Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα

PANEPISTHMIO IWANNINWN
TMHMA MAJHMATIKWN
TOMEAS ALGEBRAS KAI GEWMETRIAS
METAPTUQIAKH DIATRIBH
LAMPRINH ZOURKA
SUMPAGEIS ELAQISTIKES EPIFANEIES GENOUS MHDEN
STHN n-DIASTATH SFAIRA
IWANNINA 2008

ii
H paroÔsa Metaptuqiak Diatrib ekpon jhke sto plaÐsio twn spoud¸n gia thn
apìkthsh tou MetaptuqiakoÔ Dipl¸matoc EidÐkeushc sta
MAJHMATIKA
pou aponèmei to Tm ma Majhmatik¸n tou PanepisthmÐou IwannÐnwn.
EgkrÐjhke th Deutèra, 30-06-2008, apì thn exetastik epitrop :
ONOMATEPWNUMO BAJMIDA UPOGRAFH
Jeìdwroc Blˆqoc Anaplhrwt c Kajhght c
(Epiblèpwn Kajhght c) tou Tm matoc Majhmatik¸n
tou PanepisthmÐou IwannÐnwn
Qr stoc MpaðkoÔshc Kajhght c
tou Tm matoc Majhmatik¸n
Jwmˆc Qasˆnhc Kajhght c
tou Tm matoc Majhmatik¸n

Eisagwg
Mia isometrik embˆptish enìc poluptÔgmatoc Riemann se èna polÔptugma Rie-
mann lègetai elaqistik an to dianusmatikì pedÐo mèshc kampulìthtac eÐnai to mh-
denikì isodÔnama an to Ðqnoc thc deÔterhc jemeli¸douc morf c thc isometrik c
embˆptishc eÐnai tautotikˆ mhdèn. Apì ton tÔpo thc pr¸thc metabol c tou embadoÔ
gnwrÐzoume pwc oi isometrikèc elaqistikèc embaptÐseic eÐnai akrib¸c ta krÐsima shmeÐa
thc sunˆrthshc tou embadoÔ. Epomènwc, fusikì epakìloujo eÐnai pwc oi elaqistikèc
isometrikèc embaptÐseic apoteloÔn an¸terhc diˆstashc genÐkeush twn gewdaisiak¸n
kampul¸n kai apartÐzoun mia shmantik oikogèneia isometrik¸n embaptÐsewn.
Sthn paroÔsa ergasÐa asqoloÔmaste me elaqistikèc epifˆneiec sthn Sn, dhlad
isometrikèc elaqistikèc embaptÐseic apì èna prosanatolismèno, sunektikì, didiˆstato
polÔptugma Riemann sthn n-diˆstath monadiaÐa sfaÐra
Sn
= {(x1, x2, ..., xn+1) ∈ Rn+1
: x2
1 + x2
2 + ... + x2
n+1 = 1},
h opoÐa eÐnai efodiasmènh me th sun jh metrik Riemann , kai wc gnwstìn èqei
stajer kampulìthta tom c 1. Parìlh thn omorfiˆ pou èqei apì mình thc h melèth
elaqistik¸n epifanei¸n sth sfaÐra, èqei apodeiqjeÐ ìti sqetÐzetai me th melèth me-
monwmènwn idiazìntwn shmeÐwn elaqistik¸n embaptÐsewn ston EukleÐdeio q¸ro. O
E. Calabi sto ˆrjro [6] parat rhse ìti an èqoume èna tridiˆstato elaqistikì upo-
polÔptugma M3 ston EukleÐdeio q¸ro En+3, tìte to polÔptugma M3 ∩ Sn+2, gia
sfaÐra Sn+2 katˆllhlou kèntrou, eÐnai didiˆstato elaqistikì upopolÔptugma sthn
Sn+2. AntÐstrofa, an M2 eÐnai èna didiˆstato elaqistikì upopolÔptugma thc Sn+2,
tìte o k¸noc pou dhmiourgeÐtai apì tic hmieujeÐec me arq to kèntro thc sfaÐrac
Sn+2 kai dièrqontai apì ta shmeÐa tou M2, eÐnai tridiˆstato elaqistikì upopolÔptug-
ma ston EukleÐdeio q¸ro En+3 me memonwmèno idiˆzon shmeÐo to kèntro thc sfaÐrac an
to M2 den eÐnai olikˆ gewdaisiakì sthn Sn+2. Epomènwc, h melèth isometrik¸n ela-
qistik¸n embaptÐsewn tridiˆstatwn poluptugmˆtwn Riemann me memonwmèno idiˆzon
shmeÐo ston EukleÐdeio q¸ro, anˆgetai sth melèth isometrik¸n elaqistik¸n embaptÐ-
sewn didiˆstatwn poluptugmˆtwn Riemann sth sfaÐra. Thn idèa aut ulopoÐhse o
E. Calabi sto prwtoporiakì ˆrjro [6] melet¸ntac elaqistikèc epifˆneiec sth sfaÐra
me thn aploÔsterh dunat topologÐa, dhlad sumpageÐc elaqistikèc epifˆneiec gènouc
mhdèn isodÔnama omoiomorfikèc me thn S2. Autì to ˆrjro èdwse to ènausma gia th
melèth elaqistik¸n epifanei¸n sth sfaÐra. LÐgo metˆ thn emfˆnish tou ˆrjrou tou
E. Calabi, o S.S. Chern parousÐase mia pio gewmetrik prosèggish twn elaqistik¸n
epifanei¸n sth sfaÐra kˆnontac qr sh twn jemeliwd¸n morf¸n anwtèrac tˆxewc. H
melèth aut suneqÐsthke apì ton J.L.M. Barbosa.
iii

iv
O stìqoc thc ergasÐac eÐnai na apodeÐxoume ta apotelèsmata tou E. Calabi sto
[6], me ton trìpo pou ta anadiatÔpwse o S.S. Chern sto [8], kaj¸c kai tou J.L.M.
Barbosa sto [4].
GnwrÐzoume ìti ta olikˆ gewdaisiakˆ m-diˆstata upopoluptÔgmata thc Sn, ìpou
2 ≤ m ≤ n − 1, eÐnai oi mègistec m-sfaÐrec thc Sn, dhlad tomèc thc Sn me (m + 1)-
diˆstatouc upoq¸rouc tou Rn+1. Mia isometrik embˆptish f : (M, , ) −→ Sn, ìpou
(M, , ) eÐnai polÔptugma Riemann, kaleÐtai koresmènh (full) an h eikìna thc den
perièqetai se kanèna olikˆ gewdaisiakì upopolÔptugma thc Sn.
Sthn ergasÐa aut to pr¸to apotèlesma pou ja apodeÐxoume eÐnai ìti oi sumpageÐc,
prosanatolismènec elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai koresmènec
mìno se ˆrtiac diˆstashc sfaÐra. Epiplèon ja d¸soume mia ektÐmhsh tou embadoÔ twn.
To sumpèrasma autì to apèdeixe o E. Calabi sto ˆrjro [6]. Argìtera, o S.S. Chern
sto [8] èdwse mia diaforetik prosèggish tou apotelèsmatoc to opoÐo eÐnai to ex c:
Je¸rhma I. 'Estw f : (M, , ) −→ Sn, n ≥ 3, sumpag c, prosanatolismènh, kore-
smènh, elaqistik epifˆneia gènouc mhdèn. Tìte:
(i) O arijmìc n eÐnai ˆrtioc (n = 2m).
(ii) To embadì A(M) thc epifˆneiac eÐnai akèraio pollaplˆsio tou 2π kai isqÔei
A(M) ≥ 2πm(m + 1).
Sth sunèqeia ja apodeÐxoume ìti oi sumpageÐc, prosanatolismènec, koresmènec,
elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra eÐnai ˆkamptec (rigid), èna apotèlesma
pou ofeÐletai ston J.L.M. Barbosa [4]:
Je¸rhma II. 'Estwsan f : (M, , ) −→ S2m, f : (M, , ) −→ S2m sumpageÐc,
prosanatolismènec, koresmènec, elaqistikèc epifˆneiec gènouc mhdèn. Tìte upˆrqei
isometrÐa τ : S2m −→ S2m ¸ste f = τ ◦ f.
Tèloc, ja deÐxoume pwc oi sumpageÐc, prosanatolismènec, koresmènec, elaqistikèc
epifˆneiec gènouc mhdèn sth sfaÐra me stajer kampulìthta Gauss taxinomoÔntai
pl rwc. Gia thn akrÐbeia apodeiknÔoume to akìloujo apotèlesma tou E. Calabi [6]:
Je¸rhma III. 'Estw f : (M, , ) −→ S2m sumpag c, prosanatolismènh, kore-
smènh, elaqistik epifˆneia gènouc mhdèn. An to polÔptugma Riemann (M, , )
èqei stajer kampulìthta Gauss K, tìte K = 2
m(m+1) kai upˆrqoun isometrÐec
F : (M, , ) −→ S2 m(m+1)
2 kai τ : S2m −→ S2m ¸ste f ◦ F−1 = τ ◦ fm,
ìpou fm : S2 m(m+1)
2 −→ S2m eÐnai h epifˆneia Veronese sthn S2m.
Gia tic apodeÐxeic aut¸n twn apotelesmˆtwn qrhsimopoioÔme ergaleÐa kai mejìdouc
apì th jewrÐa kinoumènou plaisÐou, twn jemeliwd¸n morf¸n an¸terhc tˆxhc kai th
jewrÐa epifanei¸n Riemann (Je¸rhma Riemann-Roch).
Epiplèon, h prosèggish pou akoloujoÔme mac epitrèpei na d¸soume mia apìdeixh
thc eikasÐac tou U. Simon [15] gia tic peript¸seic pou eÐnai dh gnwstì ìti isqÔei.
Parajètoume thn apìdeixh aut¸n twn peript¸sewn sto tèloc thc diatrib c.

v
EuqaristÐec
Ekfrˆzw tic jermèc mou euqaristÐec ston epiblèponta kajhght mou, k. Jeìdwro
Blˆqo gia th suneq epÐbleyh kai kajod ghs tou. EuqaristÐec ofeÐlw kai sta
ˆlla dÔo mèlh thc exetastik c epitrop c, ton k. Jwmˆ Qasˆnh kai ton k. Qr sto
MpaðkoÔsh gia tic qr simec parathr seic touc sthn ergasÐa mou.
EpÐshc, euqarist¸ touc upoy fiouc didˆktorec Qr sto Tatˆkh kai Qrusìstomo
Yaroudˆkh gia th bo jeiˆ touc.
Tèloc, idiaÐterec euqaristÐec ofeÐlw sth metaptuqiak foit tria AshmÐna MpoÔs-
mpoura gia thn polÔtimh bo jeia, upost rixh kai filÐa thc katˆ th diˆrkeia twn meta-
ptuqiak¸n spoud¸n mou.

Perieqìmena
1 Prokatarktikˆ 1
1.1 Isometrikèc embaptÐseic . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exis¸seic dom c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Jemeli¸deic morfèc an¸terhc tˆxhc . . . . . . . . . . . . . . . . . . . 9
2 Elaqistikèc epifˆneiec sthn Sn omoiomorfikèc me thn S2 15
2.1 Epifˆneiec Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn . . . . . . . . . . . . . 19
2.3 Bohjhtikˆ apotelèsmata . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 KÔria apotelèsmata 53
3.1 ApodeÐxeic twn kurÐwn apotelesmˆtwn . . . . . . . . . . . . . . . . . . 53
3.2 EikasÐa tou U. Simon . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii

Kefˆlaio 1
Prokatarktikˆ
1.1 Isometrikèc embaptÐseic
Sto parìn kefˆlaio ja anafèroume aparaÐthta stoiqeÐa apì th jewrÐa twn iso-
metrik¸n embaptÐsewn, tou kinoumènou plaisÐou kai twn jemeliwd¸n morf¸n an¸te-
rhc tˆxhc. Gia tic basikèc ènnoiec thc Diaforik c GewmetrÐac parapèmpoume sta
[10, 14, 13].
'Estw Mn diaforÐsimo polÔptugma diˆstashc n. Gia tuqìn p ∈ Mn sumbolÐzoume
me TpMn ton efaptìmeno q¸ro tou Mn sto p kai me TMn = {(p, v) : p ∈ Mn, v ∈
TpMn} thn efaptìmenh dèsmh tou Mn. To sÔnolo twn diaforÐsimwn dianusmatik¸n
pedÐwn tou Mn to sumbolÐzoume me ∆(Mn) Γ(TMn) kai to sÔnolo twn diaforÐsimwn
sunart sewn g : Mn −→ R to sumbolÐzoume me D(Mn) C∞(Mn, R).
Orismìc 1.1.1. 'Estw f : Mn −→ M
n+k
diaforÐsimh apeikìnish metaxÔ twn
diaforÐsimwn poluptugmˆtwn Mn kai M
n+k
. DiaforÐsimo dianusmatikì pedÐo katˆ
m koc thc f kaloÔme kˆje apeikìnish V h opoÐa se kˆje p ∈ Mn antistoiqeÐ èna
diˆnusma V |p ∈ Tf(p)M
n+k
kai eÐnai diaforÐsimh me thn ex c ènnoia: An (U, y)
eÐnai qˆrthc tou M
n+k
gÔrw apì to f(p) me suntetagmènec (y1, y2, ..., yn+k) kai
V |q = n+k
i=1 gi(q) ∂
∂yi
|f(q) gia kˆje q ∈ f−1(U), oi sunart seic gi, i = 1, 2, ..., n + k,
eÐnai diaforÐsimec.
SumbolÐzoume me ∆(f) to sÔnolo twn diaforÐsimwn dianusmatik¸n pedÐwn katˆ
m koc thc f. Profan¸c to ∆(f) eÐnai to sÔnolo twn pedÐwn (sections) thc epagìmenhc
dianusmatik c dèsmhc f∗(TM
n+k
) := {(p, v) : p ∈ Mn, v ∈ Tf(p)M
n+k
} bajmÐdac
(rank) n + k, dhlad ∆(f) = Γ f∗(TM
n+k
) . An X ∈ ∆(Mn), tìte h apeikìnish h
opoÐa se kˆje p ∈ Mn antistoiqeÐ to diˆnusma dfp(X|p), eÐnai diaforÐsimo dianusmatikì
pedÐo katˆ m koc thc f kai sumbolÐzetai me df(X). Epiplèon, an Y ∈ ∆(M
n+k
), tìte
to Y ◦ f eÐnai epÐshc diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f.
Orismìc 1.1.2. 'Estw f : Mn −→ M
n+k
diaforÐsimh apeikìnish kai X ∈ ∆(Mn),
X ∈ ∆(M
n+k
). Ta X, X lègontai f-susqetismèna (f-related) an X ◦ f = df(X).
1

2 Prokatarktikˆ
L mma 1.1.1. An f : Mn −→ M
n+k
eÐnai diaforÐsimh apeikìnish kai X, Y ∈
∆(Mn) eÐnai f-susqetismèna twn X, Y ∈ ∆(M
n+k
) antÐstoiqa, tìte ta ginìmena Lie
[X, Y ], [X, Y ] eÐnai f-susqetismèna.
Apìdeixh. 'Eqoume X ◦ f = df(X), ˆra gia kˆje p ∈ Mn, X|f(p) = dfp(X|p). Ja
deÐxoume ìti [X, Y ] ◦ f = df [X, Y ] , isodÔnama
X|f(p)(Y ) − Y |f(p)(X) = dfp [X, Y ]|p .
Gia kˆje ϕ ∈ D(M
n+k
) èqoume:
X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|f(p) Y (ϕ) − Y |f(p) X(ϕ)
= dfp(X|p) Y (ϕ) − dfp(Y |p) X(ϕ)
= X|p Y (ϕ) ◦ f − Y |p X(ϕ) ◦ f .
'Omwc Y (ϕ)◦f = Y (ϕ◦f), afoÔ gia kˆje p ∈ Mn isqÔei: Y |f(p)(ϕ) = dfp(Y |p)(ϕ) =
Y |p(ϕ ◦ f). Telikˆ,
X|f(p)(Y ) − Y |f(p)(X) (ϕ) = X|p Y (ϕ ◦ f) − Y |p X(ϕ ◦ f)
= (XY − Y X)|p(ϕ ◦ f)
= [X, Y ]|p(ϕ ◦ f)
= dfp [X, Y ]|p (ϕ).
Qreiazìmaste thn akìloujh prìtash, h apìdeixh thc opoÐac dÐnetai sto [13].
Prìtash 1.1.1. 'Estw Mn diaforÐsimo polÔptugma kai (M
n+k
, , ) diaforÐsimo po-
lÔptugma Riemann me sunoq Levi-Civita . An f : Mn −→ M
n+k
eÐnai diaforÐsimh
apeikìnish, tìte upˆrqei monadik apeikìnish
f
: ∆(Mn
) × ∆(f) −→ ∆(f),
(X, V ) −→ f
XV,
gia thn opoÐa isqÔoun:
(i) f
X1+X2
V = f
X1
V + f
X2
V,
(ii) f
gXV = g f
XV,
(iii) f
X(V1 + V2) = f
XV1 + f
XV2,
(iv) f
X(gV ) = X(g)V + g f
XV,
(v) f
X(Y ◦ f) = df(X)Y ,
(vi) X V1, V2 = f
XV1, V2 + V1, f
XV2 ,
(vii) f
Xdf(Y ) − f
Y df(X) = df [X, Y ] ,
ìpou X, X1, X2, Y ∈ ∆(Mn), Y ∈ ∆(M
n+k
), V, V1, V2 ∈ ∆(f) kai g ∈ D(Mn).

Isometrikèc embaptÐseic 3
H apeikìnish
f eÐnai h sunoq pou epˆgei h sunoq Levi-Civita tou M
n+k
sthn
epagìmenh dèsmh f∗(TM
n+k
).
Orismìc 1.1.3. 'Estwsan (Mn, , ) kai (M
n+k
, , ) diaforÐsima poluptÔgmata Rie-
mann. Mia diaforÐsimh apeikìnish f : (Mn, , ) −→ (M
n+k
, , ) kaleÐtai isometrik
embˆptish an gia kˆje p ∈ Mn isqÔoun ta akìlouja:
(i) to diaforikì dfp : TpMn −→ Tf(p)M
n+k
eÐnai ènesh kai
(ii) dfp(v), dfp(w) f(p) = v, w p, gia kˆje v, w ∈ TpMn.
SumbolÐzoume th sunoq Levi-Civita tou Mn me kai ton tanust kampulìthtˆc
tou me R. Me sumbolÐzoume th sunoq Levi-Civita tou M
n+k
kai me R ton tanust
kampulìthtˆc tou.
An {e1, ..., en} eÐnai topikì orjomonadiaÐo plaÐsio tou Mn, tìte
R(ei, ej)ek =
n
l=1
Rijklel,
ìpou Rijkl := R(ei, ej)ek, el .
Orismìc 1.1.4. 'Estw Mn diaforÐsimo polÔptugma. Mia apeikìnish
T : ∆(Mn
) × ... × ∆(Mn
)
r
−→ D(Mn
)
kaleÐtai (r, 0)-tanustikì pedÐo an eÐnai D(Mn)-grammik wc proc kˆje metablht thc.
Epiplèon, an E eÐnai dianusmatik dèsmh (vector bundle) uperˆnw tou Mn, tìte mia
apeikìnish
T : ∆(Mn
) × ... × ∆(Mn
)
r
−→ Γ(E),
ìpou Γ(E) eÐnai to sÔnolo twn pedÐwn (sections) thc dianusmatik c dèsmhc, kaleÐtai
(r, 1)-tanustikì pedÐo an eÐnai D(Mn)-grammik wc proc kˆje metablht thc.
EÐnai gnwstì ìti an T eÐnai (r, 0)-tanustikì pedÐo (r, 1)-tanustikì pedÐo kai
X1, ..., Xr, Y1, ..., Yr ∈ ∆(Mn) me Xi|p = Yi|p gia kˆje i ∈ {1, ..., r} se kˆpoio shmeÐo
p tou Mn, tìte T(X1, ..., Xr)|p = T(Y1, ..., Yr)|p. Autì mac epitrèpei na blèpoume
to tanustikì pedÐo T se kˆje shmeÐo p ∈ Mn wc pleiogrammik apeikìnish T|p :
TpMn × ... × TpMn −→ R T|p : TpMn × ... × TpMn −→ Ep, ìpou Ep eÐnai to n ma
(ﬁber) thc dèsmhc E uperˆnw tou p.
'Estw σ ènac didiˆstatoc upìqwroc tou TpMn. H kampulìthta tom c tou Mn sto
shmeÐo p gia to epÐpedo σ eÐnai o arijmìc K(p, σ) = R(e1, e2)e2, e1 , ìpou {e1, e2}
eÐnai tuqaÐa orjomonadiaÐa bˆsh tou σ.
O tanust c Ricci tou Mn eÐnai to summetrikì (2,0)-tanustikì pedÐo
Q : ∆(Mn
) × ∆(Mn
) −→ D(Mn
), (X, Y ) −→ Q(X, Y ) :=
n
j=1
R(ej, X)Y, ej ,

4 Prokatarktikˆ
ìpou {e1, ..., en} eÐnai topikì orjomonadiaÐo plaÐsio tou Mn. 'Estw p shmeÐo tou Mn.
H kampulìthta Ricci sto shmeÐo p kai sth monadiaÐa dieÔjunsh x ∈ TpMn eÐnai
Ric(x) = Q(x, x).
H arijmhtik kampulìthta tou Mn eÐnai Sc = n
j=1 Ric(ej).
Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M
n+k
, , ) orÐzetai o tanust c
kampulìthtac Rf thc epagìmenhc dèsmhc f∗(TM
n+k
) wc proc th sunoq
f , wc ex c:
Rf
: ∆(Mn
) × ∆(Mn
) × ∆(f) −→ ∆(f),
(X, Y, V ) −→ Rf
(X, Y )V := f
X
f
Y V − f
Y
f
XV − f
[X,Y ]V
kai o opoÐoc eÐnai D(Mn)-grammikìc wc proc kˆje metablht .
L mma 1.1.2. An f : (Mn, , ) −→ (M
n+k
, , ) eÐnai isometrik embˆptish kai
X, Y, Z ∈ ∆(Mn) eÐnai f-susqetismèna twn X, Y , Z ∈ ∆(M
n+k
) antÐstoiqa, tìte
isqÔei R(X, Y )Z ◦ f = Rf (X, Y )df(Z).
Apìdeixh. Lìgw thc Prìtashc 1.1.1 isqÔei
f
Y df(Z) = f
Y (Z ◦ f) = df(Y )Z = Y ◦f Z = ( Y Z) ◦ f,
f
X
f
Y df(Z) = f
X ( Y Z) ◦ f = df(X)( Y Z) = ( X Y Z) ◦ f.
EpÐshc, kˆnontac qr sh tou L mmatoc 1.1.1, èqoume
f
[X,Y ]df(Z) = f
[X,Y ](Z ◦ f) =
df [X,Y ]
Z = [X,Y ]◦f Z = ( [X,Y ]Z) ◦ f.
Epomènwc isqÔei
Rf
(X, Y )df(Z) = ( X Y Z) ◦ f − ( Y XZ) ◦ f − ( [X,Y ]Z) ◦ f
= R(X, Y )Z ◦ f.
Gia kˆje isometrik embˆptish f : (Mn, , ) −→ (M
n+k
, , ) kai tuqìn shmeÐo
p ∈ Mn, èqoume thn anˆlush tou efaptìmenou q¸rou Tf(p)M
n+k
sto shmeÐo f(p) ∈
M
n+k
sto ex c orjog¸nio eujÔ ˆjroisma wc proc to eswterikì ginìmeno pou orÐzei
h metrik Riemann tou M
n+k
Tf(p)M
n+k
= dfp(TpMn
) ⊕ dfp(TpMn
)
⊥
.
O efaptìmenoc q¸roc Tpf thc f sto p eÐnai o n-diˆstatoc dianusmatikìc upìqwroc
dfp(TpMn) tou (n + k)-diˆstatou dianusmatikoÔ q¸rou Tf(p)M
n+k
. H efaptìmenh
dianusmatik dèsmh Tf thc f eÐnai h Tf := {(p, v) : p ∈ Mn, v ∈ Tpf}, èqei bajmÐda
n kai to sÔnolo twn pedÐwn thc eÐnai Γ(Tf) =: ∆f (Mn). OrÐzoume ton kˆjeto q¸ro

Isometrikèc embaptÐseic 5
thc f sto p na eÐnai o k-diˆstatoc dianusmatikìc q¸roc Npf := {ξ : ξ ∈ (Tpf)⊥}. H
kˆjeth dianusmatik dèsmh Nf thc f eÐnai h ènwsh ìlwn twn kajètwn q¸rwn, dhlad
Nf := {(p, ξ) : p ∈ Mn, ξ ∈ Npf}, èqei bajmÐda k kai to sÔnolo twn pedÐwn thc eÐnai
Γ(Nf) =: ∆⊥(f). Ta pedÐa thc kˆjethc dianusmatik c dèsmhc Nf ta kaloÔme kˆjeta
dianusmatikˆ pedÐa katˆ m koc thc f.
Lìgw thc parapˆnw anˆlushc, gia tuqìn v ∈ Tf(p)M
n+k
upˆrqoun monadikˆ dia-
nÔsmata v ∈ TpMn kai v⊥ ∈ Npf ètsi ¸ste v = dfp(v ) + v⊥. Katˆ sunèpeia, gia
kˆje V ∈ ∆(f) upˆrqoun monadikˆ dianusmatikˆ pedÐa V ∈ ∆(Mn) kai V ⊥ ∈ ∆⊥(f)
¸ste
V = df(V ) + V ⊥
.
Gia X, Y ∈ ∆(Mn) èqoume epomènwc thn anˆlush:
f
Xdf(Y ) = df f
Xdf(Y ) + f
Xdf(Y )
⊥
,
ìpou
f
Xdf(Y ) ∈ ∆(Mn) kai
f
Xdf(Y )
⊥
∈ ∆⊥(f).
ApodeiknÔetai ìti
f
Xdf(Y ) = XY.
OrÐzoume thn apeikìnish
B : ∆(Mn
) × ∆(Mn
) −→ ∆⊥
(f), (X, Y ) −→ B(X, Y ) := f
Xdf(Y )
⊥
.
ApodeiknÔetai ìti h B eÐnai summetrikì (2,1)-tanustikì pedÐo kai kaleÐtai deÔterh
jemeli¸dhc morf thc f. SÔmfwna me ta parapˆnw, h teleutaÐa sqèsh gÐnetai
f
Xdf(Y ) = df( XY ) + B(X, Y ),
o legìmenoc tÔpoc tou Gauss.
Gia X ∈ ∆(Mn), ξ ∈ ∆⊥(f) èqoume thn anˆlush:
f
Xξ = df ( f
Xξ) + ( f
Xξ)⊥
,
ìpou ( f
Xξ) ∈ ∆(Mn) kai ( f
Xξ)⊥ ∈ ∆⊥(f).
H apeikìnish Weingarten sth dieÔjunsh ξ ∈ ∆⊥(f) eÐnai
Aξ : ∆(Mn
) −→ ∆(Mn
), X −→ AξX := −( f
Xξ) .
H apeikìnish Weingarten eÐnai D(Mn)-grammik wc proc ξ kai autoproshrthmèno
(1,1)-tanustikì pedÐo. ApodeiknÔetai ìti AξX, Y = B(X, Y ), ξ .
EpÐshc, orÐzoume thn apeikìnish
⊥
: ∆(Mn
) × ∆⊥
(f) −→ ∆⊥
(f), (X, ξ) −→ ⊥
Xξ := ( f
Xξ)⊥
,
h opoÐa eÐnai h sunoq thc kˆjethc dianusmatik c dèsmhc Nf. Epomènwc, apì ta
parapˆnw paÐrnoume ton tÔpo tou Weingarten :
f
Xξ = −df(AξX) + ⊥
Xξ.

6 Prokatarktikˆ
JewroÔme topikì orjomonadiaÐo plaÐsio {ξ1, ..., ξk} thc Nf. To dianusmatikì pedÐo
mèshc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai to
H =
1
n
k
α=1
(traceAξα )ξα.
To H eÐnai kalˆ orismèno kˆjeto diaforÐsimo dianusmatikì pedÐo katˆ m koc thc f.
Epiplèon isqÔei
H =
1
n
n
j=1
B(ej, ej),
ìpou {e1, ..., en} topikì orjomonadiaÐo plaÐsio tou Mn.
Orismìc 1.1.5. Mia isometrik embˆptish f lègetai elaqistik an H = 0.
O tanust c kˆjethc kampulìthtac thc isometrik c embˆptishc f orÐzetai na eÐnai
h apeikìnish
R⊥
: ∆(Mn
) × ∆(Mn
) × ∆⊥
(f) −→ ∆⊥
(f),
(X, Y, ξ) −→ R⊥
(X, Y )ξ := ⊥
X
⊥
Y ξ − ⊥
Y
⊥
Xξ − ⊥
[X,Y ]ξ.
ApodeiknÔetai ìti o R⊥ eÐnai D(Mn)-grammikìc wc proc kˆje metablht .
Tèloc, gia ξ ∈ ∆⊥(f) orÐzoume to (2,1)-tanustikì pedÐo tou Mn
( XAξ)Y := X(AξY ) − Aξ( XY ),
ìpou X, Y ∈ ∆(Mn).
Gia isometrikèc embaptÐseic isqÔoun:
(i) h exÐswsh Gauss
Rf
(X, Y )df(Z), df(W) = R(X, Y )Z, W + B(X, Z), B(Y, W)
− B(Y, Z), B(X, W) ,
(ii) h exÐswsh Codazzi
Rf
(X, Y )ξ = ( Y Aξ)X − A ⊥
Y ξX − ( XAξ)Y + A ⊥
X ξY,
(iii) h exÐswsh Ricci
Rf
(X, Y )ξ
⊥
= R⊥
(X, Y )ξ − B(X, AξY ) + B(Y, AξX),
ìpou X, Y, Z, W ∈ ∆(Mn) kai ξ ∈ ∆⊥(f).
JewroÔme topikì orjomonadiaÐo plaÐsio {e1, ..., en} tou Mn. To m koc thc deÔte-
rhc jemeli¸douc morf c orÐzetai wc h sunˆrthsh
B :=
n
j,l=1
|B(ej, el)|2.

Exis¸seic dom c 7
ApodeiknÔetai ìti eÐnai kalˆ orismènh, dhlad anexˆrthth tou plaisÐou kai ìti B 2 =
k
α=1 trace(Aξα ◦ Aξα ).
'Estw f : (Mn, , ) −→ (M
n+k
c , , ) isometrik embˆptish ìpou Mn eÐnai dia-
forÐsimo polÔptugma Riemann kai M
n+k
c eÐnai diaforÐsimo polÔptugma Riemann me
stajer kampulìthta tom c c. Gia p ∈ Mn, jewroÔme dianÔsmata x, y ∈ TpMn kai
orjomonadiaÐa bˆsh {ξ1, ..., ξk} tou Npf. Apì thn exÐswsh Gauss apodeiknÔetai ìti
gia ton tanust Ricci isqÔei
Q(x, y) =
k
α=1
(traceAξα ) Aξα x, y −
k
α=1
Aξα x, Aξα y + (n − 1)c x, y .
An x eÐnai monadiaÐo, tìte h kampulìthta Ricci sth dieÔjunsh x dÐnetai apì th
sqèsh
Ric(x) =
k
α=1
(traceAξα ) Aξα x, x −
k
α=1
|Aξα x|2
+ (n − 1)c.
Epiplèon, gia thn arijmhtik kampulìthta isqÔei
Sc = n(n − 1)c + n2
|H|2
− B 2
. (1.1)
'Estw f : M2 −→ Sn isometrik embˆptish enìc polÔptugmatoc Riemann M2
sth monadiaÐa sfaÐra Sn. JewroÔme topikì orjomonadiaÐo plaÐsio {e1, e2} tou M2.
Tìte h f eÐnai elaqistik an kai mìno an B(e1, e1) + B(e2, e2) = 0. Gia thn arijmhtik
kampulìthta èqoume Sc = 2K, ìpou K eÐnai h kampulìthta Gauss tou M2. Epomènwc,
h sqèsh (1.1) gÐnetai sthn perÐptwsh pou h f eÐnai elaqistik
2K = 2 − B 2
(1.2)
isodÔnama
K = 1 − |B(e1, e1)|2
− |B(e1, e2)|2
. (1.3)
Katˆ sunèpeia, ìtan h f eÐnai elaqistik isqÔei K ≤ 1 kai èqoume K = 1 pantoÔ
an kai mìno an h elaqistik embˆptish f : M2 −→ Sn eÐnai olikˆ gewdaisiak .
1.2 Exis¸seic dom c
'Estw f : (Mn, , ) −→ (M
n+k
, , ) isometrik embˆptish, ìpou Mn kai M
n+k
eÐnai poluptÔgmata Riemann me sunoqèc Levi-Civita , kai tanustèc kampulìthtac
R, R antÐstoiqa. JewroÔme anoiktì uposÔnolo U tou Mn, ¸ste h f|U na eÐnai
emfÔteush, kai anoiktì uposÔnolo U tou M
n+k
me f(U) = U ∩f(Mn). Gia ta epìmena
ja qrhsimopoi soume thn ex c sÔmbash deikt¸n:
1 ≤ j, l, s, t, ... ≤ n,
n + 1 ≤ α, β, γ, ... ≤ n + k,
1 ≤ A, B, C, ... ≤ n + k,

8 Prokatarktikˆ
ektìc an anafèretai diaforetikˆ.
'Estw {eA} orjomonadiaÐo plaÐsio sto U ètsi ¸ste gia kˆje shmeÐo q = f(p) tou
f(U), na isqÔei ej|q ∈ Tpf gia kˆje j. SumbolÐzoume me {ωA} to sumplaÐsio tou {eA}.
Gia V ∈ ∆(U) èqoume ωA(V ) = V, eA . OrÐzoume orjomonadiaÐo plaÐsio {ej} sto
U me ej := df−1(ej ◦ f|U ) kai èstw {ωj} to sumplaÐsiì tou. Gia X ∈ ∆(U) èqoume
ωj(X) = X, ej . An jèsoume eα := eα ◦ f|U , tìte apoktoÔme orjomonadiaÐo plaÐsio
{eA} katˆ m koc thc f me efaptìmeno mèroc {ej} kai kˆjeto mèroc {eα}.
Orismìc 1.2.1. KaloÔme r-morf epÐ enìc diaforÐsimou poluptÔgmatoc kˆje anti-
summetrikì (r, 0)-tanustikì pedÐo.
SumbolÐzoume me Λr(M
n+k
) kai me Λr(Mn) to sÔnolo twn r-morf¸n (r ≥ 0)
tou M
n+k
kai tou Mn antÐstoiqa. Gia r = 0 èqoume ta sÔnola twn diaforÐsimwn
sunart sewn D(M
n+k
) kai D(Mn) antÐstoiqa. H f epˆgei gia kˆje akèraio r ≥ 0 mia
apeikìnish f∗ : Λr(M
n+k
) −→ Λr(Mn), thn anˆsursh (pullback) thc f, pou orÐzetai
wc ex c: Gia r = 0 kai g ∈ D(M
n+k
), f∗(g) = g ◦ f. Gia r > 0 kai w ∈ Λr(M
n+k
) h
f∗(w) ∈ Λr(Mn) eÐnai h r-morf , pou sto p ∈ Mn orÐzetai wc f∗(w)|p(v1, v2, ..., vr) :=
w|f(p) dfp(v1), dfp(v2), ..., dfp(vr) , ìpou v1, ..., vr ∈ TpMn.
Oi morfèc sunoq c tou Mn gia to plaÐsio {ej}, eÐnai oi 1-morfèc
ωjl : ∆(U) −→ D(U),
X −→ ωjl(X) := Xej, el .
EpÐshc, oi morfèc sunoq c tou M
n+k
gia to plaÐsio {eA}, eÐnai oi 1-morfèc
ωAB : ∆(U) −→ D(U),
X −→ ωAB(X) := XeA, eB .
IsqÔei ωjl = −ωlj, ωAB = −ωBA kai apodeiknÔetai ìti ωj = f∗(ωj), ωjl = f∗(ωjl).
Oi exis¸seic dom c tou Mn eÐnai:
dωj =
l
ωjl ∧ ωl,
dωjl =
s
ωjs ∧ ωsl + Ωjl,
ìpou Ωjl ∈ Λ2(U), Ωjl = s<t Rjlstωs ∧ ωt, Ωjl = −Ωlj kai ∧ eÐnai to exwterikì
ginìmeno morf¸n. Oi 2-morfèc Ωjl kaloÔntai morfèc kampulìthtac tou Mn.
Oi exis¸seic dom c tou M
n+k
eÐnai:
dωA =
B
ωAB ∧ ωB,
dωAB =
C
ωAC ∧ ωCB + ΩAB,
ìpou ΩAB ∈ Λ2(U), ΩAB = C<D RABCDωC ∧ ωD, ΩAB = −ΩBA.

Jemeli¸deic morfèc an¸terhc tˆxhc 9
OrÐzoume tic 1-morfèc ωjα := f∗(ωjα), ωαj := f∗(ωαj) kai tic morfèc kˆjethc
sunoq c ωαβ := f∗(ωαβ). ApodeiknÔetai ìti ωjα(X) = Aeα X, ej , ìpou Aeα eÐnai h
apeikìnish Weingarten sth dieÔjunsh eα kai ωαβ(X) = ⊥
Xeα, eβ gia kˆje X ∈
∆(U). EpÐshc orÐzoume ωα := f∗(ωα). EÔkola prokÔptei ìti ωα = 0.
L mma 1.2.1. (Cartan) 'Estwsan oi grammikˆ anexˆrthtec 1-morfèc ϕ1, ..., ϕr (r ≤
n) tou Mn. An θ1, θ2, ..., θr eÐnai 1-morfèc tou Mn tètoiec ¸ste na isqÔei
r
j=1 ϕj ∧
θj = 0, tìte upˆrqoun sunart seic ajl ∈ D(Mn), ìpou j, l ∈ {1, ..., r} ¸ste na isqÔei
θj = r
l=1 ajlϕl kai ajl = alj.
Epeid ωα = 0 èqoume
r
j=1 ωjα∧ωj = 0 kai sÔmfwna me to L mma 1.2.1, upˆrqoun
sunart seic hα
jl ∈ D(U), me hα
jl = hα
lj, tètoiec ¸ste ωjα = n
l=1 hα
jlωl kai hα
jl =
Aeα ej, el = B(ej, el), eα .
Parat rhsh 1.2.1. H isometrik embˆptish f : (Mn, , ) → (M
n+k
, , ) eÐnai
elaqistik an gia kˆje α isqÔei
n
j=1 hα
jj = 0.
'Otan to polÔptugma M
n+k
èqei stajer kampulìthta tom c c, tìte oi exis¸seic
Gauss, Codazzi kai Ricci gÐnontai antÐstoiqa:
Ωjl =
α
ωjα ∧ ωαl − cωj ∧ ωl,
dωjα =
l
ωjl ∧ ωlα +
β
ωjβ ∧ ωβα,
dωαβ =
j
ωαj ∧ ωjβ +
γ
ωαγ ∧ ωγβ.
An ω eÐnai 1-morf tou Mn, tìte apodeiknÔetai sto [14] ìti isqÔei
dω(X, Y ) = X(ω(Y )) − Y (ω(X)) − ω([X, Y ]),
ìpou X, Y ∈ ∆(Mn).
'Estw M2 èna didiˆstato polÔptugma Riemann. Me th bo jeia kai thc parapˆnw
sqèshc apodeiknÔetai ìti
dω12 = −Kω1 ∧ ω2, (1.4)
ìpou K eÐnai h kampulìthta Gauss tou M2.
1.3 Jemeli¸deic morfèc an¸terhc tˆxhc
'Estw f : (Mn, , ) −→ (M
n+k
, , ) isometrik embˆptish, ìpou Mn, M
n+k
eÐnai
poluptÔgmata Riemann me sunoqèc Levi-Civita , antÐstoiqa. Gia p ∈ Mn kai
jetikì akèraio r, o dianusmatikìc q¸roc
Tr
p f := span dfp(Xj1 |p), f
Xl1
df(Xl2 ) |p, ..., f
Xs1
... f
Xsr−1
df(Xsr ) |p :
Xm ∈ ∆(Mn
), m ∈ {j1, l1, l2, ..., s1, ..., sr}

10 Prokatarktikˆ
kaleÐtai eggÔtatoc q¸roc r-tˆxhc (osculating space) thc f sto p kai eÐnai dianusma-
tikìc upìqwroc tou Tf(p)M
n+k
.
Profan¸c, T1
p f = dfp(TpMn) kai T1f = p∈Mn T1
p f eÐnai h efaptìmenh dianu-
smatik dèsmh thc f. An af soume to p na metabˆletai, tìte gia stajerì r > 1 oi
q¸roi Tr
p f endeqomènwc na èqoun diaforetik diˆstash. EpÐshc, o eggÔtatoc q¸roc
r-tˆxhc thc f sto p eÐnai upìqwroc tou eggutˆtou q¸rou (r + 1)-tˆxhc thc f sto
p. Epomènwc, mporoÔme na orÐsoume to orjog¸nio sumpl rwma Nr
p f tou Tr
p f ston
Tr+1
p f wc proc to eswterikì ginìmeno tou Tf(p)M
n+k
, dhlad
Tr+1
p f = Tr
p f ⊕ Nr
p f.
O Nr
p f lègetai kˆjetoc q¸roc r-tˆxhc thc f sto p. IsqÔei
Tr+1
p f = T1
p f ⊕ N1
p f ⊕ N2
p f ⊕ ... ⊕ Nr
p f.
Profan¸c, gia s = r èqoume Nr
p f ∩ Ns
p f = {0} kai v, w = 0 gia kˆje v ∈ Nr
p f,
w ∈ Ns
p f.
Orismìc 1.3.1. 'Estw f isometrik embˆptish metaxÔ twn poluptugmˆtwn Riemann
Mn kai M
n+k
c , ìpou to M
n+k
c èqei stajer kampulìthta tom c c. KaloÔme jemeli¸dh
morf (r + 1)-tˆxhc thc f sto p ∈ Mn thn apeikìnish
Br|p : TpMn
× TpMn
× ... × TpMn
r+1
−→ Nr
p f,
(x1, x2, ..., xr+1) → Br|p(x1, x2, ..., xr+1) = f
X1
f
X2
... f
Xr
df(Xr+1)|p
Nr
p f
,
ìpou me
f
X1
f
X2
... f
Xr
df(Xr+1)|p
Nr
p f
sumbolÐzoume thn probol tou dianÔsmatoc
f
X1
f
X2
... f
Xr
df(Xr+1)|p tou Tr+1
p f ston upìqwrì tou Nr
p f kai X1, ..., Xr+1 eÐnai to-
pikˆ diaforÐsima dianusmatikˆ pedÐa tou Mn pou epekteÐnoun ta x1, ..., xr+1 antÐstoiqa,
dhlad Xi|p = xi gia i = 1, ..., r + 1.
Epeid to diˆnusma
f
X1
f
X2
... f
Xr
df(Xr+1)|p an kei ston Tr+1
p f = Tr
p f ⊕ Nr
p f
kai isqÔei Tr
p f = T1
p f ⊕ N1
p f ⊕ ... ⊕ Nr−1
p f, ènac isodÔnamoc orismìc thc Br|p eÐnai
Br|p(x1, ..., xr+1) = f
X1
... f
Xs
df(Xr+1)|p
(T1
p f⊕N1
p f⊕...⊕Nr−1
p f)⊥
,
ìpou (T1
p f⊕N1
p f⊕...⊕Nr−1
p f)⊥ eÐnai to orjosumpl rwma tou T1
p f⊕N1
p f⊕...⊕Nr−1
p f
ston Tf(p)M
n+k
.
ApodeiknÔetai sto [18] ìti h Br|p eÐnai kalˆ orismènh kai summetrik , dhlad
Br|p(x1, x2, ..., xr+1) = Br|p(xσ(1), xσ(2), ..., xσ(r+1)) gia kˆje stoiqeÐo σ thc omˆdac
metajèsewn twn arijm¸n 1, 2, ..., r + 1. EpÐshc, h Br|p eÐnai R-grammik wc proc kˆje
metablht thc.
JewroÔme thn ènwsh Nrf ìlwn twn kajètwn q¸rwn r-tˆxhc thc f, dhlad
Nr
f :=
p∈Mn
Nr
p f.

An af soume to p na metabˆletai ston Orismì 1.3.1, tìte èqoume th jemeli¸dh
morf (r + 1)-tˆxhc thc f :
Br : TMn
⊕ TMn
⊕ ... ⊕ TMn
r+1
−→ Nr
f,
isodÔnama
Br : TMn
⊕ TMn
⊕ ... ⊕ TMn
r+1
−→ (T1
f ⊕ N1
f ⊕ ... ⊕ Nr−1
f)⊥
,
(p, x1), ..., (p, xr+1) −→ Br|p(x1, ..., xr+1).
Profan¸c h B1 eÐnai h deÔterh jemeli¸dhc morf thc f, dhlad èqoume B1 = B.
Gia r > 1, h Br den eÐnai en gènei (r+1,1)-tanustikì pedÐo, afoÔ to sÔnolo T1f ⊕
N1f ⊕ ... ⊕ Nr−1f de gnwrÐzoume an eÐnai dianusmatik dèsmh.
Ac shmeiwjeÐ ìti sto ex c ja grˆfoume Br eÐte gia th jemeli¸dh morf (r + 1)-
tˆxhc thc f sto p, eÐte gia th jemeli¸dh morf (r + 1)-tˆxhc thc f.
L mma 1.3.1. 'Estw f : (Mn, , ) −→ (M
n+k
c , , ) isometrik embˆptish, ìpou Mn
eÐnai polÔptugma Riemann kai M
n+k
c eÐnai polÔptugma Riemann me stajer kampu-
lìthta tom c c. Tìte gia kˆje p ∈ Mn isqÔei
Nr
p f = spanImBr = span{Br(x1, x2, ..., xr+1) : x1, x2, ..., xr+1 ∈ TpMn
}.
Apìdeixh. Profan¸c spanImBr ⊂ Nr
p f, afoÔ ImBr ⊂ Nr
p f. 'Estw ξ ∈ Nr
p f. Gnw-
rÐzoume ìti Tr+1
p f = Tr
p f ⊕ Nr
p f, epomènwc ξ ∈ Tr+1
p f kai ja grˆfetai wc
ξ =
j
αjdfp(Xj|p) +
l,m
βlm
f
Xl
df(Xm)|p + ... +
+
s1,...,sr
γs1...sr
f
Xs1
... f
Xsr−1
df(Xsr )|p
+
t1,...,tr+1
δt1...tr+1
f
Xt1
... f
Xtr
df(Xtr+1 )|p
Nr
p f
,
gia katˆllhlouc suntelestèc, ìpou Xj, ..., Xtr+1 ∈ ∆(Mn). Epeid h probol eÐnai
grammik apeikìnish, Ts
p f ⊂ Tr
p f gia kˆje s < r kai Tr+1
p f = Tr
p f ⊕ Nr
p f, èqoume
ξ =
t1,...,tr+1
δt1...tr+1
f
Xt1
... f
Xtr
df(Xtr+1 )|p
Nr
p f
=
t1,...,tr+1
δt1...tr+1 Br(Xt1 |p, Xt2 |p, ..., Xtr+1 |p).
Epomènwc, ξ ∈ spanImBr.
L mma 1.3.2. 'Estw f : (M2, , ) −→ (M
2+k
c , , ) isometrik elaqistik embˆpti-
sh, ìpou M2 eÐnai polÔptugma Riemann kai M
2+k
c eÐnai polÔptugma Riemann me
stajer kampulìthta tom c c. An {e1, e2} eÐnai topikì orjomonadiaÐo plaÐsio gÔrw
apì to tuqìn shmeÐo p ∈ M2, tìte gia kˆje x1, ..., xr−1 ∈ TpM2 isqÔei
Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = 0.

12 Prokatarktikˆ
Apìdeixh. JewroÔme X1, ..., Xr−1 ∈ ∆(M2) tètoia ¸ste Xi|p = xi, gia i = 1, ..., r−1.
Apì ton orismì thc Br èqoume:
Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) =
f
X1
... f
Xr−1
f
e1
df(e1) + f
e2
df(e2) |p
Nr
p f
.
AfoÔ h f eÐnai elaqistik isqÔei B(e1, e1) + B(e2, e2) = 0. Lìgw tou tÔpou tou
Gauss èqoume
f
e1
df(e1) + f
e2
df(e2) = df( e1 e1) + df( e2 e2) + B(e1, e1) + B(e2, e2)
= df( e1 e1 + e2 e2).
Epomènwc
f
e1 df(e1) + f
e2 df(e2) ∈ ∆f (M2), opìte
f
X1
... f
Xr−1
f
e1
df(e1) + f
e2
df(e2) |p ∈ Tr
p f,
to opoÐo shmaÐnei ìti Br(x1, ..., xr−1, e1|p, e1|p) + Br(x1, ..., xr−1, e2|p, e2|p) = 0.
L mma 1.3.3. 'Estw f : (M2, , ) −→ (M
2+k
c , , ) isometrik elaqistik embˆpti-
sh, ìpou M2 eÐnai polÔptugma Riemann kai M
2+k
c eÐnai polÔptugma Riemann me sta-
jer kampulìthta tom c c. Tìte gia kˆje shmeÐo p tou M2 h diˆstash tou kˆjetou
q¸rou r-tˆxhc Nr
p f eÐnai to polÔ 2 gia kˆje r.
Apìdeixh. 'Estw p èna shmeÐo tou M2. JewroÔme orjomonadiaÐa bˆsh {e1, e2} tou
TpM2. Lìgw tou L mmatoc 1.3.1, gia tuqìnta dianÔsmata x1, ..., xr+1 tou TpM2
to diˆnusma Br(x1, ..., xr+1) eÐnai grammikìc sunduasmìc twn Br(ei1 , ..., eir+1 ), ìpou
i1, ..., ir+1 ∈ {1, 2}. Kˆnontac qr sh thc summetrÐac thc Br kai tou L mmatoc 1.3.2
èqoume,
Br(ei1 , ..., eir+1 ) = ±Br(e1, ..., e1) ±Br(e1, ..., e1, e2).
Apì ta parapˆnw paÐrnoume
Nr
p f = spanImBr
= span{Br(x1, x2, ..., xr+1) : x1, x2, ..., xr+1 ∈ TpMn
}
= span{Br(e1, ..., e1), Br(e1, ..., e1, e2)}.
Epomènwc se kˆje shmeÐo p h diˆstash tou Nr
p f eÐnai to polÔ 2 gia kˆje r.
Ja apodeÐxoume thn parakˆtw prìtash [3, 7].
Prìtash 1.3.1. Upojètoume ìti f : (M2, , ) −→ (M
2+k
c , , ) eÐnai isometrik
elaqistik embˆptish, ìpou M2 eÐnai polÔptugma Riemann kai M
2+k
c eÐnai polÔptugma
Riemann me stajer kampulìthta tom c c. Gia kˆje shmeÐo p tou M2 h eikìna tou
monadiaÐou kÔklou tou efaptìmenou q¸rou TpM2 me kèntro to 0 mèsw thc Br, dhlad

to sÔnolo Er(p) := {Br(x, ..., x) : x ∈ TpM2, |x| = 1}, eÐnai èlleiyh, endeqomènwc
ekfulismènh. Epiplèon, to sÔnolo Er(p) eÐnai kÔkloc aktÐnac ρ ≥ 0 an gia tuqoÔsa
orjomonadiaÐa bˆsh {e1, e2} tou TpM2 isqÔei |Br(e1, ..., e1)| = |Br(e1, ..., e1, e2)| = ρ
kai Br(e1, ..., e1), Br(e1, ..., e1, e2) = 0.
Apìdeixh. JewroÔme to monadiaÐo kÔklo tou efaptìmenou q¸rou sto shmeÐo p, dhlad
to sÔnolo Sp = {x ∈ TpM2 : |x| = 1}. 'Estw {e1, e2} orjomonadiaÐa bˆsh tou TpM2.
Gia kˆje x ∈ Sp, èqoume x = cos θe1 + sin θe2, ìpou θ ∈ R. Kˆnontac qr sh thc
summetrÐac thc Br, èqoume
Br(x, ..., x) =
= Br(cos θe1 + sin θe2, ..., cos θe1 + sin θe2)
=
r+1
m=0
r + 1
m
(cos θ)r+1−m
(sin θ)m
Br(e1, ..., e1
r+1−m
, e2, ..., e2
m
).
SumbolÐzoume me I to sÔnolo twn ˆrtiwn arijm¸n tou sunìlou {1, ..., r + 1} kai
me J to sÔnolo twn peritt¸n arijm¸n tou.
Br(x, ..., x) =
=
m∈I
r + 1
m
(cos θ)r+1−m
(sin θ)m
Br(e1, ..., e1
r+1−m
, e2, ..., e2
m
)
+
r+1
m∈J
r + 1
m
(cos θ)r+1−m
(sin θ)m
Br(e1, ..., e1
r+1−m
, e2, ..., e2
m
).
'Omwc lìgw tou L mmatoc 1.3.2 eÐnai
Br(e1, ..., e1, e2, ..., e2
m
) =
(−1)lBr(e1, ..., e1), m = 2l,
(−1)lBr(e1, ..., e1, e2), m = 2l + 1.
Epomènwc, èqoume
Br(x, ..., x) =
m∈I,m=2l
r + 1
m
(cos θ)r+1−m
(sin θ)m
(−1)l
Br(e1, ..., e1)
+
m∈J,m=2l+1
r + 1
m
(cos θ)r+1−m
(sin θ)m
(−1)l
Br(e1, ..., e1, e2).
EÐnai gnwstì ìti isqÔoun oi sqèseic
cos (r + 1)θ =
m∈I,m=2l
(−1)l r + 1
m
(cos θ)r+1−m
(sin θ)m
,
sin (r + 1)θ =
m∈J,m=2l+1
(−1)l r + 1
m
(cos θ)r+1−m
(sin θ)m
.

14 Prokatarktikˆ
Telikˆ, èqoume
Br(x, ..., x) = cos (r + 1)θ Br(e1, ..., e1) + sin r + 1)θ Br(e1, ..., e1, e2).
Katˆ sunèpeia to sÔnolo
Er(p) = {cos (r + 1)θ Br(e1, ..., e1) + sin (r + 1)θ Br(e1, ..., e1, e2) : θ ∈ R}
eÐnai èlleiyh, endeqomènwc ekfulismènh kai gÐnetai kÔkloc ìtan ta dÔo dianÔsmata
Br(e1, ..., e1), Br(e1, ..., e1, e2) èqoun Ðdio m koc kai eÐnai kˆjeta metaxÔ touc.
H èlleiyh Er(p) ja anafèretai kai wc èlleiyh r-tˆxhc thc f sto p. SumbolÐzoume
me κr, µr ta m kh twn hmiaxìnwn thc èlleiyhc r-tˆxhc ¸ste κr µr 0. OrÐzoume th
sunˆrthsh K⊥
r : M2 −→ R, p −→ K⊥
r (p) := 2κr(p)µr(p) kai thn onomˆzoume kˆjeth
kampulìthta r-tˆxhc thc f. EÐnai fanerì ìti an K⊥
r (p) = 0, tìte dimNr
p f < 2 kai h
èlleiyh Er(p) ekfulÐzetai se eujÔgrammo tm ma shmeÐo. En¸ ìtan K⊥
r (p) > 0, tìte
dimNr
p f = 2 kai h èlleiyh den eÐnai ekfulismènh.
H parakˆtw prìtash ofeÐletai ston Otsuki [17].
Prìtash 1.3.2. Upojètoume ìti f : (M2, , ) −→ (M
2+k
c , , ) eÐnai isometrik
elaqistik embˆptish, ìpou M2 eÐnai sunektikì polÔptugma Riemann kai M
2+k
c eÐnai
polÔptugma Riemann me stajer kampulìthta tom c c.
(i) An upˆrqei fusikìc r me 0 < r < [k+1
2 ], ìpou [k+1
2 ] eÐnai to akèraio mèroc tou
k+1
2 , tètoioc ¸ste gia kˆje shmeÐo p tou M2 na èqoume Nr
p f = {0} kai K⊥
s (p) > 0 gia
kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei olikˆ gewdaisiakì upopolÔptugma Q2r tou M
2+k
c
¸ste f(M2) ⊂ Q2r.
(ii) An upˆrqei fusikìc r me 0 < r < [k+1
2 ] tètoioc ¸ste gia kˆje shmeÐo p tou
M2 na èqoume dimNr
p f = 1 kai K⊥
s (p) > 0 gia kˆje s ∈ {1, ..., r − 1}, tìte upˆrqei
olikˆ gewdaisiakì upopolÔptugma Q2r+1 tou M
2+k
c ¸ste f(M2) ⊂ Q2r+1.
Mia isometrik elaqistik embˆptish f : M2 −→ Sn enìc poluptÔgmatoc Riemann
M2 sthn Sn kaleÐtai koresmènh (full) an h eikìna thc f(M2) den perièqetai se kanèna
olikˆ gewdaisiakì upopolÔptugma thc Sn.
O Lawson [16] apèdeixe to ex c apotèlesma:
Prìtash 1.3.3. An f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish, ìpou M2
prosanatolismèno polÔptugma Riemann, tìte h f eÐnai analutik , dhlad an (U, ϕ)
eÐnai qˆrthc tou M2, tìte h apeikìnish f ◦ϕ−1 : ϕ(U) ⊂ R2 −→ Rn+1 eÐnai analutik .
Parat rhsh 1.3.1. 'Amesh sunèpeia twn Protˆsewn 1.3.2 kai 1.3.3, eÐnai pwc
an f : M2 −→ Sn eÐnai isometrik elaqistik embˆptish enìc prosanatolismènou
poluptÔgmatoc Riemann M2 sthn Sn me Br|U = 0, ìpou U eÐnai anoiktì sÔnolo kai
r ∈ {1, ..., [n−1
2 ]−1}, tìte upˆrqei mègisth sfaÐra S2r thc Sn ¸ste f(U) ⊂ S2r. Lìgw
analutikìthtac isqÔei f(M2) ⊂ S2r. Epomènwc, an upˆrqei r kai anoiktì sÔnolo U
tou M2 ¸ste Br|U = 0, tìte h f den eÐnai koresmènh.

Kefˆlaio 2
Elaqistikèc epifˆneiec sthn Sn
omoiomorfikèc me thn S2
2.1 Epifˆneiec Riemann
Sthn enìthta aut parajètoume stoiqeÐa apì th jewrÐa epifanei¸n Riemann, ta
opoÐa eÐnai aparaÐthta stic apodeÐxeic twn kurÐwn apotelesmˆtwn.
Orismìc 2.1.1. KaloÔme epifˆneia Riemann kˆje topologikì q¸ro Hausdorﬀ M
me arijm simh bˆsh gia thn topologÐa tou, o opoÐoc eÐnai efodiasmènoc me ènan ˆtlanta
{(Uα, ϕα)}α∈I pou plhroÐ ta parakˆtw:
(i) H oikogèneia {Uα}α∈I eÐnai anoikt kˆluyh tou M kai oi apeikonÐseic ϕα : Uα ⊂
M −→ ϕα(Uα) ⊂ C eÐnai omoiomorfismoÐ. To zeÔgoc (Uα, ϕα) onomˆzetai migadikìc
qˆrthc sÔsthma suntetagmènwn tou M.
(ii) Gia kˆje α, β ∈ I, me Uα ∩ Uβ = ∅ h migadik sunˆrthsh ϕα ◦ ϕ−1
β : ϕβ(Uα ∩
Uβ) ⊂ C −→ ϕα(Uα ∩ Uβ) ⊂ C eÐnai olìmorfh.
(iii) H oikogèneia migadik¸n qart¸n {(Uα, ϕα)}α∈I eÐnai mègisth, dhlad an (U, ϕ)
migadikìc qˆrthc me ϕα ◦ϕ−1, ϕ◦ϕ−1
α , α ∈ I, olìmorfec sunart seic, tìte o migadikìc
qˆrthc (U, ϕ) an kei sthn oikogèneia {(Uα, ϕα)}α∈I.
An (U, ϕ) eÐnai migadikìc qˆrthc, tìte gia kˆje p ∈ U, o migadikìc arijmìc
z(p) := ϕ(p) = x(p) + iy(p) dÐdei tic suntetagmènec tou p wc proc ton en lìgw
qˆrth. AxÐzei na shmei¸soume ìti kˆje epifˆneia Riemann eÐnai prosanatolismèno
didiˆstato diaforÐsimo polÔptugma.
Orismìc 2.1.2. 'Estw M epifˆneia Riemann. Mia suneq c migadik sunˆrthsh
f : M −→ C lègetai olìmorfh an gia kˆje migadikì qˆrth (U, ϕ) tou M h migadik
sunˆrthsh f ◦ ϕ−1 : ϕ(U) −→ C eÐnai olìmorfh.
'Estw (M2, , ) polÔptugma Riemann kai (U, ϕ) qˆrthc gÔrw apì to p ∈ M2 me
suntetagmènec (x, y). An isqÔei , = Edx2 + Edy2, E ∈ D(U) kai E > 0, tìte to
sÔsthma suntetagmènwn (x, y) kaleÐtai isìjermo. O Chern [9] apèdeixe to parakˆtw
apotèlesma: 'Estw (M2, , ) diaforÐsimo polÔptugma Riemann. Tìte gÔrw apì kˆje
shmeÐo p ∈ M2 upˆrqei isìjermo sÔsthma suntetagmènwn.
15

16 Elaqistikèc epifˆneiec sthn Sn omoiomorfikèc me thn S2
Prìtash 2.1.1. Kˆje prosanatolismèno didiˆstato diaforÐsimo polÔptugma gÐnetai
katˆ fusikì trìpo epifˆneia Riemann.
Apìdeixh. 'Estw M2 prosanatolismèno didiˆstato diaforÐsimo polÔptugma. Gnw-
rÐzoume ìti kˆje polÔptugma dèqetai metrik Riemann. Epomènwc mporoÔme na efo-
diˆsoume to M2 me metrik Riemann , . Lìgw tou proanaferjèntoc apotelèsmatoc
tou Chern, gÔrw apì kˆje shmeÐo tou M2 upˆrqei sÔsthma isìjermwn suntetagmènwn.
JewroÔme èna shmeÐo p tou M2. GÔrw apì to p, jewroÔme qˆrtec (U, ϕ) kai (V, ψ)
tou prosanatolismoÔ me suntetagmènec (x, y) kai (u, v) antistoÐqwc, ètsi ¸ste
∂
∂x
,
∂
∂y
= 0,
∂
∂x
,
∂
∂x
=
∂
∂y
,
∂
∂y
= E, (2.1)
∂
∂u
,
∂
∂v
= 0,
∂
∂u
,
∂
∂u
=
∂
∂v
,
∂
∂v
= E. (2.2)
Sto anoiktì sÔnolo W = U ∩ V èqoume
∂
∂x
=
∂u
∂x
∂
∂u
+
∂v
∂x
∂
∂v
,
∂
∂y
=
∂u
∂y
∂
∂u
+
∂v
∂y
∂
∂v
.
Apì tic sqèseic (2.1) kai (2.2) paÐrnoume tic sqèseic
∂u
∂x
∂u
∂y
+
∂v
∂x
∂v
∂y
= 0,
(
∂u
∂x
)2
+ (
∂v
∂x
)2
=
E
E
,
(
∂u
∂y
)2
+ (
∂v
∂y
)2
=
E
E
.
EpÐshc, h apeikìnish ψ ◦ ϕ−1 : ϕ(W) −→ ψ(W) èqei jetik Iakwbian orÐzousa afoÔ
oi qˆrtec an koun ston Ðdio prosanatolismì, dhlad
∂u
∂x
∂v
∂y
−
∂v
∂x
∂u
∂y
> 0.
Apì tic teleutaÐec tèsseric sqèseic paÐrnoume
∂u
∂x
=
∂v
∂y
,
∂u
∂y
= −
∂v
∂x
,
dhlad h apeikìnish ψ ◦ ϕ−1 eÐnai olìmorfh kai epomènwc to M2 kajÐstatai epifˆneia
Riemann.
H Prìtash 2.1.1 mac epitrèpei na jewroÔme kˆje prosanatolismèno didiˆstato
diaforÐsimo polÔptugma Riemann M2 wc epifˆneia Riemann. Sto ex c ìtan ja lème
ìti jewroÔme migadikì qˆrth (U, z), ìpou z = x + iy, ja ennooÔme qˆrth (U, ϕ)

Epifˆneiec Riemann 17
tou prosanatolismoÔ tou M2 me suntetagmènec (x, y), ètsi ¸ste
∂
∂x , ∂
∂y = 0 kai
∂
∂x , ∂
∂x = E = ∂
∂y , ∂
∂y isodÔnama , = E|dz|2.
JewroÔme èna prosanatolismèno didiˆstato diaforÐsimo polÔptugma Riemann M2
kai orjomonadiaÐo plaÐsio {e1, e2} tou prosanatolismoÔ. OrÐzetai èna (1,1)-tanustikì
pedÐo J : ∆(M2) −→ ∆(M2), ¸ste gia kˆje shmeÐo p tou M2, J|p : TpM2 −→ TpM2
eÐnai h strof katˆ gwnÐa +π
2 . To (1,1)-tanustikì pedÐo J kaleÐtai migadik dom tou
M2. MigadikopoioÔme ton efaptìmeno q¸ro sto p kai epekteÐnoume C-grammikˆ to J
sto TpM2 ⊗ C wc ex c: J(v + iw) = J(v) + iJ(w). Epeid J ◦ J = −Id, oi mìnec
idiotimèc tou J eÐnai i, −i. O idioq¸roc pou antistoiqeÐ sthn idiotim i eÐnai
TpM2
:= {v ∈ TpM2
⊗ C : J(v) = iv} = {x − iJ(x) : x ∈ TpM2
}
kai o idioq¸roc pou antistoiqeÐ sthn idiotim −i eÐnai
Tp M2
:= {v ∈ TpM2
⊗ C : J(v) = −iv} = {x + iJ(x) : x ∈ TpM2
}.
L mma 2.1.1. Se kˆje shmeÐo p tou M2 isqÔei TpM2 ⊗ C = TpM2 ⊕ Tp M2.
Apìdeixh. 'Estw v ∈ TpM2 ⊗ C, tìte v = u + iw me u, w ∈ TpM2 kai èqoume
v =
u + J(w)
2
− iJ(
u + J(w)
2
)
∈TpM2
+
u − J(w)
2
+ iJ(
u − J(w)
2
)
∈Tp M2
,
dhlad TpM2 ⊗ C = TpM2 + Tp M2. Apomènei na deÐxoume ìti to ˆjroisma eÐnai eujÔ.
'Estw x = a + ib ∈ TpM2 ∩ Tp M2, a, b ∈ TpM2. Tìte x = y − iJ(y) gia kˆpoio
y ∈ TpM2, afoÔ x ∈ TpM2 kai x = h + iJ(h) gia kˆpoio h ∈ TpM2, afoÔ x ∈ Tp M2.
Epomènwc a = y = h kai b = −J(y) = J(h), dhlad −J(y) = J(y), kai ˆra y = 0.
Katˆ sunèpeia a = b = 0 kai to ˆjroisma eÐnai eujÔ.
Lìgw tou L mmatoc 2.1.1, eÐnai fanerì ìti h migadikopoihmènh efaptìmenh dianu-
smatik dèsmh TM2 ⊗ C diaspˆtai wc ex c:
TM2
⊗ C = T M2
⊕ T M2
,
ìpou T M2 := {(p, v) : p ∈ M2, v ∈ TpM2} kai T M2 := {(p, v) : p ∈ M2, v ∈
Tp M2}.
SumbolÐzoume ta pedÐa thc migadikopoihmènhc efaptìmenhc dianusmatik c dèsmhc
me Γ(TM2 ⊗ C) kai jewroÔme to sÔnolo C∞(M2, C) twn diaforÐsimwn sunart sewn
g : M2 −→ C.
Orismìc 2.1.3. KaloÔme migadikì (r, 0)-tanustikì pedÐo kˆje apeikìnish
T : Γ(TM2
⊗ C) × ... × Γ(TM2
⊗ C)
r
−→ C∞
(M2
, C)
h opoÐa eÐnai C∞(M2, C)-grammik wc proc kˆje metablht thc.
EpÐshc, an E eÐnai dianusmatik dèsmh uperˆnw tou M2, kaloÔme migadikì (r, 1)-
tanustikì pedÐo kˆje apeikìnish
T : Γ(TM2
⊗ C) × ... × Γ(TM2
⊗ C)
r
−→ Γ(E ⊗ C)

h opoÐa eÐnai C∞(M2, C)-grammik wc proc kˆje metablht thc.
'Estw τ èna migadikì (r, 0)-tanustikì pedÐo kai σ èna migadikì (s, 0)-tanustikì
pedÐo, tìte orÐzetai to tanustikì ginìmeno τ ⊗σ twn τ, σ wc to ex c migadikì (r+s, 0)-
tanustikì pedÐo
τ ⊗ σ : Γ(TM2
⊗ C) × ... × Γ(TM2
⊗ C)
r+s
−→ C∞
(M2
, C),
(X1 + iY1, ..., Xr+s + iYr+s) −→ τ ⊗ σ(X1 + iY1, ..., Xr+s + iYr+s) =
τ(X1 + iY1, ..., Xr + iYr)σ(Xr+1 + iYr+1, ..., Xr+s + iYr+s).
'Ena ˆllo ginìmeno migadik¸n tanustik¸n pedÐwn eÐnai to exwterikì ginìmeno, to
opoÐo gia migadikˆ (1,0)-tanustikˆ pedÐa τ kai σ eÐnai to migadikì (2,0)-tanustikì pedÐo
τ ∧ σ := τ ⊗ σ − σ ⊗ τ.
'Estw M2 prosanatolismèno diaforÐsimo polÔptugma kai (U, z) migadikìc qˆrthc
tou me z = x + iy. Sto U èqoume ta (1,0)-tanustikˆ pedÐa dx, dy : ∆(U) −→ D(U),
isodÔnama dx, dy : Γ(TU) −→ C∞(U, R) ta opoÐa epekteÐnoume C-grammikˆ sth
migadikopoihmènh efaptìmenh dianusmatik dèsmh TU ⊗C. OrÐzoume to migadikì (1,0)-
tanustikì pedÐo dz : Γ(TU ⊗ C) −→ C∞(U, C), dz := dx + idy kai to suzugèc tou
dz : Γ(TU ⊗ C) −→ C∞(U, C), dz := dx − idy. EpÐshc orÐzoume touc telestèc
Wirtinger
∂
∂z
:=
1
2
∂
∂x
− i
∂
∂y
kai
∂
∂z
:=
1
2
∂
∂x
+ i
∂
∂y
gia touc opoÐouc isqÔei dz( ∂
∂z ) = 1, dz( ∂
∂z ) = 0, dz( ∂
∂z ) = 0 kai dz( ∂
∂z ) = 1. Ta
{ ∂
∂z |p, ∂
∂z |p} apoteloÔn bˆsh tou TpM2⊗C kai mˆlista TpM2 = span{ ∂
∂z |p}, Tp M2 =
span{ ∂
∂z |p}. To sÔnolo twn migadik¸n (1,0)-tanustik¸n pedÐwn eÐnai C∞(U, C)-mìdio
me prˆxeic thn prìsjesh twn (1,0)-tanustik¸n pedÐwn kai to bajmwtì pollalasiasmì
kai bˆsh tou apoteloÔn ta migadikˆ (1,0)-tanustikˆ pedÐa dz, dz. Epiplèon, an W eÐnai
migadikì (1,0)-tanustikì pedÐo, tìte W = W( ∂
∂z )dz + W( ∂
∂z )dz.
To sÔnolo twn migadik¸n (2,0)-tanustik¸n pedÐwn eÐnai C∞(M2, C)-mìdio, ìpwc
kai to sÔnolo twn migadik¸n (1,0)-tanustik¸n pedÐwn, kai èqei wc bˆsh ta migadikˆ
(2,0)-tanustikˆ pedÐa dz⊗dz, dz⊗dz, dz⊗dz kai dz⊗dz. 'Ena migadikì (2,0)-tanustikì
pedÐo W grˆfetai wc
W = W(
∂
∂z
,
∂
∂z
)dz ⊗ dz + W(
∂
∂z
,
∂
∂z
)dz ⊗ dz
+ W(
∂
∂z
,
∂
∂z
)dz ⊗ dz + W(
∂
∂z
,
∂
∂z
)dz ⊗ dz.
Genikìtera, to sÔnolo twn migadik¸n (r, 0)-tanustik¸n pedÐwn eÐnai C∞(M2, C)-
mìdio me bˆsh ta migadikˆ (r, 0)-tanustikˆ pedÐa dw1 ⊗ ... ⊗ dwr, ìpou wj = z z gia
kˆje j = 1, ..., r. Kˆje migadikì (r,0)-tanustikì pedÐo W grˆfetai wc
W =
p+q=r
W(p,q)
dzp
dzq
,

SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn 19
ìpou
W(r,0)
dzr
:= W(
∂
∂z
, ...,
∂
∂z
r
) dz ⊗ ... ⊗ dz
r
,
W(r−1,1)
dzr−1
dz := W(
∂
∂z
, ...,
∂
∂z
r−1
,
∂
∂z
) dz ⊗ ... ⊗ dz
r−1
⊗dz
+ W(
∂
∂z
, ...,
∂
∂z
r−2
,
∂
∂z
∂
∂z
) dz ⊗ ... ⊗ dz
r−2
⊗dz ⊗ dz
+ ... + W(
∂
∂z
∂
∂z
, ...,
∂
∂z
r−1
)dz ⊗ dz ⊗ ... ⊗ dz
r−1
,
.
.
.
W(0,r)
dzr
:= W(
∂
∂z
, ...,
∂
∂z
r
) dz ⊗ ... ⊗ dz
r
.
Kˆje migadikì (r, 0)-tanustikì pedÐo T thc morf c
T = f(z) dz ⊗ ... ⊗ dz
r
= f(z)dzr
,
ìpou (U, ϕ) eÐnai migadikìc qˆrthc me migadik suntetagmènh z kai f ∈ C∞(U, C) lège-
tai r-diaforikì. EÐnai fanerì ìti to r-diaforikì T eÐnai migadikì (r, 0)-tanustikì pedÐo
tètoio ¸ste an kˆpoio apì ta v1, ..., vr an kei sth dèsmh T M2, tìte T(v1, ..., vr) = 0.
To r-diaforikì T = f(z)dzr kaleÐtai olìmorfo an h f eÐnai olìmorfh sunˆrthsh.
Gia ta r-diaforikˆ isqÔei to akìloujo shmantikì apotèlesma, gnwstì kai wc Je-
¸rhma Riemann-Roch [12].
Je¸rhma 2.1.1. 'Estw M epifˆneia Riemann omoiomorfik me thn S2. An Φ eÐnai
olìmorfo r-diaforikì orismèno sthn epifˆneia M, tìte Φ = 0.
2.2 SumpageÐc elaqistikèc epifˆneiec gènouc mhdèn
To kleidÐ gia thn apìdeixh twn jewrhmˆtwn, pou anafèrontai sthn eisagwg , eÐnai
h diapÐstwsh ìti gia sumpageÐc elaqistikèc epifˆneiec gènouc mhdèn sth sfaÐra, oi el-
leÐyeic kˆje tˆxhc eÐnai kÔkloi sqedìn pantoÔ. Stìqoc thc paroÔshc paragrˆfou eÐnai
h apìdeixh aut c thc diapÐstwshc. Gia to skopì autì, orÐzoume katˆllhla r-diaforikˆ
ta opoÐa apodeiknÔoume ìti eÐnai olìmorfa kai lìgw tou Jewr matoc Riemann-Roch,
eÐnai ek tautìthtoc mhdèn.

'Estw f : M2 −→ Sn, n ≥ 3, elaqistik epifˆneia, dhlad isometrik elaqistik
embˆptish enìc prosanatolismènou, sunektikoÔ, didiˆstatou poluptÔgmatoc Riemann
(M2, , ) sthn Sn me deÔterh jemeli¸dh morf B.
MigadikopoioÔme thn efaptìmenh dèsmh TM2 kai thn kˆjeth dèsmh Nf, kaj¸c kai
kˆje upodèsmh thc. EpÐshc, epekteÐnoume C-grammikˆ kˆje tanustikì pedÐo pou ja
oristeÐ sth sunèqeia. Apì ed¸ kai sto ex c jewroÔme migadikì qˆrth (U, z) tou M2 me
z = x+iy kai , = E|dz|2. Me , sumbolÐzoume th C-grammik epèktash thc metrik c
tou M2 kaj¸c kai th sun jh metrik thc Sn. Epilègoume topikì orjomonadiaÐo plaÐsio
katˆ m koc thc f me efaptìmeno mèroc {e1, e2} ¸ste e1 = 1√
E
∂
∂x, e2 = 1√
E
∂
∂y kai
kˆjeto mèroc {eα}. Oi telestèc Wirtinger tìte eÐnai
∂
∂z = 1
2
√
E(e1 − ie2) kai
∂
∂z =
1
2
√
E(e1 + ie2).
SumbolÐzoume me {ω1, ω2} to sumplaÐsio tou {e1, e2}. Profan¸c ω1 =
√
Edx kai
ω2 =
√
Edy. Sto U orÐzoume to migadikì (1,0)-tanustikì pedÐo φ := ω1 + iω2 gia to
opoÐo isqÔoun oi sqèseic
φ =
√
Edz, (2.3)
dφ =
1
2
√
E
dE ∧ dz. (2.4)
Apì tic exis¸seic dom c tou M2 èqoume
dφ = −iω12 ∧ φ. (2.5)
Gia α = 3, ..., n orÐzoume tic sunart seic
Hα
1 : U −→ C, Hα
1 := hα
11 + ihα
12,
ìpou hα
11 = B(e1, e1), eα kai hα
12 = B(e1, e2), eα . EÐnai fanerì ìti Hα
1 ∈ C∞(U, C).
Gia kˆje fusikì s ≥ 2 kai gia kˆje α = 3, ..., n orÐzoume tic sunart seic
Hα
s : U −→ C, Hα
s := hα
(s),1 + ihα
(s),2,
ìpou hα
(s),1 := Bs(e1, ..., e1), eα kai hα
(s),2 := Bs(e1, ..., e1, e2), eα .
Tèloc, orÐzoume to m koc thc (r + 1)-jemeli¸douc morf c na eÐnai h sunˆrthsh
Br :=
2
j1,...,jr+1=1
|Br(ej1 , ..., ejr+1 )|2.
EpekteÐnontac C-grammikˆ th deÔterh jemeli¸dh morf B, èqoume to migadikì
(2,1)-tanustikì pedÐo
B : Γ(TM2
⊗ C) × Γ(TM2
⊗ C) −→ Γ(Nf ⊗ C).
Kˆje mÐa apì tic n + 1 sunist¸sec tou tanustikoÔ pedÐou B ston EukleÐdeio q¸ro
Rn+1 eÐnai migadikì (2,0)-tanustikì pedÐo. Epomènwc, gia kˆje migadikì qˆrth (U, z)
me z = x + iy kai , = E|dz|2, analÔontac kˆje sunist¸sa tou B, ìpwc sthn
parˆgrafo 2.1, èqoume thn akìloujh anˆlush gia to B sto U
B = B(2,0)
dz2
+ B(1,1)
dzdz + B(0,2)
dz2
,

ìpou
B(2,0)
dz2
:= B(
∂
∂z
,
∂
∂z
)dz ⊗ dz,
B(1,1)
dzdz := B(
∂
∂z
,
∂
∂z
)dz ⊗ dz + B(
∂
∂z
,
∂
∂z
)dz ⊗ dz,
B(0,2)
dz2
:= B(
∂
∂z
,
∂
∂z
)dz ⊗ dz
kai isqÔei B(0,2) = B(2,0).
Epeid h f eÐnai elaqistik , isqÔei B(e1, e1)+B(e2, e2) = 0. 'Omwc apì thn epilog
tou plaisÐou èqoume e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y . Epiplèon, eÐnai
∂
∂x = ∂
∂z + ∂
∂z kai
∂
∂y = i( ∂
∂z − ∂
∂z ). Katˆ sunèpeia isqÔei B( ∂
∂z , ∂
∂z ) = 0, isodÔnama B(1,1)dzdz = 0.
Epomènwc to B dèqetai thn anˆlush
B = B(2,0)
dz2
+ B(2,0)dz2
.
OrÐzoume to 4-diaforikì Φ1 := B(2,0), B(2,0) dz4.
Profan¸c B(2,0), B(2,0) ∈ C∞(U, C). Ja deÐxoume ìti to Φ1 eÐnai kalˆ orismèno,
dhlad anexˆrthto thc epilog c tou qˆrth. Gia to skopì autì, jewroÔme migadikoÔc
qˆrtec (U, ϕ) me migadik suntetagmènh z kai (V, ψ) me migadik suntetagmènh ζ ¸ste
U ∩ V = ∅. To M2 eÐnai epifˆneia Riemann ˆra ψ ◦ ϕ−1 : ϕ(U ∩ V ) −→ ψ(U ∩ V )
kai ϕ ◦ ψ−1 : ψ(U ∩ V ) −→ ϕ(U ∩ V ) eÐnai olìmorfec, dhlad
∂ζ
∂z = 0 kai
∂z
∂ζ
= 0. Sto
U ∩ V èqoume loipìn tic sqèseic
∂
∂z = ∂ζ
∂z
∂
∂ζ , dζ = ∂ζ
∂z dz, kai epomènwc isqÔei
B(
∂
∂ζ
,
∂
∂ζ
), B(
∂
∂ζ
,
∂
∂ζ
) dζ4
= (
∂ζ
∂z
)2
B(
∂
∂ζ
,
∂
∂ζ
), (
∂ζ
∂z
)2
B(
∂
∂ζ
,
∂
∂ζ
) dz4
= B(
∂
∂z
,
∂
∂z
), B(
∂
∂z
,
∂
∂z
) dz4
.
Sunep¸c, to 4-diaforikì Φ1 eÐnai kalˆ orismèno. Sth sunèqeia ja doÔme ìti to Φ1
eÐnai olìmorfo 4-diaforikì.
L mma 2.2.1. 'Estw f : M2 −→ Sn, n ≥ 3, elaqistik epifˆneia. Tìte isqÔoun:
(i) (dH
α
1 − 2iH
α
1 ω12 + β H
β
1 ωβα) ∧ φ = 0 gia kˆje α ≥ 3.
(ii) To 4-diaforikì Φ1 eÐnai olìmorfo. Eidikˆ, an h epifˆneia Riemann M2 eÐnai
omoiomorfik me th didiˆstath sfaÐra, tìte Φ1 = 0.
Apìdeixh. (i) JumÐzoume ìti isqÔoun oi sqèseic
ω1α = hα
11ω1 + hα
12ω2, ω2α = hα
12ω1 − hα
11ω2,
dω1 = ω12 ∧ ω2, dω2 = −ω12 ∧ ω1.
Qrhsimopoi¸ntac tic parapˆnw sqèseic, oi exis¸seic Codazzi
dωjα =
l
ωjl ∧ ωlα +
β
ωjβ ∧ ωβα, j = 1, 2,

gÐnontai
β
hβ
11ωβα ∧ ω2 +
β
hβ
12ωβα ∧ ω1 = −dhα
11 ∧ ω1 − dhα
12 ∧ ω2 − 2hα
11dω1 − 2hα
12dω2,
β
hβ
12ωβα ∧ ω1 −
β
hβ
11ωβα ∧ ω2 = −dhα
12 ∧ ω1 + dhα
11 ∧ ω2 − 2hα
12dω1 + 2hα
11dω2.
Epeid H
α
1 = hα
11 − ihα
12, apì ta parapˆnw kai apì tic sqèseic (2.3), (2.5) eÔkola
sumperaÐnoume ìti
(dH
α
1 − 2iH
α
1 ω12 +
β
H
β
1 ωβα) ∧ φ = 0.
(ii) UpologÐzoume:
B(2,0)
= B(
∂
∂z
,
∂
∂z
) =
E
2
B(e1, e1) − iB(e1, e2) .
To sÔnolo {eα} apoteleÐ orjomonadiaÐo plaÐsio thc kˆjethc dèsmhc, ˆra
B(2,0)
=
α
B(2,0)
, eα eα
=
E
2 α
B(e1, e1), eα eα − i
E
2 α
B(e1, e2), eα eα
=
E
2 α
hα
11eα − i
E
2 α
hα
12eα
=
E
2 α
H
α
1 eα.
Telikˆ eÐnai
Φ1 = B(2,0)
, B(2,0)
dz4
=
E2
4 α
(H
α
1 )2
dz4
.
Jètoume f1 := E2
4 α(H
α
1 )2. Ja apodeÐxoume ìti h f1 eÐnai olìmorfh.
Pollaplasiˆzontac th sqèsh pou apodeÐxame sto (i) me H
α
1 , ajroÐzontac wc proc α
kai kˆnontac qr sh twn sqèsewn (2.3), (2.4), (2.5) kai thc sqèshc
α,β H
β
1 H
α
1 ωβα ∧
φ = 0, paÐrnoume df1 ∧ dz = 0 pou shmaÐnei ìti
∂f1
∂z = 0.
'Ara to Φ1 eÐnai olìmorfo 4-diaforikì sth M2. Epiplèon, an h epifˆneia Riemann
M2 eÐnai omoiomorfik me thn S2, tìte sÔmfwna me to Je¸rhma Riemann-Roch, lam-
bˆnoume Φ1 = 0.
Apì ed¸ kai sto ex c upojètoume ìti h f : M2 −→ Sn, n ≥ 3, eÐnai sumpag c,
koresmènh, elaqistik epifˆneia gènouc mhdèn, dhlad to M2 eÐnai omoiomorfikì me
thn S2.

Epeid Φ1|p = 0 gia kˆje p ∈ M2, èqoume B(2,0), B(2,0) (p) = 0, isodÔnama
|B(e1, e1)|2
(p) − |B(e1, e2)|2
(p) − 2i B(e1, e1), B(e1, e2) (p) = 0.
Apì ed¸ sumperaÐnoume ìti se kˆje shmeÐo tou M2 ta dianÔsmata B(e1, e1)|p kai
B(e1, e2)|p eÐnai tou idÐou m kouc kai kˆjeta metaxÔ touc. SÔmfwna me thn Prìtash
1.3.1 se kˆje shmeÐo p tou M2 h èlleiyh E1(p) eÐnai kÔkloc me aktÐna
κ1(p) = |B(e1, e1)|(p) = |B(e1, e2)|(p).
Autì shmaÐnei ìti se kˆje shmeÐo p ∈ M2 èqoume dimN1
p f ∈ {0, 2}.
An dimN1
p f = 0 gia kˆje shmeÐo p ∈ M2, tìte h deÔterh jemeli¸dhc morf B
eÐnai tautotikˆ 0. Se aut thn perÐptwsh h embˆptish f eÐnai olikˆ gewdaisiak kai
sÔmfwna me thn Parat rhsh 1.3.1, to f(M2) perièqetai se èna olikˆ gewdaisiakì
didiˆstato upopolÔptugma thc Sn kai epomènwc to f(M2) eÐnai mia mègisth 2-sfaÐra
thc Sn. 'Atopo, afoÔ h embˆptish eÐnai koresmènh. Epomènwc maxq∈M2 dimN1
q f = 2,
to opoÐo shmaÐnei ìti n ≥ 4.
An n = 4, tìte h diadikasÐa stamatˆ ed¸. An n ≥ 5, tìte suneqÐzoume th diadika-
sÐa.
'Eqoume dh orÐsei to m koc thc deÔterhc jemeli¸douc morf c wc th sunˆrthsh
B : M2 −→ R, me tÔpo
B = |B(e1, e1)|2 + 2|B(e1, e2)|2 + |B(e2, e2)|2.
Profan¸c h sunˆrthsh B eÐnai suneq c kai B = 2κ1.
OrÐzoume to sÔnolo M1 := {p ∈ M2 : dimN1
p f = maxq∈M2 dimN1
q f}. Profan¸c
isqÔei
M1 = {p ∈ M2
: dimN1
p f = 2}
= {p ∈ M2
: B (p) > 0}
= {p ∈ M2
: κ1(p) > 0}.
To M1 eÐnai mh kenì, afoÔ h f eÐnai koresmènh. Epiplèon, eÐnai anoiktì upo-
sÔnolo tou M2, afoÔ eÐnai h antÐstrofh eikìna tou anoiktoÔ sunìlou (0, +∞) mèsw
thc suneqoÔc sunˆrthshc B . Epomènwc, to M1 eÐnai prosanatolismèno didiˆstato
polÔptugma Riemann kai sÔmfwna me thn Prìtash 2.1.1, eÐnai epifˆneia Riemann.
Parat rhsh 2.2.1. To sÔnolo M1 eÐnai puknì uposÔnolo tou M2.
Prˆgmati, ac upojèsoume ìti to eswterikì tou sunìlou Mc
1 = M2 − M1 den
eÐnai kenì sÔnolo kai èstw U mia sunektik sunist¸sa tou int(Mc
1). Sto U isqÔei
B|U = 0 kai sÔmfwna me thn Parat rhsh 1.3.1 to f(M2) eÐnai mia mègisth 2-sfaÐra
thc Sn ìpwc parapˆnw. 'Atopo, afoÔ h f eÐnai koresmènh.
H N1f|M1 eÐnai dianusmatik dèsmh me bajmÐda 2. Epomènwc h (T1f|M1 ⊕N1f|M1 )⊥
eÐnai dianusmatik dèsmh me bajmÐda n − 4 kai h jemeli¸dhc morf trÐthc tˆxhc thc f
periorismènh sto M1 eÐnai (3,1)-tanustikì pedÐo
B2|M1 : ∆(M1) × ∆(M1) × ∆(M1) −→ Γ (T1
f|M1 ⊕ N1
f|M1 )⊥
,

L mma 2.2.2. 'Estw f : M2 −→ Sn, n ≥ 5, sumpag c, koresmènh, elaqistik
epifˆneia gènouc mhdèn. Gia kˆje migadikì qˆrth (U, z) tou M2 me z = x + iy kai
, = E|dz|2 upˆrqei orjomonadiaÐo plaÐsio katˆ m koc thc f|M1∩U me efaptìmeno
mèroc {e1, e2}, ìpou e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y kai me kˆjeto mèroc {eα}, ìpou e3 =
1
κ1
B(e1, e1), e4 = 1
κ1
B(e1, e2). Epiplèon, sto M1 ∩ U isqÔoun
1:
(i) (d log κ1 + iω34) ∧ φ = 2iω12 ∧ φ,
(ii) ω3α(e1) = −ω4α(e2), ω3α(e2) = ω4α(e1), α = 5, ..., n,
(iii) (dH
α
2 − 3iH
α
2 ω12 + β≥5 H
β
2 ωβα) ∧ φ = 0, α = 5, ..., n,
(iv) To 6-diaforikì Φ2 eÐnai olìmorfo.
Apìdeixh. Lìgw tou L mmatoc 2.2.1 èqoume Φ1 = 0. Sunep¸c ta B(e1, e1), B(e1, e2)
eÐnai isom kh kai kˆjeta metaxÔ touc. Epomènwc, mporoÔme na epilèxoume orjo-
monadiaÐo plaÐsio {e3, e4} thc dianusmatik c dèsmhc N1f|M1∩U tètoio ¸ste e3 =
1
κ1
B(e1, e1), e4 = 1
κ1
B(e1, e2).
(i) Lìgw thc epilog c tou plaisÐou {eα}, eÐnai H3
1 |M1∩U = κ1, H4
1 |M1∩U = iκ1 kai
Hα
1 |M1∩U = 0 gia α ≥ 5. Apì to L mma 2.2.1(i) gia α = 3 lambˆnoume sto M1 ∩ U
th sqèsh
(d log κ1 + iω34) ∧ φ = 2iω12 ∧ φ.
(ii) Sto M1 ∩U lambˆnontac upìyh ton orismì thc B2 kai touc tÔpouc twn Gauss
kai Weingarten èqoume
B2|M1 (e1, e1, e1) = f
e1
f
e1
df(e1)
(T1f|M1
⊕N1f|M1
)⊥
= f
e1
B(e1, e1)
(T1f|M1
⊕N1f|M1
)⊥
= f
e1
(κ1e3)
(T1f|M1
⊕N1f|M1
)⊥
= e1(κ1)e3
∈N1f|M1
+κ1
f
e1
e3
(T1f|M1
⊕N1f|M1
)⊥
= κ1( f
e1
e3)(T1f|M1
⊕N1f|M1
)⊥
= κ1( ⊥
e1
e3)(T1f|M1
⊕N1f|M1
)⊥
.
Epeid o deÔteroc kˆjetoc q¸roc thc f eÐnai upìqwroc tou dianusmatikoÔ q¸rou pou
parˆgetai apì ta {e5, ..., en} èqoume
B2|M1 (e1, e1, e1) = κ1
n
α=5
⊥
e1
e3, eα eα = κ1
n
α=5
ω3α(e1)eα.
'Omwc, lìgw tou L mmatoc 1.3.2 eÐnai
B2|M1 (e1, e1, e1) = −B2|M1 (e2, e1, e2) = −κ1( ⊥
e2
e4)(T1f|M1
⊕N1f|M1
)⊥
= −κ1
n
α=5
ω4α(e2)eα.
1
Epeid to M1 eÐnai puknì uposÔnolo tou M2
(Parat rhsh 2.2.1) eÐnai M1 ∩ U = ∅.

Sunep¸c isqÔei
ω3α(e1) = −ω4α(e2), α = 5, ..., n.
'Omoia,
B2|M1 (e1, e1, e2) = κ1
n
α=5
ω4α(e1)eα = κ1
n
α=5
ω3α(e2)eα
kai
ω3α(e2) = ω4α(e1), α = 5, ..., n.
(iii) Sto M1 ∩ U gia α ≥ 5 eÐnai hα
(2),1 = κ1ω3α(e1) kai hα
(2),2 = κ1ω3α(e2). EÐnai
fanerì ìti oi sunart seic hα
(2),1, hα
(2),2 : M1 ∩ U −→ R eÐnai diaforÐsimec kai tètoiec
¸ste B2|M1 (e1, e1, e1) = n
α=5 hα
(2),1eα, B2|M1 (e1, e1, e2) = n
α=5 hα
(2),2eα. Epiplèon,
sto M1 ∩ U oi migadikèc sunart seic Hα
2 eÐnai diaforÐsimec. Lìgw tou (ii) isqÔei h
sqèsh Hα
2 φ = κ1ω3α + iκ1ω4α. EpÐshc, sto M1 ∩ U gia α ≥ 5 kai gia j ∈ {1, 2}
èqoume
hα
1j = B(e1, ej), eα = 0,
hα
2j = B(e2, ej), eα = 0.
Epomènwc sto M1 ∩ U isqÔei ω1α = j hα
1jωj = 0 kai ω2α = j hα
2jωj = 0 kai oi
exis¸seic Ricci dÐnoun
dω3α = ω34 ∧ ω4α +
β≥5
ω3β ∧ ωβα,
dω4α = −ω34 ∧ ω3α +
β≥5
ω4β ∧ ωβα.
ParagwgÐzontac exwterikˆ th sqèsh Hα
2 φ = κ1ω3α +iκ1ω4α, qrhsimopoi¸ntac tic
parapˆnw exis¸seic Ricci kai th sqèsh (2.5) paÐrnoume
dHα
2 ∧ φ + iHα
2 ω12 ∧ φ = Hα
2 d(log κ1) ∧ φ − iHα
2 ω34 ∧ φ +
β≥5
Hβ
2 φ ∧ ωβα.
PaÐrnoume th suzug sqèsh aut c kai lambˆnontac upìyh th sqèsh pou apodeÐxame
sto (i) èqoume to zhtoÔmeno.
(iv) Epeid
B2|
(3,0)
M1
= B2|M1 (
∂
∂z
,
∂
∂z
,
∂
∂z
) =
1
2
E
3
2
α≥5
H
α
2 eα,
èqoume
Φ2 = B2|
(3,0)
M1
, B2|
(3,0)
M1
dz6
=
1
4
E3
α,β≥5
H
α
2 H
β
2 eα, eβ dz6
=
1
4
E3
α≥5
(H
α
2 )2
dz6
.
Jètoume f2 := 1
4E3
α≥5(H
α
2 )2. Ja apodeÐxoume ìti h f2 eÐnai olìmorfh. Pollaplasi-
ˆzoume th sqèsh pou apodeÐxame sto (iii) me H
α
2 , ajroÐzoume gia α ≥ 5, kˆnoume qr sh
twn sqèsewn (2.4), (2.5) kai
α,β≥5 H
β
2 H
α
2 ωβα ∧ φ = 0 kai brÐskoume df2 ∧ dz = 0.
Autì shmaÐnei ìti
∂f2
∂z = 0 kai epomènwc to Φ2 eÐnai olìmorfo sto M1 ∩ U.

Stìqoc mac eÐnai na deÐxoume ìti to Φ2 eÐnai tautotikˆ mhdèn. Gia autì ja deÐxoume
ìti to Mc
1 eÐnai peperasmèno sÔnolo kai ìti upˆrqei upodèsmh thc kˆjethc dèsmhc me
bajmÐda 2 orismènh sto M2, h opoÐa sto M1 tautÐzetai me th dèsmh N1f|M1 . Gia to
skopì autì qreiazìmaste to epìmeno l mma to opoÐo ofeÐletai ston Chern [8] (blèpe
epÐshc [5] [11]).
L mma 2.2.3. An f1, ..., fm : U ⊂ C −→ C eÐnai migadikèc sunart seic oi opoÐec se
mia perioq U tou mhdenìc ikanopoioÔn to diaforikì sÔsthma
∂fi
∂z = j aijfj, ìpou
aij : U −→ C eÐnai diaforÐsimec sunart seic kai i, j = 1, ..., m, tìte eÐte f1 = ... =
fm = 0, eÐte oi koinèc rÐzec twn f1, ..., fm eÐnai memonwmènec kai mˆlista upˆrqei
jetikìc akèraioc l kai diaforÐsimec sunart seic f∗
1 , ..., f∗
m : U ⊂ C −→ C ¸ste na
isqÔei fi(z) = zlf∗
i (z) me f∗
1 (0), ..., f∗
m(0) = (0, ..., 0).
L mma 2.2.4. An f : M2 −→ Sn, n ≥ 5, sumpag c, koresmènh, elaqistik epi-
fˆneia gènouc mhdèn, tìte to sÔnolo Mc
1 eÐnai peperasmèno. Epiplèon, upˆrqei dia-
nusmatik upodèsmh N∗1f thc kˆjethc dèsmhc Nf me bajmÐda 2 ¸ste N∗1f|M1 =
N1f|M1 .
Apìdeixh. 'Estw p ∈ Mc
1. JewroÔme gÔrw apì to p migadikì qˆrth (U, z) me z(p) = 0,
z = x + iy, , = E|dz|2 kai orjomonadiaÐo plaÐsio katˆ m koc thc f me efaptìmeno
E
∂
∂x , e2 = 1√
E
∂
∂y , kai kˆjeto mèroc {eα}. Sto U èqoume
to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 kai tic migadikèc sunart seic Hα
1 =
hα
11 + ihα
12 gia α = 3, ..., n. Me th bo jeia twn sqèsewn
dH
α
1 =
∂H
α
1
∂z
dz +
∂H
α
1
∂z
dz,
ωαβ = ωαβ(
∂
∂z
)dz + ωαβ(
∂
∂z
)dz
kai twn sqèsewn (2.4), (2.5), apì to L mma 2.2.1(i) lambˆnoume
∂H
α
1
∂z
=
β
gαβH
β
1 ,
ìpou gαβ := −ωβα( ∂
∂z ) − δαβd log E( ∂
∂z ) eÐnai diaforÐsimec sunart seic kai δαβ eÐnai
to dèlta tou Kronecker.
SÔmfwna me to L mma 2.2.3 eÐte H
α
1 = 0 gia kˆje α = 3, ..., n, eÐte oi koinèc rÐzec
twn H
α
1 eÐnai memonwmènec kai upˆrqei jetikìc akèraioc l1 kai diaforÐsimec sunart seic
H
∗α
1 : U −→ C ¸ste na isqÔei
H
α
1 = zl1
H
∗α
1 (2.6)
gia kˆje α kai H
∗3
1 (0), ..., H
∗n
1 (0) = (0, ..., 0).
An tan H
α
1 = 0 gia kˆje α = 3, ..., n, tìte B|U = 0. 'Atopo, afoÔ sÔmfwna me
thn Parat rhsh 1.3.1 h f den ja tan koresmènh. 'Ara oi koinèc rÐzec twn H
α
1 eÐnai
memonwmènec. Oi koinèc rÐzec twn H
α
1 , α = 3, ..., n, eÐnai akrib¸c ta shmeÐa ìpou o
pr¸toc kˆjetoc q¸roc gÐnetai mhdenikìc. Autì shmaÐnei ìti to sÔnolo Mc
1 apoteleÐtai
apì memonwmèna shmeÐa kai afoÔ to M2 eÐnai sumpagèc, eÐnai peperasmèno sÔnolo.

Epeid to Mc
1 eÐnai peperasmèno sÔnolo, mporoÔme na jewr soume gÔrw apì to
tuqìn shmeÐo p ∈ Mc
1 migadikì qˆrth (U, z) me z(p) = 0 gia ton opoÐo ìmwc isqÔei
U ∩ Mc
1 = {p}. Epilègoume me ton sun jh trìpo topikì orjomonadiaÐo plaÐsio katˆ
m koc thc f. 'Eqoume dh dei ìti se kˆje shmeÐo q tou M2 h èlleiyh E1(q) eÐnai kÔkloc,
opìte ta B(e1, e1)|q, B(e1, e2)|q eÐnai tou idÐou m kouc kai kˆjeta metaxÔ touc gia kˆje
q ∈ U. Autì shmaÐnei ìti to diˆnusma B(e1, e1)|q −iB(e1, e2)|q eÐnai isotropikì, dhlad
B(e1, e1)|q − iB(e1, e2)|q, B(e1, e1)|q − iB(e1, e2)|q = 0 gia kˆje q ∈ U.
'Omwc lìgw thc (2.6) èqoume B(e1, e1) − iB(e1, e2) = α H
α
1 eα = zl1
α H
∗α
1 eα,
ˆra sto U èqoume z2l1
α H
∗α
1 eα, α H
∗α
1 eα = 0. Sto U −{p} eÐnai z = 0, epomènwc
sto U − {p} èqoume th sqèsh
α
H
∗α
1 eα,
α
H
∗α
1 eα = 0.
Lìgw sunèqeiac, h sqèsh aut isqÔei kai sto p. Epomènwc gia kˆje q ∈ U isqÔei
Re
α
H
∗α
1 eα (q) = Im
α
H
∗α
1 eα (q) = 0
kai
Re
α
H
∗α
1 eα |q, Im
α
H
∗α
1 eα |q = 0.
Gia kˆje q ∈ U orÐzoume ton didiˆstato upìqwro N∗1
q f tou kˆjetou q¸rou thc f
sto q
N∗1
q f := span Re
n
α=3
H
∗α
1 eα |q, Im
n
α=3
H
∗α
1 eα |q .
Sto U èqoume th dèsmh bajmÐdac 2
N∗1
f|U =
q∈U
N∗1
q f.
Epiplèon, epeid gia q ∈ U − {p} èqoume
N1
q f = span{B(e1, e1)|q, B(e1, e2)|q}
= span Re B(e1, e1)|q − iB(e1, e2)|q , Im B(e1, e1)|q − iB(e1, e2)|q
= span Re zl1
α
H
∗α
1 eα |q, Im zl1
α
H
∗α
1 eα |q
= span Re
α
H
∗α
1 eα |q, Im
α
H
∗α
1 eα |q ,
isqÔei
N1
q f =
span{Re( α H
∗α
1 eα)|q, Im( α H
∗α
1 eα)|q}, q ∈ U − {p},
{0}, q = p,
ˆra N∗1f|U−{p} = N1f|U−{p}.
An epanalˆboume thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou Mc
1, pou
ìpwc eÐdame eÐnai peperasmèna to pl joc, apoktoÔme th dianusmatik dèsmh N∗1f me
bajmÐda 2 sto M2 gia thn opoÐa isqÔei N∗1f|M1 = N1f|M1 .

Sto M1 èqoume to migadikì (3,1)-tanustikì pedÐo B2|M1 kai to olìmorfo 6-
diaforikì Φ2. Ja deÐxoume ìti Φ2 = 0. Gia to skopì autì orÐzoume thn apeikìnish
B∗
2 : ∆(M2
) × ∆(M2
) × ∆(M2
) −→ Γ (T1
f ⊕ N∗1
f)⊥
,
(X1, X2, X3) −→ B∗
2(X1, X2, X3) = f
X1
f
X2
df(X3)
(T1f⊕N∗1f)⊥
.
H B∗
2 eÐnai D(M2)-grammik wc proc thn pr¸th metablht thc. Ja deÐxoume ìti
eÐnai summetrik kai epomènwc ja eÐnai D(M2)-grammik wc proc ìlec tic metablhtèc
thc.
L mma 2.2.5. IsqÔoun ta akìlouja:
(i) To B∗
2 eÐnai summetrikì (3, 1)-tanustikì pedÐo. Epiplèon, gia kˆje X ∈ ∆(M2)
kai {e1, e2} tuqìn topikì orjomonadiaÐo plaÐsio tou M2 isqÔei
B∗
2(X, e1, e1) + B∗
2(X, e2, e2) = 0.
(ii) Gia X1, X2, X3 ∈ ∆(M2) isqÔei
B∗
2(X1, X2, X3) = f
X1
B(X2, X3)
(T1f⊕N∗1f)⊥
.
(iii) Gia kˆje p ∈ M2 èqoume
B∗
2|p =
B2|p, p ∈ M1,
0, p ∈ Mc
1.
Apìdeixh. (i) Lìgw tou L mmatoc 2.2.4, isqÔei N∗1f|M1 = N1f|M1 . Epomènwc apì
touc orismoÔc twn B2 kai B∗
2 eÐnai B∗
2|M1 = B2|M1 . 'Ara h B∗
2|M1 eÐnai summetrik kai
plhroÐ th sqèsh B∗
2|M1 (X, e1, e1) + B∗
2|M1 (X, e2, e2) = 0. GnwrÐzoume apì to L mma
2.2.4 ìti to Mc
1 eÐnai peperasmèno sÔnolo kai epeid h B∗
2|M1 eÐnai summetrikì (3,1)-
tanustikì pedÐo, lìgw sunèqeiac, h B∗
2 eÐnai summetrik kai plhroÐ thn en lìgw sqèsh
kai sta shmeÐa tou Mc
1.
(ii) 'Estw X1, X2, X3 ∈ ∆(M2). Qrhsimopoi¸ntac ton tÔpo tou Gauss èqoume
B∗
2(X1, X2, X3) = f
X1
f
X2
df(X3)
(T1f⊕N∗1f)⊥
= f
X1
df( f
X2
X3)
∈T2f
+ f
X1
B(X2, X3)
(T1f⊕N∗1f)⊥
= f
X1
B(X2, X3)
(T1f⊕N∗1f)⊥
.
(iii) 'Eqoume dh dei ìti B∗
2|M1 = B2|M1 . Ja deÐxoume ìti B∗
2|Mc
1
= 0. Gia to skopì
autì jewroÔme tuqìn topikì orjomonadiaÐo plaÐsio {e∗
3, e∗
4} thc dianusmatik c dèsmhc
N∗1f. Tìte lìgw twn Lhmmˆtwn 1.3.1 kai 2.2.4 èqoume
B(e1, e1) = h∗3
11e∗
3 + h∗4
11e∗
4,
B(e1, e2) = h∗3
12e∗
3 + h∗4
12e∗
4,

ìpou h∗3
11, h∗4
11, h∗3
12, h∗4
12 eÐnai diaforÐsimec sunart seic. Gia kˆje p ∈ Mc
1 isqÔei B|p = 0,
epomènwc
h∗3
11(p) = h∗4
11(p) = h∗3
12(p) = h∗4
12(p) = 0.
Kˆnontac qr sh tou tÔpou tou Weingarten, upologÐzoume
B∗
2(e1, e1, e1) = f
e1
B(e1, e1)
(T1f⊕N∗1f)⊥
= ⊥
e1
(h∗3
11e∗
3 + h∗4
11e∗
4)
(T1f⊕N∗1f)⊥
= e1(h∗3
11)e∗
3 + e1(h∗4
11)e∗
4
(T1f⊕N∗1f)⊥
+ h∗3
11( ⊥
e1
e∗
3)(T1f⊕N∗1f)⊥
+ h∗4
11( ⊥
e1
e∗
4)(T1f⊕N∗1f)⊥
= h∗3
11( ⊥
e1
e∗
3)(T1f⊕N∗1f)⊥
+ h∗4
11( ⊥
e1
e∗
4)(T1f⊕N∗1f)⊥
.
'Omoia
B∗
2(e1, e1, e2) = h∗3
12( ⊥
e1
e∗
3)(T1f⊕N∗1f)⊥
+ h∗4
12( ⊥
e1
e∗
4)(T1f⊕N∗1f)⊥
.
Sunep¸c gia kˆje shmeÐo p tou Mc
1 isqÔei B∗
2(e1, e1, e1)|p = B∗
2(e1, e1, e2)|p = 0 kai
lìgw tou (i) èqoume ìti B∗
2|p = 0.
MigadikopoioÔme tic dianusmatikèc dèsmec TM2, T1f, N∗1f kai epekteÐnoume C-
grammikˆ to (3,1)-tanustikì pedÐo B∗
2, opìte apoktoÔme to migadikì (3,1)-tanustikì
pedÐo
B∗
2 : Γ(TM2
⊗ C) × Γ(TM2
⊗ C) × Γ(TM2
⊗ C) −→ Γ (T1
f ⊕ N∗1
f)⊥
⊗ C .
To migadikì (3,1)-tanustikì pedÐo B∗
2 dèqetai thn anˆlush
B∗
2 = B
∗ (3,0)
2 dz3
+ B
∗ (2,1)
2 dz2
dz + B
∗ (1,2)
2 dzdz2
+ B
∗ (0,3)
2 dz3
.
Apì to L mma 2.2.5(i) èqoume B∗
2(X, e1, e1) + B∗
2(X, e2, e2) = 0 gia X ∈ ∆(U)
kai {e1, e2} topikì orjomonadiaÐo plaÐsio tou U. Gia e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y , epeid
∂
∂x = ∂
∂z + ∂
∂z kai
∂
∂y = i( ∂
∂z − ∂
∂z ), h teleutaÐa sqèsh gÐnetai
B∗
2(X,
∂
∂z
,
∂
∂z
) = 0.
Epomènwc
B∗
2 = B
∗ (3,0)
2 dz3
+ B
∗ (0,3)
2 dz3
,
ìpou
B
∗ (3,0)
2 = B∗
2
∂
∂z
,
∂
∂z
,
∂
∂z
, B
∗ (0,3)
2 = B∗
2
∂
∂z
,
∂
∂z
,
∂
∂z
kai isqÔei B
∗ (3,0)
2 = B
∗ (0,3)
2 .
Sto U orÐzoume to 6-diaforikì
Φ∗
2 := B
∗ (3,0)
2 , B
∗ (3,0)
2 dz6
.

To Φ∗
2 eÐnai kalˆ orismèno se ìlo to M2 kai lìgw tou L mmatoc 2.2.5(iii), isqÔei
Φ∗
2|p =
Φ2|p, p ∈ M1,
0, p ∈ Mc
1.
JewroÔme èna shmeÐo p ∈ Mc
1 kai èstw (U, z) migadikìc qˆrthc gÔrw apì to p
me z(p) = 0 kai U ∩ Mc
1 = {p}. Apì to L mma 2.2.2(iv) to Φ∗
2 eÐnai olìmorfo sto
U −{p}. Epeid eÐnai kai suneqèc sto shmeÐo p, sumperaÐnoume ìti to Φ∗
2 eÐnai olìmorfo
sto U [1]. Epanalambˆnontac thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou
Mc
1, èqoume ìti to Φ∗
2 eÐnai olìmorfo 6-diaforikì se ìlo to M2. Apì to Je¸rhma
Riemann-Roch èqoume Φ∗
2 = 0. Sunep¸c deÐxame ìti Φ2 = 0. Katˆ sunèpeia gia kˆje
p ∈ M1 isqÔei Φ2|p = 0 isodÔnama
B2(e1, e1, e1)
2
(p) − B2(e1, e1, e2)
2
(p) − 2i B2(e1, e1, e1), B2(e1, e1, e2) (p) = 0.
Apì thn teleutaÐa sqèsh, lìgw thc Prìtashc 1.3.1 prokÔptei ìti se kˆje shmeÐo
p ∈ M1 h èlleiyh E2(p) eÐnai kÔkloc me aktÐna
κ2(p) = B2(e1, e1, e1) (p) = B2(e1, e1, e2) (p).
Epomènwc, h diˆstash tou kˆjetou q¸rou deÔterhc tˆxhc sto tuqìn p ∈ M1 eÐnai
0 2. An dimN2
p f = 0 gia kˆje p ∈ M1, tìte sÔmfwna me thn Parat rhsh 1.3.1,
f(M2) ⊂ S4, ìpou S4 eÐnai mia mègisth 4-sfaÐra thc Sn. 'Atopo, afoÔ èqoume dh
anafèrei pwc exetˆzoume koresmènh elaqistik epifˆneia f : M2 −→ Sn, n ≥ 5.
Sunep¸c, maxp∈M2 dimN2
p f = 2 to opoÐo shmaÐnei ìti n ≥ 6. An n = 6, tìte h
diadikasÐa stamatˆ ed¸ kai an n ≥ 7, tìte suneqÐzei.
EÐnai t¸ra eÔlogo h diadikasÐa aut na genikeÔetai epagwgikˆ. Gia lìgouc plh-
rìthtac thc apìdeixhc twn apotelesmˆtwn, ja perigrˆyoume leptomer¸c to epagwgikì
b ma.
'Estw f : M2 −→ Sn, n ≥ 7, sumpag c, koresmènh, elaqistik epifˆneia gènouc
mhdèn kai jetikìc akèraioc r me 2 ≤ r ≤ [n−1
2 ] − 1. Upojètoume ìti gia kˆje s ∈
{2, ..., r} isqÔoun ta akìlouja:
(I) Upˆrqoun dianusmatikèc upodèsmec N∗1f, ..., N∗r−1f thc kˆjethc dèsmhc Nf
me bajmÐda 2 uperˆnw tou M2 ètsi ¸ste N∗s−1f|Ms−1 = Ns−1f|Ms−1 , ìpou
Ms−1 := {p ∈ M2
: B∗
s−1 (p) > 0}
kai
B∗
s :=
2
j1,...,js+1=1
|B∗
s (ej1 , ..., ejs+1 )|2
eÐnai to m koc tou summetrikoÔ (s + 1, 1)-tanustikoÔ pedÐou
B∗
s : ∆(M2
) × ... × ∆(M2
)
s+1
−→ Γ (T1
f ⊕ N∗1
f ⊕ ... ⊕ N∗s−1
f)⊥
,
(X1, ..., Xs+1) −→ B∗
s (X1, ..., Xs+1) = f
X1
... f
Xs
df(Xs+1)
(T1f⊕N∗1f⊕...⊕N∗s−1f)⊥
.

Jètoume B∗
1 = B.
Gia to (s + 1, 1)-tanustikì pedÐo B∗
s isqÔoun:
(i) An X1, ..., Xs−1 ∈ ∆(M2) kai {e1, e2} eÐnai opoiod pote topikì orjomonadiaÐo
plaÐsio tou M2, tìte
B∗
s (X1, ..., Xs−1, e1, e1) + B∗
s (X1, ..., Xs−1, e2, e2) = 0.
(ii) An X1, ..., Xs+1 ∈ ∆(M2), tìte
B∗
s (X1, ..., Xs+1) = f
X1
B∗
s−1(X2, ..., Xs+1)
(T1f⊕N∗1f⊕...⊕N∗s−1f)⊥
.
(iii) B∗
s |Ms−1 = Bs|Ms−1 kai B∗
s |Mc
s−1
= 0.
Epiplèon, ta sÔnola Ms−1 eÐnai mh kenˆ, anoiktˆ kai puknˆ uposÔnola tou M2,
me M1 ⊃ M2 ⊃ ... ⊃ Mr−1 kai to sumpl rwma kajenìc apì autˆ sto M2 eÐnai
peperasmèno sÔnolo.
(II) MigadikopoioÔme tic dianusmatikèc dèsmec T1f, N∗1f, ..., N∗r−1f, epekteÐnoume
C-grammikˆ ta B∗
s , opìte apoktoÔme ta migadikˆ (s + 1, 1)-tanustikˆ pedÐa
B∗
s : Γ(TM2
⊗ C) × ... × Γ(TM2
⊗ C)
s+1
−→ Γ (T1
f ⊕ N∗1
f ⊕ ... ⊕ N∗s−1
f)⊥
⊗ C .
To B∗
s wc migadikì (s + 1, 1)-tanustikì pedÐo dèqetai thn anˆlush
B∗
s =
p+q=s+1
B∗ (p,q)
s dzp
dzq
.
Se èna migadikì qˆrth (U, z) tou M2 me z = x + iy kai , = E|dz|2 jewroÔme topikì
orjomonadiaÐo plaÐsio {e1, e2} me e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y . IsqÔei h sqèsh
B∗
s (X1, ..., Xs−1, e1, e1) + B∗
s (X1, ..., Xs−1, e2, e2) = 0,
gia kˆje X1, ..., Xs−1 ∈ ∆(U). Epeid
∂
∂x = ∂
∂z + ∂
∂z , ∂
∂y = i( ∂
∂z − ∂
∂z ), h teleutaÐa
sqèsh gÐnetai
B∗
s (X1, ..., Xs−1,
∂
∂z
,
∂
∂z
) = 0.
Epomènwc, to B∗
s èqei thn anˆlush
B∗
s = B∗ (s+1,0)
s dzs+1
+ B∗ (0,s+1)
s dzs+1
,
ìpou
B∗ (s+1,0)
s = B∗
s
∂
∂z
, ...,
∂
∂z
, B∗ (0,s+1)
s = B∗
s
∂
∂z
, ...,
∂
∂z
kai isqÔei B
∗ (s+1,0)
s = B
∗ (0,s+1)
s .
Upojètoume ìti ta kalˆ orismèna migadikˆ (2s + 2)-diaforikˆ
Φ∗
s := B∗ (s+1,0)
s , B∗ (s+1,0)
s dz2s+2
eÐnai ek tautìthtoc mhdèn.

(III) Gia kˆje migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2 upˆrqei
orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr−1 me efaptìmeno mèroc {e1 = 1√
E
∂
∂x , e2 =
1√
E
∂
∂y } kai kˆjeto mèroc {eα} ètsi ¸ste
e3 =
1
κ1
B(e1, e1), e4 =
1
κ1
B(e1, e2),
e5 =
1
κ2
B2(e1, e1, e1), e6 =
1
κ2
B2(e1, e1, e2),
.
.
.
e2r−1 =
1
κr−1
Br−1(e1, ..., e1), e2r =
1
κr−1
Br−1(e1, ..., e1, e2),
ìpou κs−1 > 0, gia to opoÐo upojètoume ìti sto Ms−1 ∩ U isqÔoun oi sqèseic
(d log κs−1 + iω2s−1,2s) ∧ φ = isω12 ∧ φ
kai
dH
α
s − i(s + 1)H
α
s ω12 +
β≥2s+1
H
β
s ωβα ∧ φ = 0, α ≥ 2s + 1.
Ja deÐxoume ìti ta (I), (II), (III) isqÔoun kai gia s = r + 1.
Apìdeixh. Lìgw thc epagwgik c upìjeshc, isqÔei Φ∗
r = 0 sto M2 kai sunep¸c
Φ∗
r|Mr−1 = 0. Ja deÐxoume ìti h èlleiyh Er(p) eÐnai kÔkloc se kˆje shmeÐo p ∈ Mr−1.
To Mr−1, wc mh kenì kai anoiktì uposÔnolo tou prosanatolismènou poluptÔg-
matoc Riemann M2, eÐnai kai autì prosanatolismèno polÔptugma Riemann. 'Estw p
èna tuqìn shmeÐo tou Mr−1. GÔrw apì to p ∈ Mr−1 jewroÔme migadikì qˆrth (U, z)
me z = x + iy kai , = E|dz|2. 'Estw {e1, e2} topikì orjomonadiaÐo plaÐsio sto U me
e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y . Sto U èqoume
∂
∂z = 1
2( ∂
∂x − i ∂
∂y ) = 1
2
√
E(e1 − ie2).
Gia to (2r + 2)-diaforikì Φ∗
r isqÔei Φ∗
r|p = 0 an kai mìno an
B∗
r (
∂
∂z
, ...,
∂
∂z
)|p, B∗
r (
∂
∂z
, ...,
∂
∂z
)|p = 0,
isodÔnama lìgw thc sqèshc B∗
r |Mr−1 = Br|Mr−1 ,
Br(
∂
∂z
, ...,
∂
∂z
)|p, Br(
∂
∂z
, ...,
∂
∂z
)|p = 0.
'Omwc èqoume
Br(
∂
∂z
, ...,
∂
∂z
) = Br
√
E
2
(e1 − ie2), ...,
√
E
2
(e1 − ie2)

=
1
2r+1
E
r+1
2
r+1
m=0
r + 1
m
Br(e1, ..., e1, −ie2, ..., −ie2
m
).
SumbolÐzoume me I to sÔnolo twn ˆrtiwn arijm¸n tou sunìlou {1, ..., r + 1} kai me J
to sÔnolo twn peritt¸n arijm¸n tou, epomènwc
Br(
∂
∂z
, ...,
∂
∂z
) =
1
2r+1
E
r+1
2
m∈I,m=2l
(−1)l r + 1
m
Br(e1, ..., e1, e2, ..., e2
m
)
− i
1
2r+1
E
r+1
2
m∈J,m=2l+1
(−1)l r + 1
m
Br(e1, ..., e1, e2, ..., e2
m
).
Epeid apì to L mma 1.3.2 isqÔei
Br(e1, ..., e1) + Br(e1, ..., e1, e2, e2) = 0,
eÐnai
Br(e1, ...e1, e2, ..., e2
m
) =
(−1)lBr(e1, ..., e1), m = 2l,
(−1)lBr(e1, ..., e1, e2), m = 2l + 1,
kai epomènwc,
Br(
∂
∂z
, ...,
∂
∂z
) =
1
2r+1
E
r+1
2 Br(e1, ..., e1)
m∈I
r + 1
m
− i
1
2r+1
E
r+1
2 Br(e1, ..., e1, e2)
m∈J
r + 1
m
.
'Omwc isqÔei
m∈I
r + 1
m
=
m∈J
r + 1
m
= 2r
kai telikˆ èqoume
Br(
∂
∂z
, ...,
∂
∂z
) =
1
2
E
r+1
2 Br(e1, ..., e1) − i
1
2
E
r+1
2 Br(e1, ..., e1, e2).
Opìte, apì th sqèsh Φ∗
r|p = 0 sunˆgetai ìti sto Mr−1 isqÔoun
|Br(e1, ..., e1)| = |Br(e1, ..., e1, e2)|,
Br(e1, ..., e1), Br(e1, ..., e1, e2) = 0.
Lìgw thc Prìtashc 1.3.1 èqoume ìti h èlleiyh Er(p) eÐnai kÔkloc se kˆje shmeÐo tou
Mr−1. Autì shmaÐnei ìti dimNr
p f ∈ {0, 2} gia kˆje p ∈ Mr−1.
An upojèsoume ìti dimNr
p f = 0 gia kˆje p ∈ Mr−1, tìte apì thn Parat rhsh
1.3.1 upˆrqei mègisth 2r-sfaÐra thc Sn ¸ste f(M2) ⊂ S2r. 'Atopo afoÔ h embˆptish
eÐnai koresmènh sthn Sn. 'Ara, upˆrqei shmeÐo p ∈ Mr−1 ¸ste dimNr
p f = 2, iso-
dÔnama B∗
r (p) > 0, to opoÐo shmaÐnei ìti to sÔnolo Mr := {p ∈ M2 : B∗
r (p) > 0}
eÐnai mh kenì.

Epeid h B∗
r eÐnai suneq c sunˆrthsh, to sÔnolo B∗
r
−1 (0, +∞) = Mr eÐnai
anoiktì sto M2.
Epiplèon, afoÔ
B∗
r |p =
Br|p, p ∈ Mr−1,
0, p ∈ Mc
r−1,
èqoume Mr ⊂ Mr−1.
Sth sunèqeia, ja deÐxoume ìti to Mr eÐnai puknì sÔnolo tou M2. 'Estw int(Mc
r ) =
∅. JewroÔme èna mh kenì, anoiktì kai sunektikì uposÔnolo V tou Mc
r . 'Eqoume
B∗
r |V = 0. An V ∩ Mr−1 = ∅, tìte ja isqÔei V ⊂ Mc
r−1. AdÔnato, afoÔ to Mc
r−1
apoteleÐtai apì peperasmèna shmeÐa. To V ∩ Mr−1 eÐnai mh kenì kai anoiktì sÔnolo
sto opoÐo èqoume Br|V ∩Mr−1 = 0. Tìte, sÔmfwna me thn Parat rhsh 1.3.1, h f de ja
tan koresmènh, ˆtopo. Epomènwc to Mr eÐnai puknì uposÔnolo tou M2.
'Estw p0 ∈ Mc
r . JewroÔme gÔrw apì to p0 migadikì qˆrth (U, z) me z = x +
iy, , = E|dz|2 kai z(p0) = 0. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc
f me efaptìmeno mèroc {e1, e2}, ìpou e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y kai kˆjeto mèroc
{eα}. Jètoume h∗α
(r),1 := B∗
r (e1, ..., e1), eα , h∗α
(r),2 := B∗
r (e1, ..., e1, e2), eα kai H∗α
r :=
h∗α
(r),1 +ih∗α
(r),2, gia α ≥ 2r +1. Profan¸c, h H∗α
r eÐnai diaforÐsimh gia kˆje α ≥ 2r +1
kai
B∗
r (e1, ..., e1) =
α≥2r+1
h∗α
(r),1eα, B∗
r (e1, ..., e1, e2) =
α≥2r+1
h∗α
(r),2eα.
Sto Mr−1 ∩ U èqoume B∗
r |Mr−1∩U = Br|Mr−1∩U , sunep¸c h sqèsh
dH
α
r − i(r + 1)H
α
r ω12 +
β≥2r+1
H
β
r ωβα ∧ φ = 0
mac dÐnei th sqèsh
dH
∗α
r − i(r + 1)H
∗α
r ω12 +
β≥2r+1
H
∗β
r ωβα ∧ φ = 0 (2.7)
gia kˆje α ≥ 2r + 1. Epeid to Mc
r−1 eÐnai peperasmèno sÔnolo, lìgw sunèqeiac h
teleutaÐa sqèsh isqÔei gÔrw apì to tuqìn shmeÐo tou U.
Me th bo jeia twn sqèsewn (2.3), (2.4) kai twn sqèsewn
dH
∗α
r =
∂H
∗α
r
∂z
dz +
∂H
∗α
1
∂z
dz,
ωαβ = ωαβ(
∂
∂z
)dz + ωαβ(
∂
∂z
)dz,
h (2.7) gÐnetai
∂H
∗α
r
∂z
=
β
gαβH
∗β
r ,
sto U gÔrw apì to 0, ìpou gαβ = −ωβα( ∂
∂z ) − r+1
2 δαβd log E( ∂
∂z ) eÐnai diaforÐsimec
sunart seic kai δαβ eÐnai to dèlta tou Kronecker.

SÔmfwna me to L mma 2.2.3 eÐte H
∗α
r = 0 gia kˆje α = 2r + 1, ..., n oi koinèc
rÐzec twn H
∗α
r eÐnai memonwmènec kai upˆrqei jetikìc akèraioc lr kai diaforÐsimec
sunart seic G
∗α
r : U −→ C ¸ste na isqÔei
H
∗α
r = zlr
G
∗α
r (2.8)
gia kˆje α ≥ 2r + 1 me G
∗2r+1
1 (0), ..., G
∗n
1 (0) = (0, ..., 0).
An tan H
∗α
r = 0 gia kˆje α = 2r + 1, ..., n, tìte Br|U∩Mr−1 = 0, ˆtopo afoÔ
lìgw thc Parat rhshc 1.3.1, h f den ja tan koresmènh. Epomènwc, oi koinèc rÐzec
twn H
∗α
r eÐnai memonwmènec sto U kai to sÔnolo {p ∈ U : B∗
r |p = 0} apoteleÐtai apì
memonwmèna shmeÐa. 'Ara kai to sÔnolo Mc
r = {p ∈ M2 : B∗
r |p = 0} apoteleÐtai apì
memonwmèna shmeÐa kai afoÔ to M2 eÐnai sumpagèc, eÐnai peperasmèno.
MporoÔme na jewr soume loipìn ìti isqÔei U ∩ Mc
r = {p0}. Apì thn upìjesh,
gnwrÐzoume ìti se kˆje shmeÐo p tou U isqÔei Φ∗
r|p = 0, isodÔnama
B∗
r (e1, ..., e1) (p) = B∗
r (e1, ..., e1, e2) (p)
kai
B∗
r (e1, ..., e1), B∗
r (e1, ..., e1, e2) (p) = 0.
Sunep¸c sto U isqÔei
B∗
r (e1, ..., e1) − iB∗
r (e1, ..., e1, e2), B∗
r (e1, ..., e1) − iB∗
r (e1, ..., e1, e2) = 0.
'Omwc lìgw thc (2.8) èqoume
B∗
r (e1, ..., e1) − iB∗
r (e1, ..., e1, e2) =
α≥2r+1
H
∗α
r eα = zlr
α≥2r+1
G
∗α
r eα.
Sunep¸c sto U − {p0} epeid z = 0, isqÔei h isìthta
α≥2r+1
G
∗α
r eα,
α≥2r+1
G
∗α
r eα = 0,
h opoÐa lìgw sunèqeiac isqÔei kai sto p0. Epomènwc gia kˆje q ∈ U isqÔei
Re
n
α=2r+1
G
∗α
r eα (q) = Im
n
α=2r+1
G
∗α
r eα (q) = 0,
Re
n
α=2r+1
G
∗α
r eα , Im
n
α=2r+1
G
∗α
r eα (q) = 0.
Gia kˆje q ∈ U orÐzoume ton didiˆstato upìqwro N∗r
q f tou kajètou q¸rou thc f
sto q
N∗r
q f := span Re
n
α=2r+1
G
∗α
r eα |q, Im
n
α=2r+1
G
∗α
r eα |q .
Epomènwc sto U èqoume th dianusmatik dèsmh bajmÐdac 2
N∗r
f|U =
q∈U
N∗r
q f.

Epeid Br|Mr = B∗
r |Mr , èqoume gia q ∈ U − {p0}
Nr
q f = span{Br(e1, ..., e1)|q, Br(e1, ..., e1, e2)|q}
= span{B∗
r (e1, ..., e1)|q, B∗
r (e1, ..., e1, e2)|q}
= span Re B∗
r (e1, ..., e1)|q − iB∗
r (e1, ..., e1, e2)|q ,
Im B∗
r (e1, ..., e1)|q − iB∗
r (e1, ..., e1, e2)|q
= span Re zlr
α≥2r+1
G
∗α
r eα |q, Im zlr
α≥2r+1
G
∗α
r eα |q
= span Re
α≥2r+1
G
∗α
r eα |q, Im
α≥2r+1
G
∗α
r eα |q ,
isqÔei N∗rf|U−{p0} = Nrf|U−{p0}.
An epanalˆboume thn parapˆnw diadikasÐa gÔrw apì ìla ta shmeÐa tou Mc
r , pou
ìpwc eÐdame eÐnai peperasmèna to pl joc, apoktoÔme th dianusmatik dèsmh N∗rf me
bajmÐda 2 sto M2 gia thn opoÐa isqÔei N∗rf|Mr = Nrf|Mr .
OrÐzoume thn apeikìnish
B∗
r+1 : ∆(M2
) × ... × ∆(M2
)
r+2
−→ Γ (T1
f ⊕ N∗1
f ⊕ ... ⊕ N∗r
f)⊥
,
(X1, ..., Xr+2) −→
B∗
r+1(X1, ..., Xr+2) := ( f
X1
... f
Xr+1
df(Xr+2))(T1f⊕N∗1f⊕...⊕N∗rf)⊥
.
H B∗
r+1 eÐnai D(M2)-grammik wc proc thn pr¸th metablht thc. Ja deÐxoume
ìti h B∗
r+1 eÐnai summetrik kai epomènwc ja eÐnai D(M2)-grammik wc proc ìlec
tic metablhtèc thc. Katarq n, epeid M1 ⊃ M2 ⊃ ... ⊃ Mr, isqÔei N∗1f|Mr =
N1f|Mr , ..., N∗rf|Mr = Nrf|Mr kai epomènwc èqoume B∗
r+1|Mr = Br+1|Mr . Sunep¸c,
h B∗
r+1 eÐnai summetrik sto Mr. To Mc
r apoteleÐtai apì peperasmèna shmeÐa kai
epomènwc, lìgw sunèqeiac, h B∗
r+1 eÐnai summetrik kai sto Mc
r . Gia touc Ðdiouc lìgouc
isqÔei
B∗
r+1(X1, ..., Xr, e1, e1) + B∗
r+1(X1, ..., Xr, e2, e2) = 0,
gia X1, ..., Xr ∈ ∆(M2) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou
M2.
Gia X1, ..., Xr+2 ∈ ∆(M2) èqoume
B∗
r+1|Mr (X1, ..., Xr+2) =
= Br+1|Mr (X1, ..., Xr+2)
= f
X1
... f
Xr+1
df(Xr+2)
(T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥
= f
X1
f
X2
... f
Xr+1
df(Xr+2)
Trf|Mr
∈Tr+1f|Mr
+ f
X1
f
X2
... f
Xr+1
df(Xr+2)
Nrf|Mr

= f
X1
Br|Mr (X2, ..., Xr+2)
(T1f⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥
= f
X1
B∗
r |Mr (X2, ..., Xr+2)
(T1f|Mr ⊕N∗1f|Mr ⊕...⊕N∗rf|Mr )⊥
kai lìgw sunèqeiac, isqÔei kai sto Mc
r . 'Ara sto M2 èqoume th sqèsh
B∗
r+1(X1, ..., Xr+2) = f
X1
B∗
r (X2, ..., Xr+2)
(T1f⊕N∗1f⊕...⊕N∗rf)⊥
.
En suneqeÐa, ja deÐxoume ìti an B∗
r |p = 0, tìte B∗
r+1|p = 0. Gia to skopì autì,
jewroÔme migadikì qˆrth (U, z) tou M2 kai topikì orjomonadiaÐo plaÐsio katˆ m koc
thc f me efaptìmeno mèroc {e1, e2} kai kˆjeto mèroc {eα} ètsi ¸ste eα = e∗
α gia
α = 3, ..., 2r+2, ìpou ta e∗
3, ..., e∗
2r+2 eÐnai tètoia ¸ste N∗1f = span{e∗
3, e∗
4},...,N∗rf =
span{e∗
2r+1, e∗
2r+2}. OrÐzoume tic sunart seic
h∗2r+1
(r),1 , h∗2r+2
(r),1 , h∗2r+1
(r),2 , h∗2r+2
(r),2 : U −→ R,
oi opoÐec eÐnai tètoiec ¸ste
B∗
r (e1, ..., e1) = h∗2r+1
(r),1 e∗
2r+1 + h∗2r+2
(r),1 e∗
2r+2
kai
B∗
r (e1, ..., e1, e2) = h∗2r+1
(r),2 e∗
2r+1 + h∗2r+2
(r),2 e∗
2r+2.
Prˆgmati, autì gÐnetai diìti gia kˆje p ∈ M2 o dianusmatikìc q¸roc spanImB∗
r |p
eÐnai upìqwroc tou N∗r
p f. ToÔto isqÔei apì to L mma 1.3.1 gia kˆje p ∈ Mr−1, afoÔ
B∗
r |Mr−1 = Br|Mr−1 . An p ∈ Mc
r−1, tìte isqÔei tetrimmèna epeid B∗
r |p = 0.
Lìgw tou tÔpou tou Weingarten èqoume
B∗
r+1(e1, ..., e1) =
= f
e1
B∗
r (e1, ..., e1)
(T1f⊕N∗1f⊕...⊕N∗rf)⊥
= − df(AB∗
r (e1,...,e1)e1)
(T1f⊕N∗1f⊕...⊕N∗rf)⊥
+ ⊥
e1
B∗
r (e1, ..., e1)
(T1f⊕N∗1f⊕...⊕N∗rf)⊥
= ⊥
e1
(h∗2r+1
(r),1 e∗
2r+1 + h∗2r+2
(r),1 e∗
2r+2)
(T1f⊕N∗1f⊕...⊕N∗rf)⊥
= h∗2r+1
(r),1 ( ⊥
e1
e∗
2r+1)(T1f⊕N∗1f⊕...⊕N∗rf)⊥
+ h∗2r+2
(r),1 ( ⊥
e1
e∗
2r+2)(T1f⊕N∗1f⊕...⊕N∗rf)⊥
.
'Omoia,
B∗
r+1(e1, ..., e1, e2) = h∗2r+1
(r),2 ( ⊥
e1
e∗
2r+1)(T1f⊕N∗1f⊕...⊕N∗rf)⊥
+ h∗2r+2
(r),2 ( ⊥
e1
e∗
2r+2)(T1f⊕N∗1f⊕...⊕N∗rf)⊥
.
Upojèsame pwc B∗
r |p = 0, ˆra
h∗2r+1
(r),1 (p) = h∗2r+2
(r),1 (p) = h∗2r+1
(r),2 (p) = h∗2r+2
(r),2 (p) = 0

kai sÔmfwna me tic parapˆnw sqèseic,
B∗
r+1(e1, ..., e1)|p = B∗
r+1(e1, ..., e1, e2)|p = 0.
Apì aut th sqèsh kai epeid h B∗
r+1 eÐnai summetrik kai tètoia ¸ste na isqÔei
B∗
r+1(X1, ..., Xr, e1, e1) + B∗
r+1(X1, ..., Xr, e2, e2) = 0, èqoume B∗
r+1|p = 0. Sunep¸c
apodeÐxame ìti an p eÐnai èna shmeÐo tou Mc
r , tìte B∗
r+1|p = 0.
Telikˆ, gia thn B∗
r+1 isqÔei
B∗
r+1|p =
Br+1|p, p ∈ Mr,
0, p ∈ Mc
r .
MigadikopoioÔme tic dianusmatikèc dèsmec T1f, N∗1f, ..., N∗rf, epekteÐnoume C-
grammikˆ thn B∗
r+1 kai apoktoÔme to migadikì (r + 2, 1)-tanustikì pedÐo
B∗
r+1 : Γ(TM2
⊗ C) × ... × Γ(TM2
⊗ C)
r+2
−→ Γ (T1
f ⊕ N∗1
f ⊕ ... ⊕ N∗r
f)⊥
⊗ C) .
H B∗
r+1 wc migadikì (r + 2, 1)-tanustikì pedÐo dèqetai thn anˆlush
B∗
r+1 =
p+q=r+2
B
∗ (p,q)
r+1 dzp
dzq
.
Se migadikì qˆrth (U, z) tou M2 me z = x+iy kai , = E|dz|2 gnwrÐzoume ìti isqÔei
B∗
r+1(X1, ..., Xr, e1, e1) + B∗
r+1(X1, ..., Xr, e2, e2) = 0,
gia X1, ..., Xr ∈ ∆(U) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou U.
An jewr soume e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y , tìte epeid
∂
∂x = ∂
∂z + ∂
∂z , ∂
∂y = i( ∂
∂z − ∂
∂z ),
h teleutaÐa sqèsh gÐnetai
B∗
r+1 X1, ..., Xr,
∂
∂z
,
∂
∂z
= 0.
Epomènwc,
B∗
r+1 = B
∗ (r+2,0)
r+1 dzr+2
+ B
∗ (0,r+2)
r+1 dzr+2
,
ìpou
B
∗ (r+2,0)
r+1 = B∗
r+1
∂
∂z
, ...,
∂
∂z
, B
∗ (0,r+2)
r+1 = B∗
r+1
∂
∂z
, ...,
∂
∂z
kai isqÔei B
∗ (r+2,0)
r+1 = B
∗ (0,r+2)
r+1 .
OrÐzoume topikˆ to (2r + 4)-diaforikì
Φ∗
r+1 := B
∗ (r+2,0)
r+1 , B
∗ (r+2,0)
r+1 dz2r+4
.
Ja deÐxoume ìti eÐnai kalˆ orismèno se ìlo to M2. Gia to lìgo autì jewroÔme qˆrtec
tou M2 (U, ϕ) me migadik suntetagmènh z kai (V, ψ) me migadik suntetagmènh ζ me
U ∩ V = ∅. Epeid to M2 eÐnai epifˆneia Riemann, oi apeikonÐseic ψ ◦ ϕ−1, ϕ ◦ ψ−1

eÐnai olìmorfec, dhlad
∂ζ
∂z = 0 kai
∂z
∂ζ
= 0. Sto U ∩ V èqoume tic sqèseic
∂
∂z = ∂ζ
∂z
∂
∂ζ ,
dζ = ∂ζ
∂z dz kai epomènwc
B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
), B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
) dζ2r+4
=
= B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
), B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
) (
∂ζ
∂z
)2r+4
dz2r+4
= (
∂ζ
∂z
)r+2
B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
), (
∂ζ
∂z
)r+2
B∗
r+1(
∂
∂ζ
, ...,
∂
∂ζ
) dz2r+4
= B∗
r+1(
∂
∂z
, ...,
∂
∂z
), B∗
r+1(
∂
∂z
, ...,
∂
∂z
) dz2r+4
.
'Ara to Φ∗
r+1 eÐnai kalˆ orismèno (2r + 4)-diaforikì sto M2.
'Estw p ∈ Mr. JewroÔme gÔrw apì to p migadikì qˆrth (U, z) me z = x + iy kai
, = E|dz|2. Epilègoume orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr∩U me efaptìmeno
E
∂
∂x , e2 = 1√
E
∂
∂y kai kˆjeto mèroc {eα}, ìpou
e3 =
1
κ1
B(e1, e1), e4 =
1
κ1
B(e1, e2),
e5 =
1
κ2
B2(e1, e1, e1), e6 =
1
κ2
B2(e1, e1, e2),
.
.
.
e2r+1 =
1
κr
Br(e1, ..., e1, e1), e2r+2 =
1
κr
Br(e1, ..., e1, e2)
kai eα gia α ≥ 2r + 3 eÐnai tuqìnta. SumbolÐzoume me {ω1, ω2} to sumplaÐsio tou
{e1, e2}. Sto Mr ∩ U èqoume to migadikì (1,0)-tanustikì pedÐo φ = ω1 + iω2 =
√
Edz
gia to opoÐo isqÔoun oi sqèseic (2.4) kai (2.5).
'Eqoume
Br(e1, ..., e1) =
α≥2r+1
hα
(r),1eα,
Br(e1, ..., e1, e2) =
α≥2r+1
hα
(r),2eα
kai Hα
r = hα
(r),1 + ihα
(r),2 gia α = 2r + 1, ..., n. Epeid epilèxame wc orjomonadiaÐo
plaÐsio thc Nrf|Mr∩U to {e2r+1 = 1
κr
Br(e1, ..., e1, e1), e2r+2 = 1
κr
Br(e1, ..., e1, e2)},
èqoume
h2r+1
(r),1 = κr, hα
(r),1 = 0, α ≥ 2r + 2,
h2r+2
(r),2 = κr, hα
(r),2 = 0, α = 2r + 2

kai sunep¸c
H2r+1
r = κr, H2r+2
r = iκr.
Sto Mr ∩ U isqÔei h sqèsh (dH
α
r − i(r + 1)H
α
r ω12 + β≥2r+1 H
β
r ωβα) ∧ φ = 0
gia kˆje α ≥ 2r + 1. Gia α = 2r + 1 lambˆnoume
dκr − i(r + 1)κrω12 +
β≥2r+1
H
β
r ωβ,2r+1 ∧ φ = 0
kai an qrhsimopoi soume to gegonìc ìti gia β ≥ 2r + 3 isqÔei
Hβ
r = hβ
(r),1 + ihβ
(r),2
= Br(e1, ..., e1), eβ + i Br(e1, ..., e1, e2), eβ
= κre2r+1, eβ + i κre2r+2, eβ = 0,
paÐrnoume th sqèsh
(d log κr + iω2r+1,2r+2) ∧ φ = i(r + 1)ω12 ∧ φ. (2.9)
An periorioristoÔme sto Mr, h (T1f|Mr ⊕ N1f|Mr ⊕ ... ⊕ Nrf|Mr )⊥ eÐnai dianu-
smatik dèsmh me bajmÐda n − 2r − 2 kai h Br+1|Mr eÐnai (r + 2, 1)-tanustikì pedÐo.
Sto Mr ∩ U, lambˆnontac upìyh ton orismì thc Br+1 kai touc tÔpouc twn Gauss kai
Weingarten, èqoume
Br+1|Mr (e1, ..., e1) = f
e1
f
e1
... f
e1
df(e1)
= f
e1
Br(e1, ..., e1)
= f
e1
(κre2r+1)
= e1(κr)e2r+1
+ (κr
f
e1
e2r+1)(T1f|Mr ⊕N1f|Mr ⊕...⊕Nrf|Mr )⊥
= κr( f
e1
= κr − df(Ae2r+1 e1)
+ ( ⊥
e1
= κr
α≥2r+3
⊥
e1
e2r+1, eα eα
= κr
α≥2r+3
ω2r+1,α(e1)eα.
Epiplèon,
Br+1|Mr (e1, ..., e1) = −Br+1|Mr (e2, e1, ..., e1, e2)
= −κr
α≥2r+3
ω2r+2,α(e2)eα.

'Omoia apodeiknÔetai ìti
Br+1|Mr (e1, ..., e1, e2) = κr
α≥2r+3
ω2r+2,α(e1)eα
= κr
α≥2r+3
ω2r+1,α(e2)eα.
Epomènwc sto Mr ∩ U èqoume gia α ≥ 2r + 3
ω2r+1,α(e1) = −ω2r+2,α(e2), ω2r+2,α(e1) = ω2r+1,α(e2)
kai
hα
(r+1),1 = κrω2r+1,α(e1), hα
(r+1),2 = κrω2r+1,α(e2).
Apì tic parapˆnw sqèseic paÐrnoume sto Mr ∩ U
Hα
r+1 = hα
(r+1),1 + ihα
(r+1),2 = κrω2r+1,α(e1) + iκrω2r+1,α(e2).
EÔkola diapist¸noume ìti
Hα
r+1φ = κrω2r+1,α + iκrω2r+2,α. (2.10)
Oi exis¸seic Ricci gia α ≥ 2r + 3 dÐnoun
dω2r+1,α =
2
j=1
ω2r+1,j ∧ ωjα +
n
β=3
ω2r+1,β ∧ ωβα,
dω2r+2,α =
2
j=1
ω2r+2,j ∧ ωjα +
n
β=3
ω2r+2,β ∧ ωβα.
Gia s = 2, ..., r èqoume topikˆ sto Ms−1
Bs(e1, ..., e1) = κs−1
α≥2s+1
ω2s−1,α(e1)eα = −κs−1
α≥2s+1
ω2s,α(e2)eα
kai
Bs(e1, ..., e1, e2) = κs−1
α≥2s+1
ω2s−1,α(e2)eα = κs−1
α≥2s+1
ω2s,α(e1)eα,
ˆra ω2s−1,α(e1) = −ω2s,α(e2) kai ω2s−1,α(e2) = ω2s,α(e1) gia α = 2s + 1, ..., n. 'Omwc,
Bs(e1, ..., e1) = κse2s+1 kai Bs(e1, ..., e1, e2) = κse2s+2, epomènwc èqoume
ω2s−1,2s+1(e1) = −ω2s,2s+1(e2) =
κs
κs−1
, (2.11)
ω2s,2s+2(e1) = ω2s−1,2s+2(e2) =
κs
κs−1
(2.12)
kai
ω2s−1,α(e1) = ω2s,α(e2) = 0, (2.13)

gia α = 2s + 1,
ω2s−1,α(e2) = ω2s,α(e1) = 0, (2.14)
gia α = 2s + 2. 'Ara gia kˆje α ≥ 2r + 3 isqÔei ω2s−1,α = ω2s,α = 0.
EpÐshc, gia α ≥ 2r + 3 eÐnai ω1α = 2
j=1 hα
1jωj me hα
1j = B(e1, ej), eα = 0 kai
ω2α = 2
j=1 hα
2jωj me hα
2j = B(e2, ej), eα = 0, sunep¸c ω1α = ω2α = 0.
Telikˆ paÐrnoume
ωsα = 0, (2.15)
gia s = 1, ..., 2r, α ≥ 2r + 3.
Oi exis¸seic Ricci apì tic sqèseic (2.15) gÐnontai
dω2r+1,α = ω2r+1,2r+2 ∧ ω2r+2,α +
β≥2r+3
ω2r+1,β ∧ ωβα, (2.16)
dω2r+2,α = ω2r+2,2r+1 ∧ ω2r+1,α +
β≥2r+3
ω2r+2,β ∧ ωβα. (2.17)
ParagwgÐzontac exwterikˆ th sqèsh (2.10) kai kˆnontac qr sh twn sqèsewn (2.5),
(2.9), (2.16), (2.17) paÐrnoume th zhtoÔmenh isìthta
dH
α
r+1 − i(r + 2)H
α
r+1ω12 +
β≥2r+3
H
β
r+1ωβα ∧ φ = 0
gia kˆje α ≥ 2r + 3 sto Mr ∩ U.
MigadikopoioÔme tic dianusmatikèc dèsmec
T1
f|Mr , N1
f|Mr , ..., Nr
f|Mr ,
epekteÐnoume C-grammikˆ thn Br+1|Mr kai apoktoÔme to migadikì (r + 2, 1)-tanustikì
pedÐo
Br+1|Mr : Γ(TMr ⊗ C) × ... × Γ(TMr ⊗ C)
r+2
−→ Γ (T1
f|Mr ⊕ N1
f|Mr ⊕ ... ⊕ Nr
f|Mr )⊥
⊗ C .
H Br+1|Mr wc migadikì (r + 2, 1)-tanustikì pedÐo dèqetai thn anˆlush
Br+1|Mr =
p+q=r+2
Br+1|
(p,q)
Mr
dzp
dzq
.
Se èna migadikì qˆrth (U, z) me z = x + iy kai , = E|dz|2 tou M2, sÔmfwna me to
L mma 1.3.2, isqÔei
Br+1|Mr (X1, ..., Xr, e1, e1) + Br+1|Mr (X1, ..., Xr, e2, e2) = 0,
gia X1, ..., Xr ∈ ∆(U) kai {e1, e2} opoiod pote topikì orjomonadiaÐo plaÐsio tou U.
An jewr soume e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y , tìte epeid
∂
∂x = ∂
∂z + ∂
∂z , ∂
∂y = i( ∂
∂z − ∂
∂z ),
h teleutaÐa sqèsh gÐnetai
Br+1|Mr (X1, ..., Xr,
∂
∂z
,
∂
∂z
) = 0.

Epomènwc, h Br+1|Mr èqei thn anˆlush
Br+1|Mr = Br+1|
(r+2,0)
Mr
dzr+2
+ Br+1|
(0,r+2)
Mr
dzr+2
,
ìpou
Br+1|
(r+2,0)
Mr
= Br+1|Mr
∂
∂z
, ...,
∂
∂z
, Br+1|
(0,r+2)
Mr
= Br+1|Mr
∂
∂z
, ...,
∂
∂z
kai isqÔei Br+1|
(r+2,0)
Mr
= Br+1|
(0,r+2)
Mr
.
OrÐzoume to (2r + 4)-diaforikì
Φr+1 := Br+1|
(r+2,0)
Mr
, Br+1|
(r+2,0)
Mr
dz2r+4
.
Epeid
B∗
r+1|p =
Br+1|p, p ∈ Mr,
0, p ∈ Mc
r ,
isqÔei
Φ∗
r+1|p =
Φr+1|p, p ∈ Mr,
0, p ∈ Mc
r
kai epomènwc to Φr+1 eÐnai kalˆ orismèno se ìlo to Mr. SumbolÐzoume me {ω1, ω2}
to sumplaÐsio tou {e1, e2}. Sto Mr ∩ U èqoume to migadikì (1,0)-tanustikì pedÐo
φ = ω1 + iω2 =
√
Edz kai tic migadikèc sunart seic Hα
r+1 gia α = 2s + 3, ..., n.
Ja apodeÐxoume ìti Φr+1 = 1
4Er+2
α≥2r+3(H
α
r+1)2dz2r+4 sto Mr ∩ U kai ìti eÐnai
olìmorfo.
JewroÔme migadikì qˆrth (U, z) me z = x+iy kai , = E|dz|2 tou M2. Epilègoume
orjomonadiaÐo plaÐsio katˆ m koc thc f|Mr∩U me efaptìmeno mèroc {e1, e2}, ìpou
e1 = 1√
E
∂
∂x , e2 = 1√
E
∂
∂y kai kˆjeto mèroc {eα}, ìpou
e3 =
1
κ1
B(e1, e1), e4 =
1
κ1
B(e1, e2),
e5 =
1
κ2
B2(e1, e1, e1), e6 =
1
κ2
B2(e1, e1, e2),
.
.
.
e2r+1 =
1
κr
Br(e1, ..., e1, e1), e2r+2 =
1
κr
Br(e1, ..., e1, e2)
kai eα gia α ≥ 2r + 3 eÐnai tuqìnta. Sto Mr ∩ U, epiqeirhmatolog¸ntac ìpwc sthn
arq thc apìdeixhc (sel. 33), apodeiknÔetai ìti
Br+1(
∂
∂z
, ...,
∂
∂z
) =
1
2
E
r+2
2 Br(e1, ..., e1) − i
1
2
E
r+2
2 Br+1(e1, ..., e1, e2).

Epomènwc, èqoume
B
(r+2,0)
r+1 = Br+1
∂
∂z
, ...,
∂
∂z
=
1
2
E
r+2
2 Br(e1, ..., e1) − iBr+1(e1, ..., e1, e2)
=
1
2
E
r+2
2
α≥2r+3
hα
(r+1),1eα − i
α≥2r+3
hα
(r+1),2eα
=
1
2
E
r+2
2
α≥2r+3
H
α
r+1eα
kai sunep¸c
Φr+1 = Br+1|
(r+2,0)
Mr
, Br+1|
(r+2,0)
Mr
dz2r+4
=
1
4
Er+2
α≥2r+3
H
α
r+1eα,
α≥2r+3
H
α
r+1eα dz2r+4
=
1
4
Er+2
α≥2r+3
(H
α
r+1)2
dz2r+4
.
Jètoume
fr+1 :=
1
4
Er+2
α≥2r+3
(H
α
r+1)2
.
ApodeÐxame sto Mr ∩ U th sqèsh
dH
α
r+1 − i(r + 2)H
α
r+1ω12 +
β≥2r+3
H
β
r+1ωβα ∧ φ = 0
gia kˆje α ≥ 2r + 3. Pollaplasiˆzontac aut th sqèsh me H
α
r+1, ajroÐzontac wc
proc α ≥ 2r + 3, lambˆnontac upìyh thn (2.5) kai epeid
α,β≥2r+3
H
α
r+1H
β
r+1ωβα ∧ φ = 0,
ftˆnoume sth sqèsh
d
1
4
α≥2r+3
(H
α
r+1)2
∧ φ +
r + 2
2
α≥2r+3
(H
α
r+1)2
dφ = 0,
isodÔnama
d
fr+1
Er+2
∧ φ +
2r + 4
Er+2
fr+1dφ = 0.
Lìgw twn (2.3), (2.4) metˆ apì prˆxeic brÐskoume
∂fr+1
∂z
= 0.
Sunep¸c to Φr+1 eÐnai olìmorfo diaforikì sto U.

Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα

Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα

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Μεταπτυχιακή διατριβή στη Διαφορική Γεωμετρία: Συμπαγεις Ελαχιστικες Επιφανειες Γενους μηδεν Στην n-Διαστατη Σφαιρα