., n + 1.Epeid ta (n+ 2)-wc proc to pljoc shmea aut brskontai se genik jsh,ap ton orism petai ti opoiadpote (n + 1)-wc proc to pljoc shmea exautn ja qoun antstoiqa antiproswpeutik diansmata pou enai grammik

anexrthta. Gia pardeigma epilgoume ta grammik anexrthta antiprosw-

peutik diansmata v0,v1,...,vn. Aut ta (n+ 1)-wc proc to pljoc grammikanexrthta diansmata mwc apotelon mia bsh gia to dianusmatik qro

V diti af' enc enai grammik anexrthta, af' etrou enai se pljoc t-sa sa kai h distash tou V . 'Ara kje dinusma tou V , sunepc kai toantiproswpeutik dinusma vn+1, ja grfetai wc grammikc sunduasmc twndianusmtwn thc bshc, dhlad ja qoume

vn+1 =n

i=0

ivi

gia kpoia i R. Sthn parapnw sqsh ja prpei i 6= 0 gia la ta i =0, 1, ..., n diti sthn antjeth perptwsh pou ja eqame i = 0 gia kpoio i = k,tte ja eqame to en lgw antiproswpeutik dinusma vk wc grammik sundua-sm twn upolopwn antiproswpeutikn dianusmtwn v0, v1, ..., vk1, vk+1, ..., vnpou enai topo lgw thc genikc jshc twn arqikn shmewn. Sunepc kje

bajmwt pollaplsio ivi apotele epshc antiproswpeutik dinusma (afoenai la mh-mhdenik).

Mporome sunepc na epilxoume antiproswpeutik diansmata ttoia

ste

vn+1 =n

i=0

vi

dhlad me i = 1, i = 0, 1, ..., n. Aut h antiprospeush enai monadik ditian sque

vn+1 =n

i=0

ivi

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gia kpoia i R me i = 0, 1, ..., n, tte afairntac tic do autc parapnwsqseic kat mlh parnoume

0 =n

i=0

(i 1)vi

Ap th genik jsh twn arqikn shmewn mwc, ta antiproswpeutik dia-

nsmata v0,v1,...,vn enai grammik anexrthta, sunepc i 1 = 0, i =0, 1, ..., n, opte i = 1, i = 0, 1, ..., n.

Anlogoi isqurismo isqoun gia to dianusmatik qro W .

'Ara epilgoume antiproswpeutik diansmata vi V me i = 0, 1, ..., n+ 1gia ta shmea xi P (V ) pou ikanopoion thn

vn+1 =n

i=0

vi

kai epilgoume epshc antiproswpeutik diansmata wi W me i = 0, 1, ..., n+1 gia ta shmea yi P (W ) ta opoa epshc ikanopoion thn

wn+1 =n

i=0

wi.

Orzoume katpin th grammik apeiknish T : V W wc exc:

T (n

i=0

ivi) :=n

i=0

iwi

Ap th grammikthta thc T o parapnw orismc shmanei ti Tvi = wi giai = 0, 1, ..., n en

Tvn+1 = T (n

i=0

vi) =n

i=0

wi = wn+1

To gegonc ti h grammik apeiknish T apeikonzei ta diansmata thc bshcv0,v1,...,vn tou dianusmatiko qrou V sta diansmata thc bshc w0,w1,...,wntou dianusmatiko qrou W shmanei pwc h T enai nac isomorfismc me-tax twn dianusmatikn qrwn V kai W (apeiknish 1-1 kai ep), opte e-nai kai antistryimh. Sunepc o antstoiqoc probolikc metasqhmatismc

: P (V ) P (W ) pou prokptei ap ton parapnw grammik metasqhmati-sm T : V W apeikonzei ta shmea xi P (V ), i = 0, 1, ..., n+1, sta shmea

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yi P (W ), i = 0, 1, ..., n + 1, dhlad isqei (xi) = yi gia i = 0, 1, ..., n + 1ikanopointac tic sunjkec tou jewrmatoc.

Gia na apodexoume th monadikthta tou ergazmaste wc exc: stwh antistryimh grammik apeiknish T : V W pou orzei ton probolikmetasqhmatism : P (V ) P (W ) me (xi) = yi, pou i = 0, 1, ..., n + 1.Tte mwc gia na qoume kai pli (xi) = yi, gia i = 0, 1, ..., n + 1, japrpei T (vi) = ivi gia i = 0, 1, ..., n + 1 gia kpoia mh-mhdenik i, giai = 0, 1, ..., n+ 1. 'Omwc tte

wn+1 =n

i=0

in+1

wi

diti ta w0, w1, ..., wn+1 enai grammik exarthmna. 'Omwc ap th monadikthtathc kfrashc

wn+1 =n

i=0

wi

petai ti n+1 = i = R. 'Ara T = T opte oi T kai T orzoun tondio probolik metasqhmatism .

4.3 Proboliko Metasqhmatismo kai Sqdio

Epanerqmaste se aut thn pargrafo sto praktik prblhma pou an-

timetpize o floc mac o zwgrfoc (kallitqnhc) thc Anagnnhshc gia na

dexoume ti tan zwgrafzoume mia diaforetik ktoyh enc antikeimnou,

ousiastik aut pou epiteletai enai nac probolikc metasqhmatismc.

Arqik tautzoume to fusik 3-distato qro me to dianusmatik qro

R3. 'Estw ti o zwgrfoc (akribstera to mti tou zwgrfou) brsketaise kpoio shmeo x R3 tou qrou kai jlei na zwgrafsei mia ktoyhenc antikeimnou se na kamb pou anaparstatai majhmatik me na eppedo

pi R3 me x / pi (profanc to mti tou zwgrfou de brsketai pnw stonkamb). Gia eukola jewrome ti la ta parapnw (kallitqnhc, antikemeno

pou jlei na zwgrafsei all kai o kambc) brskontai msa ston probolik

qro P (R4). Aut enai efikt giat ap to Sqlio 3.6.2 gnwrzoume pwcP (R4) = R3

P (R3), sunepc R3 P (R4).

Afo loipn x R3 kai R3 P (R4), petai ti kai x P (R4), sune-pc to shmeo x orzei na monodistato dianusmatik upqwro Vx R4 medimVx = 1, en to eppedo pi orzei nan 3-distato dianusmatik upqwro

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Vpi R4 me dimVpi = 3 (tsi ste o probolikc qroc P (R4) P (Vpi) = pikai dimP (Vpi) = 2). Afo x / pi, tte VxVpi = {0}, opte dim(VxVpi) = 0.

Jewrome sth sunqeia to euj jroisma Vx+Vpi to opoo, efarmzontacton tpo thc Prtashc 3.1.1, qei distash

dim(Vx + Vpi) = [dim(Vx)] + [dim(Vpi)] [dim(Vx Vpi)] = 1 + 3 0 = 4

opte ap to Jerhma 3.1.1 parnoume ti

R4 = Vx + Vpi(ant gia to smbolo tou isomorfismo suqn qrhsimopoiome apl to sm-

bolo thc isthtac).

Sunepc, kje dinusma v R4 grfetai me monadik trpo sth morfv = vx + vpi me vx Vx kai vpi Vpi antstoiqa.

Orzoume th grammik apeiknish ppi : R4 Vpi me tpo ppi(v) = vpi (pro-bol sthn vpi sunistsa). Kajar Ker(ppi) = Vx diti ta stoiqea tou Vxqontac ex orismo vpi sunistsa 0 apeikonzontai msw thc ppi sto 0. Katsunpeia h ppi orzei nan probolik metasqhmatism

$pi : P (R4) P (Vx) P (Vpi).

Upenjumzoume ti P (Vx) = x, pou antistoiqe sto shmeo pou brsketai tomti tou zwgrfou en P (Vpi) = pi, pou antistoiqe ston kamb tou zwgrfou.

H apeiknish $pi enai ekolo na perigrafe: an y 6= x, tte to y qei naantiproswpeutik dinusma R4 3 v(y) = v(y)x + v(y)pi me v(y)pi 6= 0 (ditian v(y)pi = 0, tte v(y) = v(y)x, dhlad v Ker(ppi), topo). Sunepcto dinusma v(y)pi = v(y) v(y)x ja brsketai ston 2-distato dianusmatikupqwro tou R4 pou pargetai ap ta diansmata v(y) kai v(y)x (afo pwcmlic edame grfetai wc grammikc sunduasmc autn). 'Omwc tte to shme-

o $pi(y), tou opoou to v(y)pi apotele antiproswpeutik dinusma (stoiqeopou petai ap ton orism thc grammikc apeiknishc ppi pou pargei ton pro-bolik metasqhmatism $pi), ja brsketai sthn probolik eujea pou enneita shmea x kai y. 'Omwc to shmeo $pi(y) P (Vpi), sunepc aut enai toshmeo pou zwgrafzei o zwgrfoc ston kamb tou, pou apeikonzei to shmeo

y tou qrou.

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Sth sunqeia kratme stajer th jsh x tou zwgrfou kai allzoumeto eppedo probolc ap to pi se kpoio diaforetik pi (enrgeia pou isodu-name se allag ktoyhc pou jloume na zwgrafsoume to dio antikemeno).

Anloga ja qoume

R4 = Vx + Vpi = Vx + VpiJewrome tra th grammik apeiknish ppi : R4 Vpi me tpo ppi(v) =vpi . Profanc tra kai pli Ker(ppi) = Vx. Jewrome sth sunqeia tonperiorism ppi|Vpi thc ppi me pedo orismo qi olklhro to R4 all mno to Vpi,dhlad jewrome th grammik apeiknish ppi|Vpi : Vpi Vpi . Profanc traafo Vx Vpi = {0}, h grammik apeiknish ppi|Vpi (dhlad o periorismc thcppi sto Vpi) ja enai antistryimh opte orzei ton probolik metasqhmatism

$ : P (Vpi) P (Vpi)Sunepc edame pwc h anaparstash miac diaforetikc ktoyhc enc an-

tikeimnou antistoiqe se nan probolik metasqhmatism.

Paratrhsh 1: To Jerhma 4.2.1 gia thn eidik perptwsh tou proboli-

ko epipdou, mac lei pwc mia tetrda shmewn se genik jsh antistoiqe

msw enc proboliko metasqhmatismo se opoiadpote llh tetrda shme-

wn se genik jsh. An aut sunduaste me to Pardeigma 4.2.2 pou lei ti

kje tetrda shmewn sto probolik eppedo sthn opoa den uprqei kama

suneujeiak trida brskontai se genik jsh, lambnoume wc apotlesma ti

sto probolik eppedo, nac probolikc metasqhmatismc apeikonzei opoia-

dpote tetrda shmewn pou den qei suneujeiak trida se mia opoiadpote

llh tetrda shmewn pou den periqei suneujeiak trida. Me apl lgia

aut shmanei pwc la ta tetrpleura sqmata sto probolik eppedo enai

isodnama msw kpoiou proboliko metasqhmatismo. Gia pardeigma na

tetrgwno msw enc proboliko metasqhmatismo metatrpetai se tuqao

tetrpleuro, dhlad oi probolikoi metasqhmatismo den diathron anallow-

ta ote ta mkh ote tic gwnec.

Gia pardeigma, wc gnwstn o kboc qei 6 drec pou enai tetrgwna.

'Otan mwc anaparistme nan kbo sto eppedo, dhlad tan sqedizoume

mia ktoyh auto sto eppedo, kpoiec drec anaparstantai wc plgia pa-

rallhlgramma, dhlad blpoume sthn prxh nan probolik metasqhmatism

pou den diathre ta mkh kai tic gwnec.

Paratrhsh 2: En efodisoume nan pragmatik dianusmatik qro me

to gnwst Eukledeio eswterik ginmeno, tte gnwrzoume pwc oi orjognioi

metasqhmatismo diathron anallowto to eswterik ginmeno, ra ta mkh,

36

(gi' aut lgontai kai isometrec). Kat' epktash diathron anallowtec kai

tic gwnec (jumhjete ton orism tou sunhmitnou thc gwnac do dianusmtwn

wc to phlko tou eswteriko ginomnou autn dia tou ginomnou twn mtrwn

touc). To snolo twn orjogniwn metasqhmatismn apotele omda.

Uprqei mia llh kathgora grammikn metasqhmatismn oi opooi enai

eurteroi ap touc orjogniouc metasqhmatismoc kai lgontai smmorfoi

metasqhmatismo. Auto diathron anallowtec mno tic gwnec all qi ta

mkh. Gia pardeigma jumhjete thn omoithta twn sqhmtwn, lgou qrin

thn omoithta trignwn sto eppedo se antidiastol me thn isthta trignwn

pou gnwrzoume ap thn Eukledeia gewmetra, h omoithta apotele smmor-

fo metasqhmatism en h isthta apotele orjognio metasqhmatism. Do

sa trgwna enai kai moia all to antstrofo den isqei.

Oi proboliko metasqhmatismo pwc edame, enai akmh eurteroi epeid

de diathron anallowta ote ta mkh ote tic gwnec. Ja dome sth sunqeia

poiec posthtec diathrontai anallowtec ktw ap probolikoc metasqhma-

tismoc en epshc ja dome ti kai oi proboliko metasqhmatismo epshc

apotelon omda. Mia isometra tou epipdou gia pardeigma apeikonzei na

trgwno se na llo trgwno so me to arqik. 'Enac smmorfoc metasqhma-

tismc tou epipdou apeikonzei na trgwno se na llo trgwno moio me to

arqik. 'Enac probolikc metasqhmatismc sto probolik eppedo apeikonzei

na trgwno se na opoiodpote llo trgwno tou proboliko epipdou.

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5 Dusmc

5.1 Dukc Probolikc Qroc kai Arq Dusmo

Xekinme me anaskphsh orismnwn stoiqewn ap th grammik lgebra.

Orismc 1. 'Estw V pragmatikc dianusmatikc qroc me dimV = n+1.O dukc qroc V tou V orzetai wc to snolo twn grammikn apeikonsewnf : V R. Gnwrzoume ap th grammik lgebra ti o V apotele pragmati-k dianusmatik qro me thn dia distash pwc o V , dhlad dimV = dimV

(sunepc dianusmatiko qroi V kai V enai ismorfoi V = V ).

Orismc 2. 'Estw {v0, v1, ..., vn} mia bsh tou V . Tte orzetai h dukbsh {f0, f1, ..., fn} thc {v0, v1, ..., vn} tou V ap th sqsh fi(vj) = ij, pouij to gnwst dlta tou Kronecker, dhlad

fi(vj) = ij =

{1, gia i = j,0, gia i 6= j, pou i, j = 0, 1, ..., n.

Orismc 3. 'Estw V pragmatikc dianusmatikc qroc me dimV = n+1kai stw P (V ) o antstoiqoc probolikc qroc. O probolikc qroc P (V )pou apoteletai ap touc monodistatouc dianusmatikoc upoqrouc tou du-

ko dianusmatiko qrou V tou V onomzetai dukc probolikc qroc.

'Enac monodistatoc dianusmatikc upqwroc tou V apoteletai ap lata bajmwt pollaplsia f , pou R kai f V , miac mh-mhdenikcgrammikc apeiknishc f : V R. Ap th grammikthta thc f petai tiafo enai mh-mhdenik, h eikna thc Im(f) ja enai olklhro to R, sunepcdim[Ker(f)] = dimV dim[Im(f)] = dimV 1 (diti dimRR = 1).

Sunepc to Ker(f) apotele dianusmatik upqwro tou V , dhlad qou-me ti Ker(f) V me dim[Ker(f)] = dimV 1. Faner o dianusmatikcupqwroc Ker(f) tou V enai anexrthtoc ap thn epilog tou antiprosw-peutiko diansmatoc f diti Ker(f) = Ker(f) an 6= 0. 'Ara loipn natuqao shmeo tou P (V ) (me antiproswpeutik dinusma thc morfc f me R kai f V ), anaparist na dianusmatik upqwro tou V distashcdimV 1 (pou enai o Ker(f)), isodnama na tuqao shmeo tou P (V )(me antiproswpeutik dinusma thc morfc f me R kai f V ), anapa-rist na grammik upqwro tou P (V ) distashc dimP (V ) 1 (pou enai oP (Ker(f))).

Antstrofa, an W V me dimW = dimV 1, epilgoume mia bsh

38

{w0, w1, ..., wn1} touW kai thn epektenoume sthn {w0, w1, ..., wn1, wn} pouapotele mia bsh tou V . An {f0, f1, ..., fn} enai h duk thc bsh tou V ,tte ap ton orism thc dukc bshc petai ti W = Ker(fn).

Kje dianusmatikc upqwroc W V me dimW = (dimV )1 prokpteime ton parapnw trpo, sunepc uprqei mia apeiknish 1-1 kai ep metax

twn shmewn tou P (V ) kai tou sunlou twn grammikn upoqrwn tou P (V )me distash dimP (V ) 1.

Shmewsh 1: Genik gia kje qro, oi upqwroi auto me distash kat

na mikrterh ap th distash tou arqiko qrou onomzontai uperepifneiec

upereppeda.

Dhlad h arq tou dusmo mac lei pwc uprqei antistoiqa 1-1 kai ep

metax twn shmewn tou P (V ) kai twn uperepipdwn tou P (V ).

Paradegmata.

1. En dimV = 2, tte epshc ja qoume dimV = 2, en dimP (V ) =dimP (V ) = 1, probolik eujea. Ta shmea thc dukc probolikc eujeacP (V ) antistoiqon se upqwrouc thc probolikc eujeac P (V ) distashcdimP (V ) 1 = 0, dhlad ta shmea thc dukc probolikc eujeac P (V )antistoiqon se shmea thc probolikc eujeac P (V ).

Uprqei sunepc mia apeiknish 1-1 kai ep

: P (V ) P (V )h opoa ousiastik apotele nan probolik metasqhmatism. An epilxoume

mia bsh {v0, v1} tou V kai th duk thc bsh {f0, f1} tou V , tte an V 3f = a0f0 + a1f1 kai V 3 v = 0v0 + 1v1 tuqaa diansmata, ja qoumeap th grammikthta thc f kai ton orism thc dukc bshc fi(vj) = ij tif(v) = 0 0a0 + 1a1 = 0 diti:

f(v) = 0 (a0f0 + a1f1)(v) = 0 a0f0(v) + a1f1(v) = 0 a0f0(0v0 + 1v1) + a1f1(0v0 + 1v1) = 0

a00f0(v0) + a01f0(v1) + a10f1(v0) + a11f1(v1) = 0 a001 + a010 + a100 + a111 = 0 0a0 + 1a1 = 0.Sunepc to dinusma v = a1v0 a0v1 enai na antiproswpeutik dinusmatou Ker(f) diti ap grammikthta qoume:

f(v) = f(a1v0 a0v1) = a1f(v0) a0f(v1)

39

= a1(a0f0 + a1f1)(v0) a0(a0f0 + a1f1)(v1)= a1a0f0(v0) + a1a1f1(v0) a0a0f0(v1) a0a1f1(v1)= a1a01 + a1a10 a0a00 a0a11 = a1a0 a0a1 = 0Sunepc se omogenec suntetagmnec

(a0, a1) = (a1,a0)

pou apotele antistryimo grammik metasqhmatism, ra orzetai o antstoi-

qoc probolikc metasqhmatismc.

2. En dimV = 3, tte epshc ja qoume dimV = 3, en dimP (V ) =dimP (V ) = 2, probolik eppedo. Ta shmea tou duko proboliko epipdouP (V ) antistoiqon se upqwrouc tou proboliko epipdou P (V ) distashcdimP (V ) 1 = 1, dhlad ta shmea tou duko proboliko epipdou P (V )antistoiqon se eujeec tou proboliko epipdou P (V ).

3. En dimV = 4, tte epshc ja qoume dimV = 4, en dimP (V ) =dimP (V ) = 3. Ta shmea tou duko proboliko qrou P (V ) antistoiqonse upqwrouc tou proboliko qrou P (V ) distashc dimP (V ) 1 = 2, dh-lad ta shmea tou duko proboliko qrou P (V ) antistoiqon se eppedatou proboliko qrou P (V ).

O dukc tou duko qrou (V ) ja sumbolzetai aplostera V kai tau-tzetai me to dianusmatik qro V msw enc fusiko isomorfismo, V = V ,pou orzetai qrhsimopointac ton antistryimo grammik metasqhmatism

T : V V me V 3 v 7 Tv V pou orzetai wc exc:

Tv(f) = f(v)

pou v V , Tv V kai f V , dhlad f : V R grammik apeiknish.

Afo V = V , tte ja qoume tatish kai twn antstoiqwn proboliknqrwn P (V ) ' P (V ).

H arq dusmo metax twn probolikn qrwn P (V ) kai P (V ) mac leipwc ta shmea tou P (V ) antistoiqon sta upereppeda tou P (V ). An efar-msoume kai pli thn arq dusmo, aut th for metax twn qrwn P (V )kai P (V ) ja proume ti ta shmea tou P (V ) antistoiqon sta uperep-peda tou P (V ). 'Omwc o P (V ) tautzetai me ton P (V ), sunepc ta shmeatou P (V ) antistoiqon sta upereppeda tou P (V ). An sundusoume tic do

40

parapnw protseic parathrome pwc h arq dusmo sumplhrnetai kai mac

lei to exc:

Ta shmea tou P (V ) antistoiqon me trpo 1-1 kai ep me ta upereppedatou P (V ) kai antstrofa ta shmea tou P (V ) antistoiqon me trpo 1-1 kaiep me ta upereppeda tou P (V ).

5.2 Mhdenistc Dianusmatiko Qrou kai Dusmc

Orismc 1. 'Estw V (pragmatikc) dianusmatikc qroc me dimV =n + 1. Se kje dianusmatik upqwro U V mporome na antistoiqsoumenan dianusmatik upqwro U0 V pou lgetai mhdenistc tou U o opoocorzetai wc exc: U0 := {f V |f(u) = 0,u U}. Dhlad o mhdenistcU0 tou U V apoteletai ap ekenec tic grammikc apeikonseic f : V Rpou apeikonzoun kje dinusma u U sto 0.

An epilxoume mia bsh {u0, u1, ..., ur} tou U V , (stw dimU = r +1), thn epektenoume se mia plrh bsh {u0, u1, ..., ur, ur+1, ..., un} tou V kaijewrome th duk thc bsh {f0, f1, ..., fn} tou V . O mhdenistc U0 pargetaiap ta grammik anexrthta diansmata {fr, fr+1, ..., fn}, opte

dimU + dimU0 = dimV

Oi basikc idithtec tou mhdenist fanontai sthn paraktw Prtash (gia

thn apdeixh blpe kpoio biblo Grammikc 'Algebrac):

Prtash 1. 'Estw V dianusmatikc qroc peperasmnhc distashc,U V kai stw U0 o mhdenistc tou dianusmatiko upoqrou U . Tteisqoun ta exc:

(i) Msw tou fusiko isomorfismo V = V , qoume ti (U0)0 = U .(ii) An U1, U2 V me U1 U2, tte U02 U01 .(iii) (U1 + U2)

0 = U01 U02 .(iv) (U1 U2)0 = U01 + U02 .

Upenjumzoume ti h arq dusmo pou edame sthn prohgomenh par-

grafo aforose thn antistoiqa metax shmewn (upqwroi distashc 0) kaiuperepipdwn (upqwroi me distash kat na mikrterh ap th distash tou

arqiko qrou).

Oi mhdenistc qi mno mporon na dsoun mia enallaktik perigraf thc

arqc tou dusmo, all to pio shmantik enai ti mporon na genikesoun

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thn arq dusmo metax upqwrwn opoiasdpote distashc. Aut epitugq-

netai wc exc:

Edame pwc h nnoia tou mhdenist antistoiqe se kje dianusmatik up-

qwro U V na dianusmatik upqwro U0 V kai kat' epktash sekje grammik upqwro P (U) P (V ) antistoiqe na grammik upqwroP (U0) P (V ). Afo

dimU0 = dimV dimU,

petai ti

dimP (U0) = dimP (V ) dimP (U) 1.An dimP (U) = dimP (V ) 1, tte dimP (U0) = 0 opte to P (U0) enaishmeo tou P (V ). Aut enai h antistoiqa pou edame sthn prohgomenhpargrafo metax shmewn tou P (V ) kai uperepifanein tou P (V ) ekpefra-smnh me qrsh mhdenistn.

Ac dome na geniktero pardeigma pou afor dusm metax upoq-

rwn distashc na: stw dimP (V ) = 3. Tte an dimP (U) = 1, ttedimP (U0) = 1, sunepc oi eujeec tou P (V ) antistoiqon me trpo 1-1 kaiep me tic eujeec tou duko proboliko qrou P (V ).

Sqhmatik h arq dusmo perigrfetai me tic paraktw antistoiqec:

V V dimV = dimV = n+ 1

P (V ) P (V )dimP (V ) = dimP (V ) = n

U V, dimU = k, (0 < k < n+ 1) U0 V , dimU0 = n+ 1 kArq Dusmo: antistoiqa 1-1 kai ep metax P (U) P (U0)

dimP (U) = k 1 dimP (U0) = n k

Ap ton parapnw genik pnaka dusmo, blpoume gia pardeigma pwc

ta shmea tou P (V ) (ta opoa antistoiqon se upoqrouc P (U0) P (V )tou duko proboliko qrou distashc mhdn, dhlad dimP (U0) = 0 n k = 0 n = k), antistoiqon se upoqrouc P (U) P (V ) me distash

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dimP (U) = k 1 = n 1, dhlad se upereppeda tou P (V ), kti pou sum-fwne me ta parapnw.

Ta upereppeda tou P (V ) (ta opoa antistoiqon se upoqrouc P (U0) P (V ) tou duko proboliko qrou distashc n 1, dhlad dimP (U0) =n 1 n k = n 1 k = 1), antistoiqon se upoqrouc P (U) P (V )me distash dimP (U) = k 1 = 1 1 = 0, dhlad se shmea tou P (V ),prgma pou sumfwne me sa qoume dh pe.

Tloc parathrome ti an dimP (V ) = n = 3, tte oi eujeec tou proboli-ko qrou P (V ) pou antistoiqon se upoqrouc P (U) P (V ) me distashdimP (U) = k 1 = 1 k = 2, antistoiqon se upoqrouc P (U0) P (V )tou duko proboliko qrou distashc n k = 1, dhlad se eujeec touduko proboliko qrou.

5.3 Efarmogc Dusmo: Jerhma Desargues, Jerh-ma Pppou

H qrhsimthta thc (genikeumnhc) arqc dusmo enai meglh diti ja

dome pwc gia kje jerhma apoktme automtwc kai th duk morf tou.

'Estw probolikc qroc P (V ), me dimP (V ) = 2, sunepc milme gia toprobolik eppedo. Oi mnoi grammiko upqwroi auto enai ta shmea kai oi

eujeec. Ap ton pnaka dusmo prokptei mesa ti an (P (U) =)x P (V )kpoio shmeo, tte (P (U0) =)x0 P (V ) ja qei distash 1 opte ja apo-tele eujea en an a P (V ) eujea, tte to a0 P (V ) ja qei distashmhdn opte ja apotele shmeo. Ap thn Prtash 5.2.1.(ii) petai ti anx a, tte a0 x0. Omowc, an x, y a do shmea thc eujeac a, ttepetai ti a0 x0 all kai a0 y0, pou shmanei ti h duk eikna miaceujeac a pou periqei ta shmea x, y enai ti oi eujeec x0 kai y0 tmnontaisto shmeo a0. Me bsh ta parapnw ja apodexoume to Jerhma Desargues :

Jerhma 1. 'Estw P,A,A, B,B, C, C ept diaforetik shmea se nanprobolik qro P (V ) ttoia ste oi eujeec AA, BB kai CC na enai dia-foretikc kai sundierqmenec ap to shmeo P . Tte ta shmea tomc R,S, Ttwn zeugn twn eujein AB kai AB, BC kai BC , CA kai C A antstoiqa,enai suneujeiak.

Apdeixh: Epilgoume antiproswpeutik diansmata p, a, a, b, b, c, c

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V gia ta shmea P,A,A, B,B, C, C P (V ) antstoiqa. Ap to Pardeigma4.2.1 gnwrzoume pwc mia trida diaforetikn shmewn se mia eujea brskon-

tai pnta se genik jsh, sunepc mporome na epilxoume antiproswpeutik

diansmata pwc sthn apdeixh tou Jewrmatoc 4.2.1 entc twn trin disdi-

statwn dianusmatikn upoqrwn tou V twn opown oi antstoiqoi probolikoqroi enai oi eujeec PAA, PBB kai PCC ston P (V ). Me lla lgiamporome na epilxoume diansmata a, a, b, b, c, c V tsi ste

p = a+ a

p = b+ b

p = c+ c

Afairntac tic do prtec exisseic kat mlh parnoume 0 = a+abb b a = a b = r, pou r kpoio dinusma. Afo to dinusma r brsketaisto eppedo pou pargoun ta a, b (afo grfetai wc grammikc sunduasmcautn), tte to r anaparist na shmeo R P (V ) pou brsketai sthn eujeaAB, dhlad R AB. 'Omwc to r epshc ankei kai sto eppedo pou pargounta diansmata a, b, sunepc ja enai epshc kai shmeo thc eujeac AB, dh-lad R AB. 'Ara to R enai to monadik shmeo tomc twn eujein ABkai AB.

Entelc anloga afairntac tic do teleutaec exisseic kat mlh pro-

kptei ti 0 = b+ b c c c b = b c = s, pou s kpoio dinusma pouanaparist thn tom twn eujein BC kai BC en afairntac thn prth apthn teleutaa exswsh parnoume 0 = c+caa ac = ca = t, pout kpoio dinusma pou anaparist thn tom twn eujein CA kai C A. 'Omwcr+s+t = ab+bc+ca = 0 opte to r brsketai sto dianusmatik qropou pargoun ta diansmata s, t (diti ta r, s, t enai grammik exarthmna),sunepc to R brsketai pnw sthn eujea ST , opte ta shmea R,S, T enaisuneujeiak.

Gia na proume th duk kdosh tou parapnw jewrmatoc, arke na para-

thrsoume pwc an dimP (V ) = 2, tte ja asqolhjome me to duk probolikqro P (V ) parathrntac pwc tra ta ept shmea ja gnoun ept eujeec,opte ja qoume to Duk Jerhma Desargues :

Jerhma 1'. 'Estw pi, , , , , , ept diaforetikc eujeec se naprobolik eppedo P (V ) (dhlad dimP (V ) = 2) ttoiec ste ta shmea tomctouc an zegh , kai na enai diaforetik kai na brskontaisthn eujea pi. Tte oi eujeec pou ennoun ta shmea tomc kai ,

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kai , kai enai sundierqmenec (tmnontai se na shmeo).

Gia to parapnw Jerhma 1' den apaitetai apdeixh kajc prokptei ap

to Jerhma 1 me efarmog thc arqc tou dusmo. Shmeinoume epshc pwc

sthn perptwsh aut to duk jerhma apotele sthn ousa to antstrofo

jerhma (arqzoume ap thn eujea RST kai prosdiorzoume to shmeo P ).

Eidik perptwsh tou Jewrmatoc Desargues apotele to Jerhma Pp-pou:

Jerhma 2. 'Estw A,B,C kai A, B, C do tuqaec diatetag-mnec tridec suggrammikn shmewn pou brskontai se diaforetikc eujeec

kai antstoiqa. Tte ta shmea tomc twn diastaurwmnwn ensewn (dh-lad ta tra shmea ABAB, AC AC kai BC BC), enai suneujeiak.

Shmewsh 1: Ap thn arq dusmo prokptei mesa ti isqei kai to

duk Jerhma tou Pppou.

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6 Kamplec b' bajmo (Quadrics)

6.1 Digrammikc Morfc

Orismc 1. 'Estw V dianusmatikc qroc me sma F peperasmnhc di-stashc dimFV = n + 1. Mia summetrik digrammik morf B ston V enaimia apeiknish B : V V F ttoia ste:(i) B(v, w) = B(w, v) (h idithta aut anafretai sto qarakthrism summe-trik)

(ii) B(1v1+2v2, w) = 1B(v1, w)+2B(v2, w), v, w V kai 1, 2 F.Apaitome h B na enai grammik mno sthn arister metablht opte lgwthc (i) ja enai grammik kai sth dexi metablht.

H summetrik digrammik morf B ja lgetai mh-ekfulismnh an isqeiB(v, w) = 0 w V v = 0.

To klasik pardeigma miac summetrikc digrammikc morfc enai to gnw-

st eswterik ginmeno twn dianusmtwn tou qrou B : R3 R3 R meB(x ,y ) = x y . An epilxoume na Kartesian Ssthma Suntetagmnwnkai an wc proc aut to ssthma suntetagmnwn ta diansmata qoun sunte-

tagmnec

x = (x1, x2, x3) kaiy = (y1, y2, y3), ttex y = x1y1+x2y2+x3y3.Msw tou eswteriko ginomnou orzontai sth sunqeia ta mkh

|x | :=x x

kai oi gwnec metax twn dianusmtwn

cos(x ,y ) :=x y|x ||y |'Estw T : V V antistryimoc grammikc metasqhmatismc. O T meta-sqhmatzei mia summetrik digrammik morf B pwc h parapnw se mia llhsummetrik digrammik morf BT pou orzetai wc exc:

BT (v, w) = B(Tv, Tw).

An epilxoume mia bsh {v0, v1, ..., vn} tou V , tte h B orzei na summetrikpnaka Bij = B(vi, vj). O summetrikc pnakac Bij qei mh-mhdenik orzousaen kai mno en h summetrik digrammik morf B enai mh-ekfulismnh.

En to dinusma v V wc proc th bsh {v0, v1, ..., vn} qei suntetagmnec(0, 1, ..., n), tte h digrammik morf B kajorzetai ap thn tetragwnik

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morf

B(v, v) =n

i,j=0

Bijij

O kajorismc thc digrammikc morfc msw thc tetragwnikc morfc isqei

mno an to sma F den qei qarakthristik 2, dhlad an 1 + 1 6= 0, pou 0kai 1 ta oudtera stoiqea thc prsjeshc kai tou pollaplasiamo sto smaantstoiqa. Shmeinoume pwc tso to sma R twn pragmatikn arijmn sokai to sma C twn migadikn arijmn den qoun qarakthristik 2.

To paraktw Jerhma jewretai gnwst ap th Grammik 'Algebra kai

ja mac enai qrsimo se sa ja akoloujsoun:

Jerhma 1. 'Estw B(v, v) mia tetragwnik morf se na dianusmatikqro V peperasmnhc distashc me sma F touc pragmatikoc touc miga-dikoc arijmoc. Tte:

(i) An F = C, tte uprqei mia bsh tou V wc proc thn opoa h tetragwnikmorf mpore na grafe wc

B(v, v) = 20 + 21 + ...+

2r

pou r = rank(Bij) h txh tou summetriko pnaka Bij kai profanc r dimCV .(ii) An F = R, tte uprqei mia bsh tou V wc proc thn opoa h tetragwnikmorf mpore na grafe wc

B(v, v) = 20 + 21 + ...+

2r 2r+1 ... 2spou s dimRV h txh kai r (s r) h upograf thc digrammikc morfc.

Shmewsh 1. H tetragwnik morf ( isodnama h summetrik digrammik

morf) enai mh-ekfulismnh en kai mno en r = dimCV (migadik perptw-sh) s = dimRV (pragmatik perptwsh).

Orismc 2. 'Estw V dianusmatikc qroc peperasmnhc distashc kaiP (V ) o antstoiqoc probolikc qroc. Mia tetragwnik kamplh ( kamplhb' bajmo, quadric), ston P (V ) enai to snolo twn shmewn tou probolikoqrou twn opown ta antiproswpeutik diansmata ikanopoion th sqsh

B(v, v) = 0, pou B mia summetrik digrammik morf tou V .

Shmewsh 2. Epeid gia tic tetragwnikc morfc isqei B(v, v) =2B(v, v), petai ti to snolo twn shmewn thc tetragwnikc kamplhc enai

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anexrthto ap thn epilog antiproswpeutikn dianusmtwn.

'Estw T : V V antistryimoc grammikc metasqhmatismc kai :P (V ) P (V ) o antstoiqoc probolikc metasqhmatismc. O probolikcmetasqhmatismc apeikonzei mia tetragwnik kamplh se mia llh tetra-gwnik kamplh diti

B(v, v) = B(T1Tv, T1Tv) = BT1(Tv, Tv)

miac kai BT1(Tv, Tv) = 0 en B(v, v) = 0.

Shmewsh 3. Ap to Jerhma 6.1.1.(i) prokptei ti kje migadik te-tragwnik kamplh msw enc proboliko metasqhmatismo enai isodnamh

me mia tetragwnik kamplh thc morfc

B(v, v) = 20 + 21 + ...+

2r.

H paraktw prtash ousiastik apotele to antstrofo thc Shmewshc

2 parapnw:

Prtash 1. An do summetrikc digrammikc morfc B kai B se na mi-gadik dianusmatik qro V peperasmnhc distashc dimCV = n+1 orzounthn dia tetragwnik kamplh ston probolik qro P (V ), tte B = B, giakpoio C.

Apdeixh: An F = C, tte ap to Jerhma 6.1.1.(i) mporome na upoj-soume ti gia kpoia bsh stw {v0, v1, ..., vn} tou V isqei

0 = B(v, v) = 20 + 21 + ...+

2r (4)

pou to dinusma v V wc proc th bsh {v0, v1, ..., vn} qei suntetagmnec(0, 1, ..., n), me i C gia i = 0, 1, ..., n, en

0 = B(v, v) =n

i,j=0

Bijij (5)

diti en gnei den mporome na upojsoume ti to Jerhma 6.1.1.(i) ika-nopoietai tautqrona gia th B kai th B wc proc thn dia bsh.

[Parapomp: upenjumzoume to exc sqetik ap th jewra pinkwn: do

tetragwniko n n pnakec A, B diagwniopoiontai tautqrona en kai mno

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en metatjentai, dhlad en AB = BA. En oi pnakec diagwniopoiontaitautqrona aut shmanei pwc ja qoun ta dia idiodiansmata, sunepc oi

pnakec apokton diagnia morf wc proc thn dia bsh pou apoteletai ap

ta koin idiodiansmata].

Epilgoume ta antiproswpeutik diansmata vi V twn shmewn thctetragwnikc kamplhc pou orzetai ap thn tetragwnik morf B(v, v) = 0ta opoa (antiproswpeutik diansmata) wc proc th bsh {v0, v1, ..., vn} touV qoun suntetagmnec (0, 1, ..., n) = (1,i, 0, ..., 0) oi opoec profancikanopoion th sqsh 4. An antikatastsoume tic suntetagmnec autc sth

sqsh 5 ja proume:

(+i) : B11 B22 + 2iB12 = 0

(i) : B11 B22 2iB12 = 0ap tic opoec prokptei ti B11 = B22 kai B12 = 0. Epanalambnoume thnparapnw diadikasa metatopzontac diadoqik th jsh thc suntetagmnhc ikai parnoume B11 = B22 = ... = Brr kai Bij = 0 gia i < j r. An r < nparnoume diansmata me suntetagmnec (0, 0, ..., r+1, r+2, ..., n) opte apo-ktme Bij = 0 an i j r (profanc i, j, r n+ 1).

6.2 Paradegmata Kampuln b' bajmo

Prin anaferjome sta Paradegmata, enai skpimo na upenjumsoume ap

th Grammik 'Algebra ti oi summetrikc digrammikc morfc lgontai kai me-

trikc (sth gewmetrik orologa, sunjwc sth gewmetra oi metrikc apaito-

me na enai kai jetik (hmi-)orismnec gia na enai ta mkh pou metrme jetiko

arijmo, allic lgontai yeudometrikc). 'Enac isomorfismc pou epiprsje-

ta diathre mia summetrik digrammik morf ( metrik) anallowth lgetai

isotima sthn orologa twn pinkwn ( isometra sth gewmetrik orologa-gia

pardeigma oi strofc sto eppedo kai sto qro apotelon klasik paradeg-

mata isometrin diti den metablloun ta mkh twn dianusmtwn).

Stouc pragmatikoc dianusmatikoc qrouc (peperasmnhc distashc),

ap to Jerhma Adrneiac tou Sylvester, pou ousiastik enai Prisma touJewrmatoc 6.1.1.(ii), gnwrzoume pwc h kathgoriopohsh twn mh-ekfulismnwnsummetrikn digrammikn morfn wc proc th sqsh isodunamac thc isotimac

gnetai msw thc anallowthc posthtac thc upografc (sth genik perptw-

sh pou h summetrik digrammik morf den enai aparathta mh-ekfulismnh,

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apaitetai tso h txh so kai h upograf).

Ac dome sth sunqeia orismna paradegmata.

1. Jewrome wc sma touc migadikoc arijmoc. 'Estw dimCV = 2 optedimCP (V ) = 1 (migadik probolik eujea). Ap to Jerhma 6.1.1.(i) kaith Shmewsh 6.1.3 prokptei pwc kje mh-ekfulismnh tetragwnik kamplh

thc migadikc probolikc eujeac enai isodnamh msw enc proboliko meta-

sqhmatismo me to snolo twn shmewn me omogenec suntetagmnec (0, 1),pou 0, 1 C, pou ikanopoion th sqsh

20 + 21 = 0

dhlad

0 = i1Apoteletai sunepc ap 2 mlic shmea, aut me anomogenec suntetagmnec

i. Sunepc, uprqei ma kai monadik probolik klsh mh-ekfulismnwntetragwnikn kampuln sth migadik probolik eujea apotelomenh ap 2

mno shmea (kje llh mh-ekfulismnh tetragwnik kamplh sundetai msw

kpoiou proboliko metasqhmatismo me autn).

1'. Ac dome ti sumbanei sthn pragmatik probolik eujea dimRP (V ) =1, sunepc dimRV = 2. Uprqoun mno do istimec klseic mh-ekfulismnwnsummetrikn digrammikn morfn, ma me upograf 2 kai ma me upograf 0.

Faner, ap thn pragmatik summetrik digrammik morf me upograf

stw +2, prokptei h tetragwnik kamplh me exswsh stw

20 + 21 = 0

pou 0, 1, 2 R omogenec suntetagmnec sthn pragmatik