Διαρμονικές υποπολλαπλότητες της τρισδιάστατης σφαίρας

PANEPISTHMIO PATRWN TMHMA MAJHMATIKWNDIARMONIKES UPOPOLLAPLOTHTES THS SFAIRAS S3 STELLA SEREMETAKH MAJHMATIKOS ErgasÐa gia metaptuqiakì dÐplwma eidÐkeushc sta Jewrhtikˆ Majhmatikˆ Epiblèpwn : Lèktorac Andrèac Arbanitoge¸rgoc PATRA 2006

EuqaristÐec Jewr¸ kaj kon, na ekfrˆsw tic eilikrineÐc kai pio jermèc mou euqaristÐecston epiblèponta Lèktora Andrèa Arbanitogèwrgo kaj¸c kai kai sta ˆl-la dÔo mèlh thc TrimeloÔc Sumbouleutik c Epitrop c, Kajhght  BasÐleioPapantwnÐou kai Kajhght  Ajanˆsio Kotsi¸lh gia th sumbol  touc sthnteleiopoÐhsh aut c thc metaptuqiak c ergasÐac.Jewr¸ epÐshc upoqrèws  mou na euqarist sw thn oikogèneiˆ mou gia thnhjik  kai oikonomik  upost rixh pou mou prìsferan katˆ th diˆrkeia twnspoud¸n mou. Stèlla Seremetˆkh Pˆtra, Septèmbrioc 2006

PerieqìmenaPrìlogoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii1 Basikèc 'Ennoiec 12 Logismìc twn metabol¸n 132.1 Eisagwg  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Armonikèc kai diarmonikèc apeikonÐseic 233.1 Armonikèc apeikonÐseic . . . . . . . . . . . . . . . . . . . . . . 233.2 Diarmonikèc apeikonÐseic . . . . . . . . . . . . . . . . . . . . . 384 Diarmonikèc Upopollaplìthtec 414.1 Eisagwg  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Diarmonikèc kampÔlec sthn S3 . . . . . . . . . . . . . . . . . . 424.3 Diarmonikèc epifˆneiec sthn S3 . . . . . . . . . . . . . . . . . 49bibliografÐa 61 i

ii PERIEQŸOMENAPrìlogoc Skopìc thc ergasÐac aut c eÐnai h anaz thsh twn diarmonik¸n upopol-laplot twn M m, m = 1, 2, thc sfaÐrac S3.H mèjodoc pou efarmìzoume sundèetai me thn arq  logismoÔ twn metabol¸nwc mia mèjodoc sullog c twn bèltistwn antikeimènwn apì ènan q¸ro X meton ex c trìpo:(1) Sullègoume ìla ta antikeÐmena sto q¸ro X.(2) Epilègoume mia katˆllhlh sunˆrthsh E ston X. Tìte ta mègista  elˆqista thc sunˆrthshc eÐnai ta bèltista antikeÐmena pou anazhtˆme.Pio sugkekrimèna, ja parousiˆsoume me ekten  trìpo apotelèsmata apì ticergasÐec [6] , [8] twn R. Caddeo, S. Montaldo, C. Oniciuc, oi opoÐec aforoÔndiarmonikèc upopollaplìthtec thc sfaÐrac S3.Analutikìtera, h diˆrjrwsh thc ergasÐac èqei ¸c ex c:Sto kefˆlaio 1 parousiˆzontai sunoptikˆ orismoÐ kai ènnoiec apì th jewrÐapollaplot twn pou apaitoÔntai gia thn parousÐash thc metaptuqiak c er-gasÐac.Sto kefˆlaio 2 parousiˆzetai o sun jhc trìpoc prosèggishc tou logismoÔtwn metabol¸n kaj¸c kai kˆpoiec gnwstèc jewrÐec pou phgˆzoun apì ticmejìdouc metabol¸n.Sto kefˆlaio 3 orÐzontai oi ènnoiec thc armonik c kai diarmonik c apeikì-nishc metaxÔ duo pollaplot twn Riemann kai dÐnontai paradeÐgmata tètoiwnapeikonÐsewn.Sto kefˆlaio 4 anazhtoÔme tic diarmonikèc kampÔlec kai tic diarmonikèc epifˆneiecthc sfaÐrac S3. Oi kentrikèc mac anaforèc eÐnai oi ergasÐec [8] , [11] twn R.Caddeo, S. Montaldo, C. Oniciuc kai J. Eells, L. Lemaire.

Kefˆlaio 1Basikèc 'Ennoiec Orismìc 1.1. Onomˆzoume topologik  pollaplìthta diˆstashcn ènan sunektikì topologikì q¸ro tou Hausdorff me thn idiìthta se kˆjeshmeÐo tou na upˆrqei perioq  omoiomorfik  me èna anoiktì uposÔnolo touRn.Tètoia pollaplìthta gia parˆdeigma eÐnai o q¸roc Rn.Orismìc 1.2. Topikìc qˆrthc pˆnw se mia n-diˆstath topologik  pol-laplìthta M lègetai kˆje duˆda (U, φ) ìpou φ eÐnai h omoiomorfik  apeikì-nish φ : U ⊆ M n → V ⊆ Rn, ìpou U eÐnai èna anoiktì uposÔnolo thc M nkai V èna anoiktì uposÔnolo tou EukleÐdiou q¸rou Rn.JewroÔme èna qˆrth (U, φ) miac topologik c pollaplìthtac M n. Tìte kˆjeshmeÐo p ∈ U kajorÐzetai apì tic suntetagmènec {x1(p), x2(p), ..., xn(p)} toushmeÐou φ(p) ∈ Rn. Dhlad , xi(p) = xi(φ(p)) = (xi ◦ φ)(p), i = 1, 2, ..., n.An to sÔnolo U eÐnai sunektikì, tìte oi arijmoÐ xi(p) lègontai topikècsuntetagmènec tou shmeÐou p wc proc to qˆrth (U, φ) kai h n-ˆda twnsunart sewn xi : U ⊆ M → R 1

2 KEFŸALAIO 1. BASIKŸES ŸENNOIES p → xi(p) = (φ(p))i , i = 1, 2, ..., nlègetai sÔsthma topik¸n suntetagmènwn sto U wc proc to qˆrth(U, φ), ìpou h i-suntetagmènh tou p eÐnai h i-suntetagmènh tou φ(p).Epomènwc kˆje topikìc qˆrthc thc M orÐzei èna topikì sÔsthma suntetag-mènwn aut c.Orismìc 1.3. Onomˆzoume ˆtlanta diˆstashc n kai klˆshc Cr pˆn-w se mia n-diˆstath topologik  pollaplìthta M, mia oikogèneia topik¸nqart¸n Uα = {, Uαφα}α∈I (ìpou I eÐnai èna sÔnolo deikt¸n), pou ikanopoieÐta parakˆtw axi¸mata :(1) Ta sÔnola Uα kalÔptoun thn topologik  pollaplìthta M , dhlad  Uα = M α∈I(2) An Uα ∩ Uβ = ∅, oi omoiomorfismoÐ φα kai φβ eÐnai tètoioi ¸ste o o- moiomorfismìc φβ ◦ φ−α 1 : φα(Uα ∩ Uβ) ⊆ Rn → φβ(Uα ∩ Uβ) ⊆ Rn na eÐnai amfidiaforÐsimoc klˆshc Cr. Orismìc 1.4. Oi qˆrtec c1 = (Uα, φα) kai c2 = (Uβ, φβ) klˆshcCr, pˆnw se mia n-diˆstath topologik  pollaplìthta M , onomˆzonai Cr-sumbibastoÐ, an(1) Uα ∩ Uβ = ∅,   efìson Uα ∩ Uβ = ∅,(2) h apeikìnish φβ ◦ φ−α 1 : φα(Uα ∩ Uβ) ⊆ Rn → φβ(Uα ∩ Uβ) ⊆ Rn na eÐnai klˆshc Cr.

3Orismìc 1.5. DÔo Cr-ˆtlantec U1,U2 diˆstashc n miac topologik c pol-laplìthtac M onomˆzontai Cr-sumbibastoÐ, an(1) U1 ∪ U2 eÐnai pˆli ènac Cr-ˆtlantac thc M kai(2) An c1 ∈ U1 kai c2 ∈ U2 eÐnai dÔo tuqaÐoi qˆrtec, tìte oi qˆrtec autoÐ eÐnai Cr-sumbibastoÐ. Orismìc 1.6. DiaforÐsimh pollaplìthta diˆstashc n kai klˆshcCr, onomˆzoume kˆje n-diˆstath topologik  pollaplìthta M, efodiasmènhme mia klˆsh isodÔnamwn Cr-sumbibast¸n atlˆntwn pˆnw sth M .Upojètoume ìti M eÐnai mia diaforÐsimh pollaplìthta diˆstashc n, tˆxhcdiaforisimìthtac r(  klˆshc Cr) kai ìti A eÐnai èna anoiktì uposÔnolo thcM.Orismìc 1.7. H sunˆrthsh f : A ⊆ M → R onomˆzetai diaforÐsimhtˆxhc r (  klˆshc Cr ) pˆnw sto A an h sunˆrthsh f ◦ φ−1 : φ(U ∩ A) ⊆ Rn → ReÐnai diaforÐsimh gia kˆpoio qˆrth (U, φ) pˆnw sth M.To sÔnolo twn diaforÐsimwn sunart sewn klˆshc Cr, pou orÐzontai sth n-diˆstath pollaplìthta M klˆshc Cr, sumbolÐzetai me Dr(M ), en¸ to sÔnolotwn diaforÐsimwn sunart sewn pou orÐzontai sthn pollaplìthta M , klˆshcC∞, sumbolÐzetai me D0(M ).

4 KEFŸALAIO 1. BASIKŸES ŸENNOIES Orismìc 1.8. H apeikìnish f : A ⊆ Mn → Nm onomˆzetai dia-forÐsimh klˆshc Cr sto shmeÐo p ∈ A, an gia kˆje qˆrth (U, φ) thc Mkai (V, ψ) thc N tètoio ¸ste p ∈ U kai f (p) ∈ V , h apeikìnish F = ψ ◦ f ◦ φ−1 : φ(U ∩ f −1(V )) ⊆ Rn → Rmna eÐnai diaforÐsimh klˆshc Cr sto shmeÐo φ(p) ∈ Rn.'Estw to sÔnolo Dr(M, p) ìlwn twn diaforÐsimwn sunart sewn klˆshc Crsto shmeÐo p ∈ M . To sÔnolo Dr(M, p) apoteleÐ dianusmatikì q¸ro, oopoÐoc gÐnetai ˆlgebra an orÐsoume wc deÔtero nìmo eswterik c sÔnjeshcton pollaplasiasmì sunart sewn.Orismìc 1.9. An p eÐnai èna tuqaÐo shmeÐo thc n-diˆstathc pollaplìthtacM kai X = (X1, X2, ..., Xn) èna diˆnusma sto shmeÐo p, onomˆzoumeEfaptìmeno diˆnusma sto shmeÐo p thc n-diˆstathc pollaplìthtac Mthn apeikìnish Xp : Dr(M, p) → Rme tim  n Xp(φ) = ( ∂φ )Xpi ∂xi ipou ikanopoieÐ tic parakˆtw sunj kec :(1) Xp(λf + µg) = λXpf + µXpg(2) Xp(f g) = f (p)Xg(f ) + g(p)Xp(f ), gia kˆje f, g ∈ Dr(M, p), λ, µ ∈ RTo sÔnolo twn efaptìmenwn dianusmˆtwn sto shmeÐo p miac diaforÐsimhc pol-laplìthtac M , apoteleÐ dianusmatikì q¸ro. Ton dianusmatikì autì q¸ro ton

5lème efaptìmeno q¸ro thc M sto shmeÐo p kai ja ton sumbolÐzoumeme TpM.Orismìc 1.10. O duikìc q¸roc tou TpM eÐnai o grammikìc q¸roc pouapoteleÐtai apì to sÔnolo twn grammik¸n apeikonÐsewn me pedÐo orismoÔ toq¸ro TpM kai timèc sto sÔnolo R. O q¸roc autìc sumbolÐzetai me Tp∗M ,eÐnai isomorfikìc me ton TpM kai onomˆzetai sunefaptìmenoc q¸roc thc Msto p. H sullog  ìlwn twn efaptìmenwn (sunefaptìmenwn) q¸rwn thc Mse kˆje shmeÐo aut c sumbolÐzetai me T M (T ∗M antÐstoiqa) kai lègetai e-faptìmenh dèsmh (sunefaptìmenh dèsmh antÐstoiqa), T M = TpM = (p, Xp); p ∈ M, Xp ∈ TpM p∈MkaiT ∗M = Tp∗M p∈MOrismìc 1.11. Mia diaforik  morf  pr¸thc tˆxhc   diaforik  1-morf epÐ thc diaforÐsimhc pollaplìthtac M onomˆzetai h apeikìnish ω : M → Tp∗M p∈Mh opoÐa se kˆje shmeÐo p ∈ M antistoiqeÐ to sunefaptìmeno diˆnusma ωptou sunefaptìmenou q¸rou Tp∗M . Dhlad  gia kˆje p ∈ M h antÐstoiqh di-aforik  1-morf  eÐnai mia grammik  morf  pˆnw ston TpM , (ωp : TpM → R).Ean D1(M ) eÐnai to sÔnolo twn dianusmatik¸n pedÐwn epÐ thc M kai D1(M )to duikì tou sÔnolo, tìte wc diaforÐsimec 1-morfèc orÐzontai ta stoiqeÐa tou

6 KEFŸALAIO 1. BASIKŸES ŸENNOIESD1(M ) ìpou D1(M ) = ω; ω : D1(M ) → D0(M ) kai h ω eÐnai diaforÐsimhgrammik  apeikìnish en¸ D0(M ) eÐnai o q¸roc twn diaforÐsimwn sunart sewn.'Estw M , N dÔo diaforÐsimec pollaplìthtec kai φ mia apeikìnish apì thM sth N .Orismìc 1.12. H apeikìnish dφp : TpM → Tφ(P )Nme tim  dφp : Xp → dφp(Xp)onomˆzetai diaforikì thc apeikìnishc φ : M → N sto shmeÐo p. Sum-bolÐzetai epÐshc kai me φ∗p kai eÐnai mia grammik  apeikìnish tou efaptìmenouq¸rou TpM ston efaptìmeno q¸ro Tφ(p)N , ìpou, dφp(Xp) : D0(N ) → R dφp(Xp) : g → dφp(Xp)g = Xp(g ◦ ϕ)Orismìc 1.13. An φ : M → N eÐnai mÐa diaforÐsimh apeikìnish, todiaforikì dφp thc φ sto p ∈ M eÐnai mÐa grammik  apeikìnish dφp : TpM → Rme tim  pou orÐzetai mèsw thc apeikìnishc dφp : Xp → dφp(Xp)

7kai exaitÐac thc isomorfik c taÔtishc Tφ(p)R ≡ R èqoume dφp(Xp) = Xp(φ)Orismìc 1.14. 'Estw φ : M → N mia diaforÐsimh apeikìnish metaxÔ twnpollaplot twn M kai N kai èstw ω ∈ T ∗M . Onomˆzoume apeikìnishepistrof c (pull back) thc ω mèsou thc φ thn apeikìnish φ∗ω : Tφ(p)N → TpMme tim  φ∗ω(u1, u2, u3) = ω(φ∗(u1), ..., φ∗(un))gia kˆje ui ∈ TpM , i = 1, 2, ..., n kai p ∈ M .Orismìc 1.15. O metrikìc tanust c Riemann eÐnai ènac sunalloÐ-wtoc tanust c tÔpou (0,2), tètoioc ¸ste se kˆje shmeÐo p ∈ M antistoiqeÐthn apeikìnish , : TpM × TpM → Rme tic akìloujec idiìthtec:1. (i) vp + wp, zp kai (ii) λvp, wp = λ vp, wp2. vp, wp = wp, vp3. vp, vp ≥ 0 me vp, vp = 0 an kai mìno an vp = 0,gia kˆje vp, wp, zp ∈ TpM .Orismìc 1.16. Kˆje pollaplìthta M efodiasmènh me mia metrik  Riemann, , lègetai pollaplìthta Riemann.Orismìc 1.17. Sunoq    sunalloÐwth parˆgwgo ∇ se mia C∞−pollaplìthta M kaloÔme thn apeikìnish : ∇ : D1(M ) × D1(M ) → D1(M )

8 KEFŸALAIO 1. BASIKŸES ŸENNOIES (X, Y ) → ∇XYpou ikanopoieÐ tic akìloujec sunj kec :(1) ∇X(Y + Z) = ∇XY + ∇XZ(2) ∇X+Y Z = ∇X Z + ∇Y Z(3) ∇fX Y = f ∇X Y(4) ∇X(f X) = (Xf )Y + f ∇XYgia kˆje f ∈ C∞(M ) kai X, Y ∈ D1(M )(me D1(M ) sumbolÐzoume to sÔnolo ìlwn twn dianusmatik¸n pedÐwn epÐ thcM .)Orismìc 1.18. Gia kˆje dianusmatikì pedÐo X, Y ∈ D1(M ) to diaforÐsimodianusmatikì pedÐo thc M , [X, Y ] = XY − Y Xpou dra sto q¸ro D0(M ) twn diaforÐsimwn sunart sewn thc M me tim  [X, Y ]f = X(Y f ) − Y (Xf )pou eÐnai epÐshc mia diaforÐsimh sunˆrthsh, gia kˆje f ∈ D0(M ), lègetaiagkÔlh tou Lie twn dianusmatik¸n pedÐwn X, Y tou D1(M ).Je¸rhma 1.1. 'Estw (M, g) mia C∞− pollaplìthta Riemann diˆstashcn. H sunoq  ∇ pou ikanopoieÐ th sqèsh2g(∇XY, Z) = X(g(Y, Z))+Y (g(Z, X))−Z(g(X, Y ))+g(Z, [X, Y ])+g(Y, [Z, X])−g(X, [Y, Z])gia kˆje X, Y, Z ∈ D1(M ), kaleÐtai Levi-CivitaEpiplèon h sunoq  Levi-Civita ikanopoieÐ tic sunj kec(1) X(g(Y, Z)) = g(∇XY, Z) + g(Y, ∇XZ)(2) ∇XY − ∇Y X = [X, Y ]Antistrìfwc, kˆje sunoq  pou ikanopoieÐ tic (1) kai (2) eÐnai Levi-Civita.

9 Orismìc 1.19. Tanustikì pedÐo kampulìthtac R miac pol-laplìthtac M efodiasmènhc me mia sÔndesh ∇ kaleÐtai to tanustikì pedÐotÔpou (1,3) me tim R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z, gia kˆje X, Y, Z ∈ D1(M ).To tanustikì pedÐo kampulìthtac ikanopoieÐ tic akìloujec sqèseic: (1) R(X, Y )Z = −R(Y, X)Z (2) R(X1 + X2, Y ) = R(X1, Y ) + R(X2, Y ) (3) R(X, Y1 + Y2) = R(X, Y1) + R(X, Y2) (4) R(f X, gY )hZ = f ghR(X, Y )ZEˆn h sunoq  ∇ eÐnai summetrik , tìte isqÔoun oi tautìthtec (5) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0 (6) (∇XR)(Y, Z)W + (∇Y R)(Z, X)W + (∇ZR)(X, Y )W = 0gia kˆje X, Y, Z ∈ D1(M ) kai f, g, h ∈ D0(M ).Oi tautìthtec (5) kai (6) kaloÔntai pr¸th kai deÔterh tautìthta tou BianchiantÐstoiqa.Orismìc 1.20. SunalloÐwto tanustikì pedÐo twn Cristoffel-Riemann lègetai h tetragrammik  apeikìnishR : D1(M ) × D1(M ) × D1(M ) × D1(M ) → D0(M ) me tim R(X, Y, Z, W ) = g(R(X, Y )Z, W ) gia kˆje X, Y, Z, W ∈ D1(M ).To sunalloÐwto tanustikì pedÐo twn Cristoffel-Riemann ikanopoeÐ tic akìlou-jèc idiìthtec: (1) R(X, Y, Z, W ) = −R(Y, X, Z, W ) = −R(X, Y, W, Z) = R(Y, X, Z, W )

10 KEFŸALAIO 1. BASIKŸES ŸENNOIES(2) R(X, Y, Z, W ) + R(X, Z, W, Y ) + R(X, W, Y, Z) = 0 (3) (∇XR)(Y, Z, W, V ) + (∇Y R)(Z, X, W, V ) + (∇ZR)(X, Y, W, V ) = 0'Ena ˆllo tanustikì pedÐo pou orÐzetai apì to tanustikì pedÐo twn Cristoffel-Riemann , eÐnai to tanustikì pedÐo tou Ricci kai gia kˆje shmeÐo thcpollaplìthtac M o antÐstoiqoc tanust c tou Ricci.'Estw p ∈ M tuqaÐo shmeÐo thc pollaplìthtac M kai TpM o efaptìmenocq¸roc aut c sto shmeÐo p. JewroÔme thn apeikìnish R(−, X)Y : TpM → TpMme tim  R(−, X)Y : Z → R(Z, X)YOrismìc 1.20. O tanust c tou Ricci orÐzetai wc to Ðqnoc thc apeikìni-shc, R(−, X)Y , gia kˆje X, Y ∈ TpM , kai sumbolÐzetai me S(X, Y )   Ric(X, Y ).Me th bo jeia topikoÔ sust matoc suntetagmènwn {xi}ni=1, o tanust c touRicci grˆfetai wc èxhc: nn S(X, Y ) = Ric(X, Y ) = g(R(ei, X)Y, ei) = R(ei, X, Y, ei) i=1 i=1ìpou X, Y ∈ TpM kai {e1, e2, ..., en} eÐnai mia orjokanonik  bˆsh tou TpM wcproc to topikì sÔsthma suntetagmènwn {xi}ni=1.Orismìc 1.21. Metasqhmatismìc tou Ricci   telest c kampu-lìthtac tou Ricci sto shmeÐo p ∈ M , wc proc to efaptìmeno diˆnusmaX ∈ TpM , lègetai h apeikìnish Sx = R(−, X)X : TpM → TpMme tim  Sx : Y → Sx(Y ) = R(Y, X)X

11Orismìc 1.22. Bajmwt  kampulìthta thc pollaplìthtac (M, g)lègetai h sunˆrthsh sg h opoÐa orÐzetai apì th sustol  twn deikt¸n toutanustikoÔ pedÐou tou Ricci kai dÐnetai apì th sqèsh nn n sg = gijSij = Ric(ei, ei) = g(Qei, ei) i,j=1 i=1 i=1ìpou Q = n R(−, ei)ei eÐnai o metasqhmatismìc tou Ricci kai {e1, e2, ..., en} i=1eÐnai mia orjokanonik  bˆsh tou TpM.Orismìc 1.23. Gia dÔo grammik¸c anexˆrthta dianÔsmata u, v touefaptìmenou q¸rou TpM sto p thc pollaplìthtac Riemann (M, g) o arijmìc K(u, v) = g(R(u, v)v, u) g(u, u)g(v, v) − g(u, v)2lègetai kampulìthta tom c thc (M, g) wc proc to zeÔgoc u, v . H(M, g) èqei jetik  (arnhtik ) kampulìthta tom c eˆn gia kˆje p ∈ M kaigia duo grammik¸c anexˆrthta dianÔsmata u, v tou TpM , K(u, v) ≥ 0,(K(u, v) ≤ 0).

12 KEFŸALAIO 1. BASIKŸES ŸENNOIES

Kefˆlaio 2Logismìc twn metabol¸n2.1 Eisagwg  O logismìc twn metabol¸n eÐnai mia jewrÐa pou basÐzetai sthn idèa ìti eÐnaidunatìn na ermhneujoÔn pollˆ fainìmena sta majhmatikˆ kai sth fusik  wckrÐsima shmeÐa sunarthsoeid¸n. Sto kefˆlaio autì ja parousiˆsoume merikècshmantikèc jewrÐec twn majhmatik¸n kai thc fusik c, oi opoÐec proèrqontaiapì mejìdouc tou logismoÔ metabol¸n (variational methods) kaj¸c kai pa-radeÐgmata twn mejìdwn aut¸n. H arq  tou logismoÔ twn metabol¸n eÐnai miamèjodoc sullog c twn bèltistwn apì mia sullog  majhmatik¸n antikeimènwnme ton ex c trìpo: (1) Sullègoume ìla ta antikeÐmena apì èna q¸ro X. (2) Epilègoume mia katˆllhlh sunˆrthsh E ston X. Ta mègista   taelˆqista thc sunˆrthshc aut c eÐnai ta bèltista antikeÐmena pou anazhtoÔme.ArketoÐ epist monec, ìpwc oi I. Newton, G.W. Leibnitz, P.L. Maupertuis, L.Euler kai J.L. Lagrange asqol jhkan me to logismì metabol¸n. 13

14 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWN O sunhjismènoc trìpoc prosèggishc tou logismoÔ twn metabol¸n eÐnai oex c: (1) Sto q¸ro X jewr¸ to diaforikì E thc sunˆrthshc E. (2) Eˆn to x ∈ X eÐnai èna apì ta bèltista majhmatikˆ antikeÐmena tìteautì epitugqˆnei thn elaqistopoÐhsh   th megistopoÐhsh thc sunˆrthshc E.Epomènwc h parˆgwgoc thc E mhdenÐzetai sto x, dhlad  E (x) = 0. (3) To shmeÐo x pou ikanopoieÐ th sqèsh E (x) = 0 kaleÐtai krÐsimoshmeÐo. H parapˆnw sqèsh antistoiqeÐ sthn exÐswsh twn Euler-Lagrange. (4) Skopìc eÐnai na lujeÐ h exÐswsh aut . Kˆpoiec forèc sqediˆzoume thn antÐstrofh diadikasÐa: (1) Jèloume na lÔsoume tic diaforikèc exis¸seic kˆpoiou problhmatìc mac (2) Gia na pragmatopoihjeÐ autì, jewroÔme ènan q¸ro X kai mia sunˆrthshE ston X ètsi ¸ste h exÐswsh twn Euler-Lagrange na antistoiqeÐ sthn exÐsw-sh tou problhmatìc mac. (3) ArkeÐ tìte na brejeÐ èna elˆqisto   mègisto thc sunˆrthshc E ston X.Sto mèso thc dekaetÐac tou 1960, oi R. Palais kai S. Smale dieukrÐnhsan kˆtwapì poièc sunjhkèc h sunˆrthsh E èqei elˆqista. H sunj kh aut  kaleÐtaisunj kh twn Palais-Smale (P-S) kai perigrˆfetai wc ex c : Upojètoume ìti(M, g) eÐnai mia Ck+1-pollaplìthta Riemann kai f : M → N mia Ck+1-sunˆrthsh ( k ≥ 1) kai èstw S èna uposÔnolo thc M . H f ikanopoieÐ thsunj kh (P-S) eˆn isqÔoun ta ex c: (1) H f eÐnai fragmènh sto S kai (2) inf ∇f (x) : x ∈ S = 0 Tìte upˆrqei shmeÐo x sth j kh S¯ tou S, ètsi ¸ste to x na eÐnai krÐsimoshmeÐo thc f , dhlad  ∇fx= 0. (∇f : M → ∇fx ∈ TxM gia kˆje x ∈ M ).

2.1. EISAGWGŸH 15Gia na exhg soume th sunj kh (P-S) jewroÔme to ex c parˆdeigma :'Estw duo sunart seic f kai g ston M = R me tÔpouc, (1) f (x) = x2, −∞ < x < ∞ (2) g(x) = ex3, −∞ < x < ∞Kai oi duo sunart seic èqoun infima mhdèn. H pr¸th èqei elˆqisto stoshmeÐo (0, 0), en¸ h deÔterh den èqei elˆqisto.PoÔ ofeÐletai to parapˆnw fainìmeno;H apˆnthsh eÐnai ìti h sunˆrthsh f (x) ikanopoieÐ th sunj kh (P-S), en¸h sunˆrthsh g(x) ìqi. SumbaÐnei wstìso, gia kˆpoia probl mata pou denikanopoioÔn th sunj kh (P-S) h sunˆrthsh E na èqei elˆqisto.1.1. Mèjodoc twn metabol¸n kai jewrÐec pedÐou H mèjodoc twn metabol¸n brÐskei efarmog  sth fusik , kurÐwc stic jew-rÐec pedÐou (field theories). Se aut n th parˆgrafo ja d¸soume mia eikìnatwn armonik¸n apeikonÐsewn kai ˆllwn jewri¸n pedÐou. EÐnai gnwstì ìtisth fÔsh upˆrqoun tessˆrwn eid¸n dunˆmeic, h barÔthta (gravitation), h h-lektromagnhtik  dÔnamh (electromagnetism), h asjènhc allhlepÐdrash (weakinteraction) kai h isqur  allhlepÐdrash (strong interaction). Eqoun gÐneiprospˆjeiec na sumperilhfjoÔn oi dunˆmeic autèc se mia enwpoihmènh jewrÐapedÐou. H barÔthta èqei perigrafeÐ apì th jewrÐa sqetikìthtac tou Einsteinkai o hlektromagnhtismìc apì th jewrÐa tou Maxwell. Autèc oi tèssericdunˆmeic èqoun katagrafeÐ apì touc fusikoÔc wc jewrÐec bajmÐdac.Ja perigrˆyoume tic jewrÐec autèc ìpwc phgˆzoun apì tic mejìdouc metabol¸n. Metrikèc tou Einstein'Estw M mia pollaplìthta diˆstas c m kai X o q¸roc ìlwn twn metrik¸nRiemann g sth M pou èqoun ìgko monˆda. 'Estw E h sunˆrthsh ston X,

16 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWNpou dÐdetai apì th sqèshE(g) = Sgvg, g ∈ X, Mìpou Sg h bajmwt  kampulìthta thc g kai vg to stoiqeÐo ìgkou pou dÐnetaiapì th sqèsh vg = det(gij).dx1...dxmH sunˆrthsh E onomˆzetai sunarthsoeidèc olik c kampulìthtac.JewroÔme mia tuqaÐa metabol  (deformation) gt , (− < t < ) , g0 = g thc g.Tìte h g eÐnai krÐsimo shmeÐo thc E ston X an kai mìno an d E(gt) = 0 dt t=0to opoÐo apodeiknÔetai ìti isodunameÐ me thn exÐswsh Ric(g) = cgìpou Ric(g) eÐnai o tanust c Ricci thc g kai c mia stajerˆ.Mia metrik  g pou ikanopoieÐ th parapˆnw exÐswsh kaleÐtai metrik  tou Ein-stein.Sunoqèc Yang - Mills (Yang - Mills Connections)Estw E mia dianusmatik  dèsmh se mia sumpag  pollaplìthta Riemann(M, g). JewroÔme to q¸ro X ìlwn twn sunoq¸n ∇ thc dianusmatik c dèsmhcE kai th sunˆrthsh E ston X me tÔpoE(∇) = 1 ∇R 2vg, ∇∈X 2 MO R∇ eÐnai o tanust c kampulìthtac thc sunoq c ∇ sth dianusmatik dèsmh E. JewroÔme mia metabol  (deformation) ∇t , (− < t < ), ∇0 = ∇thc ∇.Tìte h sunoq  ∇ apoteleÐ krÐsimo shmeÐo thc E an kai mìno an d E(∇t) = 0 dt t=o

2.1. EISAGWGŸH 17Ta krÐsima shmeÐa tou parapˆnw sunarthsoeidoÔc kaloÔntai sunoqèc Yang-Mills. Armonikèc apeikonÐseic 'Estw dÔo sumpageÐc pollaplìthtec Riemann (M, g) kai (N, h) kai èstwto sÔnolo X ìlwn twn leÐwn apeikonÐsewn apì th M sth N , dhlad  X =C∞(M, N ). 'Estw h sunˆrthsh E ston X pou dÐnetai apì th sqèsh E(φ) = 1 |dφ|2vg, φ ∈ X 2 Mìpou h apeikìnish dφ : T M → T N eÐnai to diaforikì thc φ.'Estw mia tuqoÔsa metabol  φt , (− < t < ) , φ0 = φ , thc φ.(Bl. sq ma 2.1)Tìte, h φ eÐnai armìnikh apeikìnish an kai mìno an eÐnai krÐsimo shmeÐothc E, dhlad  an kai mìno an d E(φt) = 0 dt t=0Parˆdeigma : Kleistèc gewdaisiakèc sth sfaÐra'Estw mia kleist  diaforÐsimh kampÔlh φ(x) = (φ1(x), φ2(x), φ3(x)), x ∈[0, 2π] ston R3 me perÐodo 2π. (Periodikìthta shmaÐnei ìti: φ(x + 2π) = φ(x),dhlad  φi(x + 2π) = φi(x), i = 1, 2, 3). AnazhtoÔme tic kampÔlec ekeÐnec pouapoteloÔn krÐsima shmeÐa tou sunarthsoeidoÔc thc enèrgeiac E(φ) = 1 2π 3 dφi 2 2 0 i=1 dx dx'Estw φε(x) = (φε,1(x), φε,2(x), φε,3(x)), x ∈ [0, 2π] mia metabol  thc φ meφ0 = φ kai φε(x + 2π) = φε(x), x ∈ [0, 2π] 'Eqoume ìtid E(φε) = 1 2π d 3 dφε,i 2 2π 3 d dφε,i(x) dφi(x) dxdε 2 0 dε ε=0 i=1 dx dx dx dx = dε ε=0 0 i=1 ε=0

18 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWN= 3d φε,i(x) dφi(x) x=2π 2π 3 d φε,i(x) d2φi(x) dx dε dx 0 i=1 dε dx2 ε=0 − ε=0 i=1 x=0Epeid  oi φε,i kai φi eÐnai periodikèc me perÐodo 2π o pr¸toc ìroc tou deÔteroumèlouc mhdenÐzetai, opìte prokÔptei ìti d E(φε) = 2π 3 d φε,i(x) d2 φε,i(x) dx dε 0 i=1 dε dx2 ε=0 ε=0Epiplèon, epeid  h φε(x) = (φε,1(x), φε,2(x), φε,3(x)) eÐnai mia leÐa metabol thc φ tìte kai h d φε(x) = d φε,1(x), d φε,2(x), d φε,3(x) dε dε dε dε ε=0 ε=0 ε=0 ε=0eÐnai leÐa periodik  apeikìnish .Epomènwc h φ eÐnai krÐsimo shmeÐo thc enèrgeiac an kai mìno an d E(φε) = 0, dε ε=0  isodÔnama d2φi(x) dx2 = 0, i = 1, 2, 3H lÔsh twn exis¸sewn eÐnai φi(x) = Bix + Ai, i = 1, 2, 3ìpou ta Ai, Bi eÐnai stajerèc. ExaitÐac thc periodikìthtac twn φi(x) èqoumeìti (x + 2π)Bi + Ai = xBi + Ai,dhlad  Bi = 0, opìte φi(x) = Ai gia kˆjex ∈ [0, 2π]. Epeid  oi lÔseic pou lambˆnoume sth perÐptwsh aut  eÐnai mìnooi tetrimmènec, eisˆgoume ton ex c periorismì: ApaitoÔme oi kampÔlec φ nabrÐskontai sth monadiaÐa sfaÐra S2 = (y1, y2, y3) ∈ R3; y12 + y22 + y32 = 1 kaianazhtoÔme ta krÐsima shmeÐa thc E, metaxÔ twn kampul¸n aut¸n.

2.1. EISAGWGŸH 19Me ton Ðdio trìpo pou perigrˆyame parapˆnw, jewroÔme mia metabol  φε(x)thc φ , x ∈ [0, 2π] . Tìte h φ ∈ S2 eÐnai krÐsimo shmeÐo an kai mìno an d E(φε) = 0 dε ε=0  isodÔnama 2π 3 d φε,i(x) d2φi(x) = 0 0 i=1 dε dx2 ε=0Sto shmeÐo autì prèpei na lˆboume upìyhn to periorismì φε(x) ∈ S2, x ∈[0, 2π]. Gia to lìgo autì, jewroÔme ton efaptìmeno q¸roTyS2 = V ∈ R3; V, y = 0 thc S2 se èna y ∈ S2, pou eÐnai to kˆjetoepÐpedo sto diˆnusma y.Kˆje diˆnusma V ∈ R3 mporeÐ na analujeÐ se duo sunist¸sec, mia sto kˆ-jeto q¸ro (TyS2)⊥ kai mia ston TyS2, dhlad  V = V, y y + (V − V, y y)ExaitÐac thc sunj khc φε(x) ∈ S2 gia kˆje x ∈ [0, 2π], to φε(x), φε(x) = 1.ParagwgÐzontac th teleutaÐa sqèsh sto ε = 0 kai lambˆnontac upìyh ìtiφ0(x) = φ(x) èqoume ìti ( d ) φε(x), φε(x) =0 dε ε=0dhlad  ( d ) φε(x) ∈ Tφ(x) S 2 dε ε=0Lìgw thc sqèshc V = V, y y + (V − V, y y) to diˆnusma d2φ = d2φ1 , d2φ2 , d2φ3 dx dx2 dx2 dx2analÔetai wc ex c : d2φ = d2φ(x) , φ(x) φ(x) + d2φ(x) − d2φ(x) , φ(x) φ(x) dx2 dx2 dx2 dx2kai epeid  o deÔteroc ìroc an kei ston Tφ(x)S2 autìc eÐnai mhdèn.'Ara, d2φ(x) d2φ(x) dx2 dx2 = , φ(x) φ(x)

20 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWNParagwgÐzoume th sqèsh φ(x), φ(x) = 1 gia kˆje x sto [0, 2π] kai èqoume dφ(x) , φ(x) =0 dxParagwgÐzontac xanˆ paÐrnoume d2φ(x) , φ(x) + dφ(x) , dφ(x) =0 d2(x) dx dx  isodÔnama d2φ(x) , φ(x) =− dφ(x) , dφ(x) dx2 dx dxExaitÐac thc teleutaÐac sqèshc h d2φ(x) = d2φ(x) , φ(x) φ(x) dx2 dx2paÐrnei th morf  d2φ(x) + dφ(x) , dφ(x) φ(x) =0 d2(x) dx dxSth sunèqeia paragwgÐzoume to eswterikì ginìmeno dφ(x) , dφ(x) dx dxkai èqoume d dφ(x) , dφ(x) =2 d2φ(x) , dφ(x) . dx dx dx dx2 dxLìgw thc d2φ(x) = d2φ(x) , φ(x) φ(x) dx2 dx2h parapˆnw sqèsh gÐnetaid dφ(x) , dφ(x) =2 d2φ(x) , dφ(x) = −2 dφ(x) , dφ(x) φ(x), dφ(x)dx dx dx dx2 dx dx dx dxkai lìgw thc dφ(x) , φ(x) =0 dx

2.1. EISAGWGŸH 21èqoume telikˆ ìtid dφ(x) , dφ(x) =2 d2φ(x) , dφ(x) = −2 dφ(x) , dφ(x) φ(x), dφ(x) =dx dx dx dx2 dx dx dx dx −2 dφ(x) , dφ(x) φ(x), dφ(x) =0 dx dx dxEpomènwc to eswterikì ginìmeno dφ(x) , dφ(x) dx dxeÐnai stajerì gia kˆje x ∈ [0, 2π]. Jètoume dφ(x) , dφ(x) = c2, c > 0 dx dxkai h sqèsh d2φ(x) d2(x) + dφ(x) , dφ(x) φ(x) =0 dx dxgÐnetai d2φ(x) dx2 + dφ(x) , dφ(x) φ(x) =0 dx dxIsodÔnama d2φi(x) dx2 + c2φi = 0, i = 1, 2, 3H genik  lÔsh tou sust matoc eÐnai φi(x) = Ai cos(cx) + Bi sin(cx) ⇔ φ(x) = A cos(cx) + B sin(cx)ìpou ta A kai B eÐnai dianÔsmata ston R3.Ikan  kai anagkaÐa sunj kh ¸ste h kampÔlh φ(x), x ∈ [0, 2π] na eÐnaiperiodik  me perÐodo 2π, na keÐtai sth sfaÐra S2 kai na apoteleÐ krÐsimo shmeÐothc E eÐnai : A, A = B, B = 1, A, B = 0 kai c = m (akèraioc) Mia tètoiakampÔlh eÐnai ènac mègistoc kÔkloc thc sfaÐrac S2 kai diagrˆfetai m forèckaj¸c to x metabˆletai apì to 0 èwc to 2π. (Eˆn to m eÐnai arnhtikì o kÔklocdiagrˆfetai sthn antÐjeth kateÔjunsh ).

22 KEFŸALAIO 2. LOGISMŸOS TWN METABOLŸWNSumpèrasma : Apì ìlec tic leÐec periodikèc kampÔlecφ(x) = (φ1(x), φ2(x), φ3(x)), x ∈ [0, 2π] me perÐodo 2π, oi opoÐec brÐskontaisth sfaÐraS2 = (y1, y2, y3) ∈ R3; y12 + y22 + y32 = 1 ta krÐsima shmeÐa thcE(φ) = 1 2π 3 dφi 2 2 0 i=1 dx dxeÐnai oi lÔseic thc diaforik c exÐswshcd2φ(x) + dφ(x) , dφ(x) φ(x) =0 dx dx dxAutèc oi lÔseic eÐnai mègistoi kÔkloi thc S2 pou diagrˆfontai m forèc kaj¸cto x metabˆletai apì to 0 èwc to 2π.

Kefˆlaio 3Armonikèc kai diarmonikècapeikonÐseic3.1 Armonikèc apeikonÐseic Orismìc 3.1.1. Mia leÐa apeikìnish φ ∈ C∞(M, N ) metaxÔ duo pol-laplot twn Riemann (M, g) kai (N, h) kaleÐtai armonik  an kai mìno aneÐnai krÐsimo shmeÐo tou sunarthsoeidoÔc thc enèrgeiacE(φ) = 1 |dφ|2vg 2 MH apeikìnish dφ : T M → T N eÐnai to diaforikì thc φ ∈ C∞(M, N ) kaivg = det(gij)dx1dx2...dxm to stoiqeÐo ìgkou thc metrik c g.H φ eÐnai krÐsimo shmeÐo thc E eˆn gia opoiad pote leÐa apeikìnishF : (−ε, ε) × M → N me tim  F (t, x) = φt(x), gia kˆje t ∈ (−ε, ε) kai giakˆje x ∈ M me F (0, x) = φ0(x) = φ(x) isqÔei h sqèshd E(φt) = 0dt t=0 23

24 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEIS Orismìc 3.1.2. Mia C1−kampÔlh γ : I → M thc pollapìthtac Monomˆzetai gewdaisiak  an ∇γ γ = 0, gia kˆje shmeÐo tou anoiqtoÔ di-ast matoc I.'Estw èna topikì sÔsthma suntetagmènwn {xi}ni=1 thc M .Tìte, γ(t) = (γ1(t), γ2(t), ..., γn(t)) kai γ (t) = n γi (t) ∂ γ(t) i=1 ∂xiEpomènwc h sqèsh ∇γ γ = 0 isodÔnama gÐnetai d2γi n dγj dγk dt2 dt dt + Γijk = 0, i = 1, 2, ..., n. j,k=1Jètoume ξi = dγi kai katal goume sto ex c sÔsthma diaforik¸n exis¸sewn: dt dξi n dt = − Γjikξjξk, i = 1, 2, ..., n. j,k=1Eˆn dojoÔn oi arqikèc timèc γ(0) = (γ1(0), γ2(0), ..., γn(0)) kaidγ (0) = dγ1 (0), dγ2 (0), ..., dγn (0) gia t = 0 to sÔsthma èqei monadik  lÔshdt dt dt dtgia ìla ta t sthn perioq  tou mhdenìc. Autì shmaÐnei ìti gia opoiod poteshmeÐo p thc M kai gia opoiod pote efaptìmeno diˆnusma u sto shmeÐo p touefaptìmenou q¸rou T M pou ikanopoioÔn tic sunj kec(1) γ(0) = p kai(2)γ (0) = uupˆrqei monadik  gewdaisiak  γ(t) gia t kontˆ sto mhdèn.SumbolÐzoume γ(t) = expp(tu) kai dÐnoume ton parakˆtw orismì.Orismìc 3.1.3. Ekjetik  apeikìnish sto shmeÐo p miac pollaplìthtacM , lègetai h apeikìnish expp : TpM → M me tim  ekeÐno to shmeÐo thc Mpou orÐzetai apì to γ(1), dhlad  γ(1) = expp ugia kˆje u ∈ TpM kai tètoio ¸ste na orÐzetai to γ(1). Autì shmaÐnei ìti tomètro tou efaptìmenou dianusmatoc u prèpei na eÐnai arketˆ mikrì, dhlad  to

3.1. ARMONIKŸES APEIKONŸISEIS 25t na paÐrnei timèc se mia perioq  tou mhdenìc sto q¸ro TpM .Orismìc 3.1.4. 'Estw mia tuqoÔsa C∞− apeikìnish V : M → T Nme V (x) ∈ Tφ(x)N, x ∈ M kai φt : M → N h ekjetik  C∞− apeikìnishme tim  φt(x) = expφ(x)(tV (x)), x ∈ M. Onomˆzoume to dianusmatikì pedÐoV (x) = d t=0 φt(x) dianusmatikì pedÐo metabol c katˆ m koc thc φ dt(variation vector field along φ).Antistrìfwc eˆn jewr soume mia tuqoÔsa leÐa metabol  φt ∈ C∞(M, N ) thcφ, ( <t< ) kai φ0 = φ, jètontac V (x) = d t=0 φt(x) orÐzetai mia C∞− dtapeikìnish V apì thn pollaplìthta M sthn efaptìmenh dèsmh T N me tim V (x) ∈ Tφ(x)N, x ∈ M .Orismìc 3.1.5. 'Estw duo Ck− pollaplìthtec E kai N kai π : E → Nmia Ck− apeikìnish. H π : E → N onomˆzetai Ck−dianusmatik  dèsmhepÐ thc N eˆn :(1) Gia kˆje x ∈ N o q¸roc π−1(x) = Ex o kaloÔmenoc n ma epÐ tou x eÐnaidianusmatikìc q¸roc diˆstashc k(2) upˆrqei anoiqt  geitoniˆ U thc N sto x, kai ènac diaforomorfismìcφ : π−1(U ) → U × Rk tou opoÐou o periorismìc sto π−1(ψ) eÐnai ènac i-somorfismìc epÐ tou ψ × Rk gia kˆje ψ ∈ U .Orismìc 3.1.6. DÐnetai mia Ck dianusmatik  dèsmh p : E → N kai mi-a Ck− apeikìnish φ : M → N metaxÔ duo Ck− pollaplot twn M kai N .Kataskeuˆzoume thn dianusmatik  dèsmh π : E → M , ìpou E = (p , u) ∈M × E; φ(p ) = π(u) , π ((p , u)) = p . SumbolÐzw th dianusmatik  dèsmh Eme φ∗E   φ−1E kai thn onomˆzw epag¸menh dianusmatik  dèsmh thcdianusmatik c dèsmhc E mèsw thc φ.

26 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEISSqhmatikˆ èqoume to diˆgramma:φ−1E /Eπ π  φ /  M N'Estw h C∞− dianusmatik  dèsmh π : T N → N me π(u) = φ(x) gia kˆjex ∈ M . OrÐzoume thn epag¸menh dèsmh φ−1T N thc efaptìmenhc dèsmhc T Nmèsw thc φ wc to sÔnoloφ−1T N = (x, u) ∈ M × T N ; π(u) = φ(x), x ∈ M = x∈M Tφ(x)NH sqhmatik  parˆstash èqei wc ex c:φ−1T N / TNπ π  φ / M Nìpou π : (M, T N ) → M eÐnai h C∞− dianusmatik  dèsmh me π (x, u) = x, x ∈M.Orismìc 3.1.7. Mia C∞−tom  (section) thc epag¸menhc dèsmhc φ−1T Nmèsw thc φ : M → N eÐnai h C∞− apeikìnish V : M → T N me V (x) ∈Tφ(x)N, x ∈ M .SumbolÐzoume to sÔnolo ìlwn twn C∞−tom¸n meΓ(φ−1T N ) = V ∈ C∞(M, T N ), V (x) ∈ Tφ(x)N, x ∈ M .ParathroÔme ìti to sÔnolo Γ(φ−1T N ) eÐnai to sÔnolo ìlwn twn dianus-matik¸n pedÐwn metabol c katˆ m koc thc φ.Gia kˆje f ∈ C∞(M ) ,V, V1, V2 ∈ Γ(φ−1T N ) kai x ∈ M orÐzoume sto sÔnoloφ−1T N touc ex c nìmouc :+ : Γ(E) × Γ(E) → Γ(E)(V1, V2) → V1 + V2

3.1. ARMONIKŸES APEIKONŸISEIS 27me tim  (V1 + V2)(x) = V1(x) + V2(x)ìpou E = φ−1T N kai · : C∞(M ) × Γ(E) → Γ(E) (f, V ) → f.Vme tim  (f.V )(x) = f (x).V (x)Me ton prosjetikì nìmo (+) to Γ(E) kajÐstatai abelian  omˆda. Epiplèon,isqÔoun oi ex c idiìthtec :(1) ((f + g)V )(x) = (f V )(x) + (gV )(x)(2) ((f.g)V )(x) = (f.(g.V ))(x)(3) (f.(V1 + V2))(x) = (f.V1)(x) + (f.V2)(x)Me tic parapˆnw idiìthtec h abelian  omˆda (Γ(E), +) kajÐstatai èna prìtupo(module) epÐ thc C∞(M ).Prin d¸soume ton orismì thc epag¸menhc sunoq c ∇ sthn epag¸menh dèsmhφ−1T N thc efaptìmenhc dèsmhc T N mèsw thc φ, dÐnoume touc epìmenoucorismoÔc.Orismìc 3.1.8. H apeikìnish σ : R → M me tim  σ(t) ∈ M gia kˆje t ∈ ReÐnai mia C1− kampÔlh thc M . Gia t = 0 èqoume(1) σ(0) = x kai(2) σ (0) = Xx ⇔ d t=0 σ(t) = Xx dtìpou Xx ∈ TxM kai h kampÔlh σt me tim  σt(s) = σ(s) eÐnai o periorismìc thcσ ìtan to 0 ≤ s ≤ t.Orismìc 3.1.9. To dianusmatikì pedÐo X lègetai parˆllhlo katˆ m kocthc C1− kampÔlhc γ : [a, b] ⊂ R → M an ta dianÔsmata tou pedÐou X se

28 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEISopoiad pote dÔo diaforetikˆ shmeÐa thc kampÔlhc eÐnai parˆllhla metaxÔ touc,dhlad  ∇γ X = 0'Estw èna topikì sÔsthma suntetagmènwn {xi}ni=1 se mia perioq  U thc M .Tìte, grˆfoume X(t) = n ξi(t) ∂ , ìpou X(t) ∈ Tγ(t)M , gia kˆje i=1 ∂xi γ(t)t ∈ [a, b] kai γ(t) = (γ1(t), γ2(t), ..., γn(t)), opìte γ (t) = n γi(t) ∂ . i=1 ∂xi γ(t)Epomènwc, apì th sqèsh ∇γ X = 0isodÔnama èqoumedξi(t) n dγj (t) dt dt + Γijk (γ(t)) ξk (t) = 0, i = 1, 2, ...n. j,k=1Eˆn dojeÐ h kampÔlh γ(t) kai dojeÐ h arqik  tim  (ξ1(α), ξ2(α), ..., ξn(α)) stoshmeÐo p = γ(α) tìte ta ξi eÐnai monadikˆ orismèna, efìson to sÔsthma twndiaforik¸n exis¸sewn èqei monadik  lÔsh.Epomènwc, h tim  (ξ1(b), ξ2(b), ..., ξn(b)) sto q = γ(b) kai katˆ sunèpeia toX(b) orÐzontai monadikˆ. 'Eqoume dhlad  thn antistoiqÐa Tγ(α)M X(α) → X(b) ∈ Tγ(b)M.Orismìc 3.1.10. Onomˆzoume thn apeikìnish Pγ : Tγ(α)M → Tγ(b)Mparˆllhlh metaforˆ katˆ m koc thc kampÔlhc γ wc proc th Levi-Civitasunoq  ∇ sth pollaplìthta (M, g).H apeikìnish Pγ eÐnai ènac grammikìc isomorfismìc kai epiplèon,gγ(b) (Pγ(u), Pγ(v)) = gγ(α) (u, v) , u, v ∈ Tγ(α)M.

3.1. ARMONIKŸES APEIKONŸISEIS 29 SumbolÐzoume me ∇ kai N ∇ tic sunoqèc Levi-Civita stic pollaplìthtec(M, g) kai (N, h) antÐstoiqa, kai dÐnoume ton akìloujo orismì.Orismìc 3.1.11. Gia kˆje C∞− dianusmatikì pedÐo X thc M onomˆzoumeepag¸menh sunoq  ∇ sthn epag¸menh dèsmh φ−1T N thc efaptìmenhcdèsmhc T N mèsw thc f , thn apeikìnish∇X : N Pφ−◦1σtV (σ(t))Γ(φ−1T N ) → Γ(φ−1T N )V → ∇XVgia kˆje V ∈ Γ(φ−1T N ), me tim ∇X V (x) = N ∇φ∗X V = d ,x ∈ M dt t=0H ∇ ikanopoieÐ tic ex c idiìthtec :(1)∇fX+gY V = f ∇X V + g∇Y V(2) ∇X (V1 + V2) = ∇X V1 + ∇2V2(3)∇X(f V ) = X(f )V + f ∇XVgia kˆje f, g ∈ C∞(M ) gia kˆje X, Y ∈ D1(M ) kai V, V1, V2 ∈ Γ(φ−1T N ).H apeikìnish N Pφ◦σt : Tφ(x)N → Tφ(σ(t))N eÐnai h kaloÔmenh parˆllhlhmetaforˆ katˆ m koc thc C1−kampÔlhc φ ◦ σt wc proc th Levi-Civitasunoq  ∇N sth pollaplìthta (N, h).Sth sunèqeia ja apodeÐxoume endeiktikˆ thn trÐth katˆ seirˆ apì tic idiìthtecthc epag¸menhc sÔnoq c ∇. Ja apodeÐxoume dhlad  ìti∇X(f V ) = X(f )V + f ∇XV

30 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEIS ApìdeixhGia kˆje x ∈ M èqoume ∇X (f V )(x) = d N Pφ−◦1σt(f (σ(t))V (σ(t))) dt t=0= d f (σ(t)) N Pφ−◦1σt(V (σ(t))) + f (x) d N Pφ−◦1σt V (σ(t)) dt dt t=0 = Xx(f )V (x) + f (x)(∇XV )(x)H epag¸menh dèsmh φ−1T N epidèqetai èna eswterikì ginìmeno proerqìmenoapì th metrik  h sth pollaplìthta N pou sumbolÐzetai me hφ(x) kai eÐnai hapeikìnish hφ(x) : Tφ(x)N × Tφ(x)N → RLambˆnontac upìyh thn isometrik  diìthta :hϕ(σ(t))(V1(σ(t)), V2(σ(t)) = hφ(x)(N Pφ−◦1σtV1(σ(t)), N Pφ−◦1σtV2(σ(t)) = hφ(x)(V1(x), V2(x))thc apeikìnishc N Pφ−◦1σt V (σ(t)) : Tϕ(σ(t))N → Tφ(x)Nja deÐxoume ìti h epag¸menh sunoq  ∇ eÐnai sumbat  me th metrik  hφ(x) ìpwcthn orÐsame parapˆnw.Prˆgmati, Xxhφ(x)(V1, V2) = d hφ(σ(t))(V1(σ(t)), V2(σ(t)) dt t=0 = d hφ(x)(N Pφ−◦1σt V1(σ(t)), N Pφ−◦1σt V2(σ(t)) dt t=o= hφ(x) d N Pφ−◦1σtV1(σ(t)), V2(x) +hφ(x) V1(x), d N Pφ−◦1σt V2(σ(t)) dt dt t=0 t=o = hφ(x)(∇Xx V1, V2) + hφ(x)(V1, ∇Xx V2)gia kˆje X ∈ D1(M ), V1, V2 ∈ Γ(φ−1T N ) kai x ∈ M .

3.1. ARMONIKŸES APEIKONŸISEIS 31 O H. Urakawa sto biblÐo tou [25] anafèrei to parakˆtw je¸rhma metabo-l c:Je¸rhma 3.1.1.'Estw φ ∈ C∞(M, N ) kai φt mia tuqaÐa leÐa metabol  thc φ, ìpou − < t <, φ0 = φ kai V (x) = d t=0 φt(x), x ∈ M to C∞− dianusmatikì pedÐo dtmetabol c katˆ m koc thc φ.Tìte d E(φt) = − h(V, τ (φ))vg dt t=0 Mìpou to τ (φ) eÐnai stoiqeÐo tou Γ(φ−1T N ) pou kaleÐtai pedÐo èntashc thcφ (tension field) kai dÐdetai apì th sqèsh m τ (φ) = (∇eidφ(ei) − dφ(∇eiei) i=1Sumpèrasma: h φ ∈ C∞(M, N ) eÐnai armonik  an kai mìno an d E(φt) = 0 ⇔ τ (φ) = 0 dt t=0H exÐswsh τ (φ) = 0 kaleÐtai exÐswsh twn Euler-Lagrange.ParadeÐgmata armonik¸n apeikonÐsewn(1) Stajerèc apeikonÐseic'Estw dÔo sumpageÐc pollaplìthtec Riemann (M, g) kai (N, h) kai q ∈ Nèna stajerì shmeÐo. Kˆje stajer  apeikìnish φ : M → N me tim  φ(x) =q, x ∈ M , eÐnai armonik  kai antistrìfwc.ApìdeixhH φ eÐnai stajer  an kai mìno an to sunarthsoeidèc thc puknìthtacthc enèrgeiac e(φ) = 1 |dφ|2 thc φ eÐnai mhdèn, dhlad  an kai mìno an 2e(φ) = 0. 'Omwc to sunarthsoeidèc thc enèrgeiac thc φ dÐnetai apì th sqèshE(φ) = M e(φ)vg.Epomènwc, e(φ) = 0 ⇔ E(φ) = 0 ⇔ d t=o E(φ) = 0 ⇔ τ (φ) = 0 dt

32 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEISDhlad  h φ eÐnai stajer  an kai mìno an eÐnai armonik .(2) Kleistèc gewdaisiakèc sth sfaÐra S2'Estw mia kleist  diaforÐsimh kampÔlh φ(x) = (φ1(x), φ2(x), φ3(x)), x ∈[0, 2π] ston R3 me perÐodo 2π, dhlad  φ(x + 2π) = φ(x)   φi(x + 2π) =φi(x), i = 1, 2, 3. AnazhtoÔme tic kampÔlec pou apoteloÔn krÐsima shmeÐa thcenèrgeiac 1 2π 3 dφi 2 0 dx E(φ) = ( )2dx i=1kai brÐskontai sth monadiaÐa sfaÐra S2 = (y1, y2, y3); y12 + y22 + y32 = 1Autèc oi kampÔlec eÐnai mègistoi kÔkloi thc sfaÐrac S2 pou strèfontai mforèc kaj¸c to x metabˆletai apì to 0 èwc to 2π.(Analutik  parousÐash ègine sthn parˆgrafo 2.2.)Sth sunèqeia ja sundèsoume thn armonikìthta me tic pollaplìthtec elˆqi-sthc èktashc.

3.1. ARMONIKŸES APEIKONŸISEIS 33 Orismìc 3.1.12. 'Estw duo diaforÐsimec pollaplìthtec (M, g) kai(N, h). Mia leÐa apeikìnish φ : M → N onomˆzetai isometrik  embˆ-ptish (isometric immersion) eˆn :(1) to diaforikì thc φ sto p ∈ M , dhlad  h apeikìnish dφp : TpM → Tφ(p)NeÐnai 1 − 1 gia kˆje x ∈ M,(2) xp, yp M = dφ(xp), dφ(yp) N gia kˆje xp, yp ∈ TpM.Orismìc 3.1.13. 'Otan mia embˆptish φ : M → N eÐnai 1 − 1, tìte hφ lègetai emfÔteush thc M sthn N . Sthn perÐptwsh aut  lème ìti hpollaplìthta M eÐnai emfuteumènh mèsa sth N mèsou thc φ,   ìti h M eÐnaimia emfuteumènh upopollaplìthta thc N .Orismìc 3.1.14. Mia m−diˆstath pollaplìthta M onomˆzetai upopol-laplìthta thc n−diˆstathc pollaplìthtac N ìtan :(1) M ⊂ N (h M eÐnai topologikìc upìqwroc thc N .)(2) H tautotik  apeikìnish i : M → N eÐnai mia emfÔteush thc pollaplìthtacM sthn pollaplìthta N .Eˆn dimN − dimM = 1, tìte h M lègetai uperepifˆneia thc N.'Estw M mia m− diˆstath upopollaplìthta thc n− diˆstathc pollaplìth-tac Riemann N (m < n).An h eÐnai h metrik  Riemann thc N, tìte h epag¸menh metrik  thc MeÐnai h g = i∗h, ìpou(1) h i : M → N eÐnai leÐa(2) h i : M → N eÐnai tautotik  me tim  i(x) = x(3) h i : M → N eÐnai 1-1(4) h di : TpM → Ti(p)N eÐnai èna proc èna kai tautotik .H M efodiasmènh me th g kajistˆ thn i : M → N isometrik  : g(xp, yp) =

34 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEISi∗h(x, y) = h(di(xp), di(yp)) gia kˆje xp, yp ∈ TpM .'Ena diˆnusma ξp ∈ TpN, x ∈ M lègetai kˆjeto sthn M sto shmeÐo p anh(ξp, xp) = 0 gia kˆje xp ∈ TpM . An T M ⊥ eÐnai to sÔnolo ìlwn twn kˆjetwndianusmˆtwn se kˆje shmeÐo p ∈ M tìte TN = TM ⊕ TM⊥'Estw X, Y duo dianusmatikˆ pedÐa thc M kai X, Y oi epektˆseic aut¸n sthnpollaplìthta N , dhlad  ta dianusmatikˆ pedÐa thc N ta opoÐa ìtan perior-isjoÔn sthn pollaplìthta M eÐnai ta dianusmatikˆ pedÐa X, Y antÐstoiqa.'Estw ∇ h sunoq  thc pollaplìthtac Riemann N . Tìte h tim  tou dianus-matikoÔ pedÐou ∇XY sto p ∈ M den exartˆtai apì tic epektˆseic X, Y twnX, Y antÐstoiqa kai to dianusmatikì pedÐo [X, Y ] thc N eÐnai epèktash toudianusmatikoÔ pedÐou [X, Y ] thc M . 'Etsi grˆfoume ∇XY antÐ ∇XY kaianalÔoume autì to dianusmatikì pedÐo thc N se duo sunist¸sec, mia efap-tìmenh thc M , thn ∇XY kai mia kˆjeth sth M , thn B(X, Y ). 'Epomènwc, ∇XY = ∇XY + B(X, Y )O tÔpoc autìc onomˆzetai tÔpoc tou Gauss. H apeikìnish ∇ : TM ×TM → TM (X, Y ) → ∇XYorÐzei mia sunoq  sth M pou lègetai epag¸menh sunoq  sthn upopol-laplìthta M . EpÐshc h apeikìnish B : TM × TM → TM⊥ (X, Y ) → B(X, Y )eÐnai summetrik , digrammik  kai legetai deÔterh jemeli¸dhc morf (second fundamental form) thc upopollaplìthtac M.

3.1. ARMONIKŸES APEIKONŸISEIS 35'Estw ξ èna dianusmatikì pedÐo thc N kˆjeto sth M . To dianusmatikì pedÐo∇Xξ analÔetai se mia efaptìmenh sunist¸sa thn −AξX kai mia kˆjeth thn∇X⊥ ξ opìte isqÔei o akìloujoc tÔpoc tou Weingarten ∇X ξ = −AξX + ∇X⊥ ξH apeikìnish ∇⊥ : T M × T M ⊥ → T M ⊥ (X, ξ) → ∇X⊥ ξèqei tic idiìthtec miac sunoq c kai lègetai kˆjeth sunoq  (normal conne-ction) thc upopollaplìthtac M .H apeikìnish Aξ : T M → T M X → AξXeÐnai grammik  wc proc X kai ξ kai autosuzug c, dhlad , gia kˆje X, Y ∈ T MisqÔei: AξX, Y M = X, AξY M kai kaleÐtai telest c sq matoc (shapeoperator)   deÔterh jemeli¸dhc morf  sth kˆjeth dieÔjunshξ ∈ T M ⊥ (the second fundamental form in the normal direction ξ).Jewr¸ th diaforÐsimh kampÔlh a : I ⊂ R → M sthn pollaplìthta M metim  a(t) ∈ M pou ikanopoieÐ tic sunj kec a(t0) = p kai a (t0) = xp ∈ TpM.To AξX = −(∇xpξ) = −(ξ ◦ a) (t0) metrˆei thn allag  kateÔjunshc tou ξkaj¸c autì dièrqetai apì to p katˆ m koc thc kampÔlhc a. O efaptìmenocq¸roc Ta(t)M thc M sto a(t) strèfetai kaj¸c to kˆjeto diˆnusma ξ strèfe-tai. 'Epomènwc to AξX ekfrˆzei èna mètro strof c tou efaptìmenou q¸routhc M sto p kaj¸c to ξ dièrqetai apì to p katˆ m koc thc a. 'Ara o telest csq matoc mac dÐnei plhroforÐec gia to sq ma thc M .

36 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEIS Prìtash 3.1.1. Gia kˆje dianusmatikì pedÐo ξ thc N kˆjeto sth Mkai gia X, Y ∈ T M èqoume AξX, Y M = B(X, Y ), ξ MApìdeixh X ξ, Y M = ∇X ξ, Y M + ξ, ∇X Y M ⇔ 0 = ∇X ξ + ∇X⊥ ξ, Y M + ξ, ∇X Y + B(X, Y ) M ⇔0 = −AξX, Y M + ∇⊥X ξ, Y M + ξ, ∇X Y )M + ξ, B(X, Y ) M ⇔ AξX, Y M = B(X, Y ), ξ MGia èna monadiaÐo kˆjeto diˆnusma ξ thc M sto p o telest c sq matoc AξeÐnai grammikìc kai autosuzug c opìte mporoÔme na epilèxoume orjokanonik bˆsh e1, e2, ..., em thc M ìpou ta stoiqeÐa thc na apoteloÔn idiodianÔsmatatou Aξ, dhladh Aξ(ei) = λiei, i = 1, 2, ..., m. Ta λi ∈ R kaloÔntai kÔrieckampulìthtec (principal curvatures) thc M wc proc thn kˆjeth dieÔjunshξ kai ta idiodianÔsmata ei kaloÔntai kÔriec dieujÔnseic (principal directions).Oi kÔriec kampulìthtec mac dÐnoun mia perigraf  tou topikoÔ telest  sq -matoc thc emfuteumènhc pollaplìthtac M .Orismìc 3.1.15. 'Estw φ : M m → N n mia isometrik  embˆptish metaxÔduo pollaplot twn M kai N . To dianusmatikì pedÐo mèshc kampulìth-tac H thc φ eÐnai h apeikìnish H : M → TM⊥ x → H(x) ∈ TxM ⊥me tim  m H (x) = 1 B(ei, ei) ⇔ H (x) = 1 traceB m m i=1

3.1. ARMONIKŸES APEIKONŸISEIS 37ìpou ei m mia orjokanonik  bˆsh tou q¸rou TxM . i=1'Estw ξa m mia orjokanonik  bˆsh tou TM⊥ sto x. Tìte a=1 traceB = B(ei, ei), ξa M a,ikai lìgw thc sqèshc AξX, Y M = B(X.Y ), ξ M èqoume traceB = Aξa(ei), ei M = traceAξa a,i a'Ara to dianusmatikì pedÐo mèshc kampulìthtac gÐnetai wc ex c: H (x) = 1 traceAξa ⇔ H (x) = 1 (traceA)ξ m m aOrismìc 3.1.16. H φ kaleÐtai elˆqisth isometrik  embˆptish kaih upopollaplìthta M elaqÐsthc èktashc (minimal submanifold) eˆnH = 0.Apì to tÔpo tou Gauss èqoume ∇XY = ∇XY + B(X, Y )gia kˆje X, Y ∈ T M . Gia X = Y = ei ∈ T M, i = 1, 2, ..., m o tÔpoc gÐnetaiwc ex c: ∇eiei = ∇eiei + B(ei, ei) ⇔ B(ei, ei) = ∇eiei − ∇eieiìpou ∇ eÐnai h sÔndesh sthn epag¸menh dèsmh φ−1T N thc efaptìmenhc dèsmh-c TN.Epomènwc to dianusmatikì pedÐo mèshc kampulìthtac 1 m m H = B(ei, ei) i=1

38 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEISgÐnetai wc ex c: m H = 1 (∇ei ei − ∇ei ei) m i=1'Omwc m τ (φ) = (∇eidφ(ei) − dφ(∇eiei) i=1Lìgw tou tautotikoÔ isomorfismoÔ Tφ(x)N ∼= Nφ(x) èqoume H = 1 τ (φ) mSunep¸c H = 0 an kai mìno an τ (φ) = 0Prìtash 3.1.2. An h φ : M → N eÐnai isometrik  embˆptish tìte h M eÐnaielˆqisthc èktashc an kai mìno an to pedÐo èntashc τ (φ) thc φ mhdenÐzetai.3.2 Diarmonikèc apeikonÐseicOrismìc 3.2.1. MÐa leÐa apeikìnish φ ∈ C∞(M, N ) metaxÔ duo pol-laplot twn Riemann (M, g) kai (N, h) kaleÐtai diarmonik  an kai mìno aneÐnai krÐsimo shmeÐo tou sunarthsoeidoÔc thc enèrgeiac deÔterhc tˆxhc(bienergy) E2(φ) = 1 |τ (φ)|2vg 2 MH φ eÐnai krÐsimo shmeÐo thc E2 an gia opoiad pote metabol  φt ∈ C∞(M, N )(− < t < ), φ0 = φ thc φ isqÔei h sunj kh d E2(φt) = 0 dt t=0Stic ergasÐec [14] , [15] o J. Jiang èdwse gia thn pr¸th metabol  tou sunarth-soeidoÔc E2 ton akìloujo tÔpo d E2(φt) = − h(τ2(φ), V )vg dt t=0 M

3.2. DIARMONIKŸES APEIKONŸISEIS 39Je¸rhma 3.2.1. 'Estw φ ∈ C∞(M, N ) kai φt mia tuqaÐa leÐa metabol thc φ, ìpou (− <t< ), φ0 = φ kai V (x) = d |t=0 φt(x), x ∈ M to C∞− dtdianusmatikì pedÐo metabol c katˆ m koc thc φ.Tìte, d E2(φt) = − h(τ2(φ), V )vg dt t=0 Mìpou τ2(φ) = Jφ(τ (φ)) eÐnai to pedÐo tˆshc deÔterhc tˆxhc kai Jφ eÐnai ènac au-tosuzug c, diaforikìc telest c pou dra sto sÔnolo twn dianusmatik¸n pedÐ-wn metabol c katˆ m koc thc φ, onomˆzetai telest c tou Jacobi(Jacobioperator) kai orÐzetai wc ex c : Jφ = ¯ φ − RφO diaforikìc telest c ¯ φ onomˆzetai Laplasian  (rough Laplacian), drasta dianusmatikˆ pedÐa metabol c katˆ m koc thc φ kai orÐzetai wc ex c : m ¯ φV = − (∇ei ∇ei − ∇∇ei ei )V i=1ìpou V ∈ Γ(φ−1T N ), ei m orjokanonik  bˆsh wc proc th metrik  g sth i=1M kai (m = dimM ).Tèloc o diaforikìc telest c Rφ dra epÐshc sta dianusmatikˆ pedÐa metabol ckatˆ m koc thc φ kai dÐnetai apì th sqèsh m RφV = N R(V, dφ(ei))dφ(ei) i=1ìpou V ∈ Γ(φ−1T N ) kai N R eÐnai to tanustikì pedÐo kampulìthtac thc suno-q c N ∇ sthn pollaplìthta (N, h).Sumpèrasma : H φ ∈ C∞(M, N ) eÐnai diarmonik  an kai mìno an d E(φt) = 0 ⇔ τ2(φ) = 0 ⇔ J(τ (φ)) = 0 dt t=0H exÐswsh τ2(φ) = 0 kaleÐtai diarmonik  exÐswsh.

40 KEFŸALAIO 3. ARMONIKŸES KAI DIARMONIKŸES APEIKONŸISEIS Idiìthtec twn diarmonik¸n apeikonÐsewnPrìtash 3.2.1. H φ eÐnai diarmonik  an kai mìno an to τ (φ) an kei stonpur na tou telest  Jφ, dhlad  an kai mìno an τ (φ) ∈ KerJφ.ApìdeixhKerJφ = V ∈ Γ(φ−1T N ); Jφ(V ) = 0τ (φ) ∈ KerJφ ⇔ Jφ(τ (φ)) = 0 ⇔ τ2(φ) = 0, dhlad  h φ eÐnai diarmonik .Prìtash 3.2.2. Eˆn h φ ∈ C∞(M, N ) eÐnai armonik  tìte eÐnai kai diar-monik .ApìdeixhJèlw na deÐxw ìti h φ eÐnai diarmonik , dhlad  ìti d t=0 E2(φt) = 0 gia kˆje dtleÐa metabol  φt, (− <t< ), φ0 = φ thc φ, ìpou E2(φ) = 1 M |τ φ)|2vg 2to sunarthsoeidèc thc enèrgeiac deÔterhc tˆxhc (bienergy). Apì thn upì-jesh èqw pwc h φ eÐnai diarmonik , dhlad  τ (φ) = 0, ˆra E2(φ) = 0, ˆrad t=0 E2(φt) = 0, ˆra h φ eÐnai diarmonik .dtPrìtash 3.2.3. Mia armonik  apeikìnish elaqistopoieÐ to sunarthsoeidècE2(φ) = 1 M |τ (φ)|2vg . 2ApìdeixhH φ eÐnai armonik , dhlad  τ (φ) = 0. Epomènwc E2(φ) = 0.

Kefˆlaio 4DiarmonikècUpopollaplìthtec4.1 Eisagwg  O B.Y. Chen sthn ergasÐa tou [4] anafèrei thn ex c eikasÐa:EikasÐa tou ChenKˆje diarmonik  upopollaplìthta tou eukleÐdeiou q¸rou En eÐnai armonik ,dhlad  eÐnai elˆqisthc èktashc.Eˆn o q¸roc den eÐnai eukleÐdeioc h eikasÐa tou Chen genikˆ den epalhjeÔetai.'Ena antiparˆdeigma anafèrei o G.Y. Jiang sthn ergasÐa tou [15] kai prìkeitaigia to genikeumèno tìro tou Clifford Sp( √1 ) × S q ( √1 ) ⊂ S m+1 me p + q = m 2 2kai p = q.Orismìc 4.1.1. Tìroc tou Clifford lègetai h eikìna f (S1 × S1) thcapeikìnishc f : S1 × S1 → R4 me tim  f (u, v) = (cosu, sinu, cosv, sinv). Otìroc T 2 = S1 ×S1 diagrˆfetai apì thn peristrof  tou kÔklou S1 me exÐswsh(x1 − a)2 + x23 = r2, r < a sto epÐpedo x10x3 gÔrw apì ton ˆxona x3. 41

42 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESSth sunèqeia oi G.Y. Jiang kai C. Oniciuc stic ergasÐec touc [7] kai [21]apèdeixan tic parakˆtw protˆseic:Prìtash 4.1.1. Eˆn M eÐnai mia sumpag c upopollaplìthta Riemannmiac pollaplìthtac N me kampulìthta tom c RiemN ≤ 0, tìte h φ : M → NeÐnai diarmonik  an kai mìno an eÐnai armonik , dhlad  elˆqisthc èktashc.Prìtash 4.1.2. Eˆn h φ : M → N eÐnai isometrik  embˆptish me |τ (φ)|stajerì kai h kampulìthta tom c thc pollaplìthtac N eÐnai RiemN ≤ 0,tìte h φ eÐnai diarmonik  an kai mìno an eÐnai armonik , dhlad  elˆqisthc èk-tashc.Oi parapˆnw protˆseic mac odhgoÔn sth genikeumènh eikasÐa tou Chen.Genikeumènh eikasÐa tou ChenOi mìnec diarmonikèc upopollaplìthtec miac pollaplìthtac N me kampulìth-ta tom c RiemN ≤ 0 eÐnai oi elˆqisthc èktashc, dhlad  oi armonikèc.Stìqoc mac eÐnai na anazht soume tic diarmonikèc kampÔlec kai tic diarmonikècepifˆneiec thc sfaÐrac S3. Oi kentrikèc mac anaforèc eÐnai oi ergasÐec [3],[4]twn R. Caddeo, S. Montaldo, C. Oniciuc kai h ergasÐa [11] twn J. Eells, L.Lemaire.4.2 Diarmonikèc kampÔlec sthn S3 Arqikˆ ja anazht soume tic diarmonikèc kampÔlec miac trisdiˆstathc pol-laplìthtac M .Jewr¸ (M 3, g) mia tridiˆstath pollaplìthta Riemann me stajer  kampulìth-ta tom c K kai mia diaforÐsimh kampÔlh γ : I ⊂ R → (M 3, g) parametrikopoih-mènh wc proc to m koc tìxou thc. 'Estw T, N, B èna orjokanonikì pedÐo

4.2. DIARMONIKŸES KAMPŸULES STHN S3 43plaisÐwn efaptìmeno sthn M 3 katˆ m koc thc γ, ìpou :• T = γ eÐnai to monadiaÐo dianusmatikì pedÐo efaptìmeno sth γ• N to monadiaÐo kˆjeto dianusmatikì pedÐo sth dieÔjunsh tou ∇T T• B to dianusmatikì pedÐo kˆjeto sta T kai N epÐ thc γ epilegmèno ¸ste h T, N, B na apoteleÐ jetikˆ prosanatolismènh bˆsh.KaloÔme to T, N, B paidÐo plaisÐwn tou Frenet epÐ thc γ. Eˆn hkampÔlh γ eÐnai monadiaÐac taqÔthtac, dhlad  |γ (t)| = 1,tìte kg = |∇T T | =|τ (γ)|. H kg onomˆzetai gewdaisiak  kampulìthta kai ekfrˆzei thntaqÔthta metabol c thc dieÔjunshc tou efaptomenikoÔ pedÐou sth kampÔlhanˆ monˆda m kouc tìxou. H sunˆrthsh τg pou perilambˆnetai stouc parakˆtwtÔpouc onomˆzetai gewdaisiak  strèyh kai ekfrˆzei thn taqÔthta metabo-l c thc dieÔjunshc tou dianusmatikoÔ pedÐou B. IsqÔoun oi parakˆtw exis¸-seic tou Frenet : ∇T T = kgN ∇T N = −kgT + τgB ∇T B = −τgNH kampÔlh γ eÐnai diarmonik  an kai mìno an τ2(γ) = 0 ⇔ ∇T3 T − R(T, kgN )T = 0 ⇔ (−3kgkg)T + (kg − kg3 − kgτg2 + kgK)N + (2kgτg + kgτg)B = 0ìpou K = K(T, N ) = g(R(T, N )N, T ) = g(T, T )g(N, N ) − g(T, N )2 g(R(T, N )N, T ) = R(T, N, N, T ) = −R(T, N, T, N )

44 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESeÐnai h kampulìthta tom c thc M 3 wc proc to zeÔgoc T, N , h opoÐa èqoumeupojèsei ìti eÐnai stajer .Epomènwc h kampÔlh γ eÐnai diarmonik  an kai mìno an τ2(γ) = 0.IsodÔnama(1) kgkg = 0(2) kg − kg3 − kgτg2 + kgK = 0(3) 2kgτg + kgτg = 0AnazhtoÔme diarmonikèc mh gewdaisiakèc kampÔlec, dhlad  diarmonikèckampÔlec me gewdaisiak  kampulìthta kg = 0. 'Eqontac wc upìjesh ìtikg = 0 èqw ta ex c :Apì thn exÐswsh (1) sunepˆgetai ìti kg = c1, ìpou h c1 eÐnai mia mh mhdenik pragmatik  stajerˆ.Apì thn (2) sunepˆgetai ìti kg2 + τg2 = K.Apì thn (3) sunepˆgetai ìti τg = c2, ìpou h c2 eÐnai mia mh mhdenik  prag-matik  stajerˆ.Epomènwc katal goume sth parakˆtw prìtash.Prìtash 4.2.1 Oi diarmonikèc mh gewdaisiakèc kampÔlec thc pollaplìth-tac M eÐnai ekeÐnec pou èqoun stajer  gewdaisiak  kampulìthta kai strèyhkai pou ikanopoioÔn th sunj kh kg2 + τg2 = K.Sthn perÐptwsh pou h kampulìthta tom c eÐnai mikrìterh   Ðsh tou mhdenìc(K ≤ 0) h sunj kh kg2 + τg2 = K den mporeÐ na isqÔei parˆ mìno ìtankg = τg = 0. Tìte h γ eÐnai gewdaisiak , dhlad  elˆqisthc èktashc (mini-mal). Epomènwc epibebai¸netai h genikeumènh eikasÐa tou Chen .

4.2. DIARMONIKŸES KAMPŸULES STHN S3 45Sth sunèqeia ja anazht soume diarmonikèc mh gewdaisiakèc kampÔlec sthsfaÐra S3. Oi kentrik  mac anaforˆ eÐnai h ergasÐa [6].Prìtash 4.2.2. 'Estw γ : I → S3 ⊂ R4 mia mh gewdaisiak  diarmonik kampÔlh parametrikopoihmènh wc proc to m koc tìxou thc. Tìte isqÔei hexÐswsh γIV + 2γ + (1 − kg2)γ = 0ApìdeixhPaÐrnoume th sunalloÐwth parˆgwgo wc proc T thc exÐswshc ∇T N = −kgT + τgBtou Frenet kai èqoume ∇T2 N = ∇T (∇T N ) = −kg∇T T + τg∇T BExaitÐac kai twn upoloÐpwn exis¸sewn tou Frenet, h parapˆnw sqèsh gÐnetai ∇T2 N = −kg(kgN ) + τg(−τgN ) = −(kg2 + τg2)N = −KNEpeid  h kampulìthta tom c thc sfaÐrac eÐnai K = 1, h parapˆnw sqèshgÐnetai ∇2T N = −N ⇔ ∇2T + N = 0H exÐswsh tou Gauss gia tuqaÐo dianusmatikì pedÐo X thc S3 katˆ m koc thcγ èqei wc ex c : ∇T X = X + T, X γEfarmìzontac thn parapˆnw sqèsh gia to dianusmatikì pedÐo N èqoume ∇T N = N + T, N γ

46 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESEpomènwc, ∇T2 N = ∇T (∇T N ) = ∇T (N + T, N γ) = ∇T N = N + T, N γ = N + T, ∇T N γ = N + T, −kgT + τgB γ = N + (−kg T, T + τg T, B )γ = N − kgγ ⇔ ∇T2 N = N − kgγ'Omwc N = Tìpou kg T = ∇T T = ∇T γ = γ + T, ∇T γ γ ⇔ T = γ + γ ,γ γEpeid  h kampÔlh γ èqei monadiaÐa taqÔthta, dhlad  |γ | = 1 ⇔ γ , γ = 1èqoume telikˆ ìti T =γ +γOpìte N = γ +γ kgParagwgÐzoume thn parapˆnw sqèsh kai paÐrnoume N =γ +γ kgParagwgÐzoume xanˆ N = γIV + γApì tic sqèseic kg ∇T2 N + N = 0 ∇2T N = N − kgγ