Διαρμονικές υποπολλαπλότητες της τρισδιάστατης σφαίρας

4.2. DIARMONIKŸES KAMPŸULES STHN S3 47 N = γIV + γ kgèqoume telikˆ γIV + 2γ + (1 − kg2)γ = 0.'Ara oi mh gewdaisiakèc diarmonikèc kampÔlec thc S3 eÐnai lÔseic thc diafori-k c exÐswshc γIV + 2γ + (1 − kg)γ = 0.ApodeÐxame prohgoumènwc ìti isqÔoun oi sunj kec kg = στ αθ. = 0 τg = στ αθ. = 0 kg2 + τg2 = Kgia tic mh gewdaisiakèc diarmonikèc kampÔlec γ : I → (M 3, g).Ean h pollaplìthta M eÐnai h sfaÐra S3 tìte h kampulìthta tom c isoÔtaime th monˆda, dhlad  K = 1, opìte h trÐth sunj kh gÐnetai kg2 + τg2 = 1Apì th teleutaÐa sqèsh sunepˆgetai ìti kg ≤ 1.Je¸rhma 4.2.1. 'Estw mia mh gewdaisiak  diarmonik  kampÔlh γ : I → S3parametrikopoihmènh wc proc to m koc tìxou thc. IsqÔoun ta ex c :(1) Eˆn kg = 1, tìte h γ eÐnai kÔkloc aktÐnac √1 2(2) Eˆn 0 < kg < 1, tìte h γ eÐnai gewdaisiak  tou tìrou tou CliffordS1( √1 ) × S1( √1 ) 2 2ApìdeixhPr¸th perÐptwsh :H sqèsh γIV + 2g + (1 − kg2)γ = 0 ìtan to kg = 1 gÐnetai γIV + 2γ = 0

48 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESH exÐswsh eÐnai grammik , tètarthc tˆxhc, omogen c, me stajeroÔc sunte-lestèc. To qarakthristikì polu¸numo eÐnai f (ρ) = ρ4 + 2ρ2 kai oi rÐzec tou √√eÐnai oi ρ = 0 (dipl  pragmatik ), ρ1 = i 2, ρ2 = −i 2. 'Ara h genik  lÔshthc diaforik c exÐswshc eÐnai h γ(t) = e0t(c1 √ + c2 √ + c3t + c4 ⇔ cos( 2t) sin( 2t) √√ γ(t) = c1 cos( 2t) + c2 sin( 2t) + c3t + c4GnwrÐzontac ìti |γ|2 = 1 kai |γ |2 = 1 kai me efarmog  twn sqèsewn touFrenet, èqoume c3 = 0, |c1|2 = |c2|2 = |c4|2 = 1 . 2Epomènwc, h genik  lÔsh eÐnai h γ(t) = √1 √ √1 √ 0, √12 cos( 2t), sin( 2t), 22  isodÔnama γ(t) = √1 √ √1 √ d1, d2 2 cos( 2t), 2 sin( 2t),ìpou d21 + d22 = 1 2'Ara h kampÔlh γ eÐnai kÔkloc aktÐnac ρ = √1 2DeÔterh perÐptwsh :LÔnoume th diaforik  exÐswsh γIV + 2γ + (1 − kg2)γ = 0 ìtan to 0 < kg < 1Aut  eÐnai grammik , tètarthc tax c, omogen c, me stajeroÔc suntelestèc.To qarakthristikì polu¸numo eÐnai f (ρ) = ρ4 + 2ρ2 + (1 − kg2). Oi rÐzectou eÐnai oi ρ1 = i 1 + kg, ρ2 = −i 1 + kg kai oi ρ3 = i 1 − kg, ρ4 =−i 1 − kg.Epomènwc, h genik  lÔsh thc diaforik c exÐswshc eÐnai hγ(t) = e0t c1 cos( 1 + kg)t+c2 sin( 1 + kg)t +e0t c3 cos( 1 − kg)t+c4 sin( 1 − kg)t

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 49  isodÔnamaγ(t) = c1 cos( 1 + kg)t+c2 sin( 1 + kg)t+c3 cos( 1 − kg)t+c4 sin( 1 − kg)tGnwrÐzontac ìti |γ|2 = |γ |2 = 1 kai me efarmog  twn tÔpwn tou Frenet, èqwìti |ci|2 = 1 gia kˆje i = 1, 2, 3, 4. 2Epomènwc, h lÔsh eÐnai h γ(t) = √1 cos(At), √1 sin(At), √1 cos(Bt), √1 sin(Bt) 2 22 2ìpou A = 1 + kg kai B = 1 − kgH parapˆnw kampÔlh γ eÐnai gewdaisiak  tou tìrou tou CliffordS 1( √1 ) × S1( √1 ) ⊂ S3 ⊂ R4 2 24.3 Diarmonikèc epifˆneiec sthn S3 Prin anaferjoÔme stic diarmonikèc epifˆneiec thc sfaÐrac S3 ja parousiˆ-soume kˆpoia genikˆ apotelèsmata pou aforoÔn upopollaplìthtec sth sfaÐraSn.'Estw (M, , ) mia upopollaplìthta diˆstashc m thc Sn kai i : M → Sn hapeikìnish ègklishc. SumbolÐzoume me: • B th deÔterh jemeli¸dh morf  thc M • A to telest  sq matoc thc M • H to dianusmatikì pedÐo mèshc kampulìthtac thc M • ∇⊥ thn orjog¸nia sÔndesh, dhlad  th sÔndesh sthn orjog¸nia dèsmh T M ⊥ thc M

50 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTES • ⊥ th Laplasian  sthn orjog¸nia dèsmh T M ⊥ thc MJe¸rhma 4.3.1. H apeikìnish ègklishc i : M → Sn eÐnai diarmonik  an kaimìno an(1) − ⊥H − traceB(−, AH−) + mH = 0(2) 2traceA∇⊥(−)H (−) + m grad(|H |2 ) = 0 2ApìdeixhGnwrÐzoume ìti traceRSn(di, τ (i))di = trace ∇dSin, ∇τS(ni) di − ∇[Sdni,τ(i)]di = −mτ (i)H apeikìnish i eÐnai diarmonik  an kai mìno an τ2(i) = 0. 'Omwcτ2(i) = J(τ (i)) = − (τ (i)) − traceRSn(di, τ (i))di = trace∇dτ (i) + mτ (i)'Ara h i eÐnai diarmonik  an kai mìno an τ2(i) = trace∇dτ (i) + mτ (i) = 0.Gia mia isometrik  emfÔteush i èqoumeH = 1 τ (i) ⇒ 1 dτ (i) = dH ⇒ 1 ∇dτ (i) = ∇dH ⇒ 1 trace∇dτ (i) = trace∇dH ⇒ m m m m trace∇dτ (i) = mtrace∇dHApì tic duo teleutaÐec sqèseic èqoumeτ2(i) = mtrace∇dH+mτ (i) = mtrace∇dH+m.mH = m trace∇dH+mH = 0'An xi m eÐnai èna sÔsthma orjog¸niwn suntetagmènwn sth perioq  tou i=1tuqaÐou shmeÐou p ∈ M kai ei = ∂ m èna orjog¸nio sÔsthma suntetag- ∂xi i=1mènwn ston efaptìmeno q¸ro TpM thc M tìte m trace∇dH = ∇Sein ∇Sein H i=1

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 51Apì ton tÔpo tou Weingarten èqoume ìti trace∇dH = m ∇eSin −AH (ei) + ∇⊥ei H i=1ìpou ∇e⊥i : T M ⊥ → T M ⊥ H → ∇⊥eiH ∈ T M ⊥kai −AH(ei) ∈ T MApì to tÔpo tou Gauss ∇SeinAH (ei) = ∇eiAH (ei) + B(ei, AH (ei))ìpou ∇eSin AH (ei) ∈ T Sn ∇eiAH (ei) ∈ T M B(ei, AH (ei)) ∈ T M ⊥kai apì ton tÔpo tou Weingarten ∇Sein (∇e⊥i H) = −A∇⊥ei H (ei) + ∇⊥ei (∇⊥ei H)ìpou ∇Sein (∇e⊥i H) ∈ T SnEpomènwc, −A∇e⊥i H (ei) ∈ T M ∇e⊥i (∇e⊥i H) ∈ T M ⊥ m trace∇dH = ∇Sein −AH (ei) + ∇e⊥i H = i=1

52 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTES m ∇e⊥i(∇e⊥iH) − A∇e⊥iH (ei) − ∇eiAH (ei) − B(ei, AH (ei)) i=1'Omwc m m ⊥H = − ∇e⊥i (∇⊥ei H) − ∇⊥ ei H =− ∇⊥ei ∇e⊥i H i=1 ∇⊥ei i=1'Ara, mtrace∇dH = − ⊥H − traceB(−, AH−) − A∇e⊥i H (ei) + ∇ei AH (ei) i=1Sth sunèqeia ja apodeÐxoume mia prìtash pou mac eÐnai qr simh sthn olokl -rwsh thc apìdeixhc tou jewr matoc.JewroÔme ton mousikì isomorfismì : T M → T ∗M mèsw tou opoÐoutautÐzontai ta dianusmatikˆ pedÐa me tic 1-morfèc. H apeikìnish aut  orÐze-tai wc ex c: 'Estw V ∈ T M kai V ∗ oi 1-morfèc sto q¸ro T ∗M ètsi ¸steV ∗(X) = V, X gia kˆje X ∈ T M .Prìtash'Estw V ∈ T M kai V ∗ oi 1-morfèc sto q¸ro T ∗M ètsi ¸ste V ∗(X) = V, Xgia kˆje X ∈ T M .Tìte h apeikìnish : T M → T ∗M eÐnai ènac isomorfismìc.ApìdeixhGia na deÐxoume ìti h apeikìnish eÐnai isomorfismìc prèpei na deÐxoume ìti aut eÐnai 1-1 kai epÐ.Gia na deÐxoume ìti eÐnai 1-1 arkeÐ na deÐxoume ìti an V ∗(X) = W ∗(X) giakˆje X ∈ T M tìte V = W . 'H isodÔnama an V, X = W, X gia kˆjeX ∈ T M tìte V = W . Prˆgmati, èstw U = V − W . ArkeÐ na deÐxw ìti eˆn Up, Xp = 0 gia kˆje p ∈ M kai X ∈ T M tìte U = 0. Autì ìmwc isqÔeiapì ton orismì tou metrikoÔ tanust  Riemann.

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 53Gia na deÐxoume ìti eÐnai epÐ, prèpei na deÐxoume ìti dojeÐshc miac 1-morf cθ ∈ T ∗M upˆrqei monadikì dianusmatikì pedÐo V ∈ T M tètoio ¸ste θ(X) = V, X gia kˆje X ∈ T M .Prˆgmati, jewroÔme èna topikì sÔsthma suntetagmènwn {xi}mi=1 kai mia to-pik  orjokanonik  bˆsh {∂i}mi=1 tou q¸rou T M , kai {dxi}im=1 thn antÐstoiqhorjokanonik  bˆsh tou duikoÔ q¸rou T ∗M .Tìte, h 1-morf  θ kai to dianusmatikì pedÐo V grˆfontai wc ex c :θ = i θidxi kai V = i,j gijθi∂j.Tìte, èqoume V, ∂k M = i,j gij θi∂j , ∂k = i,j gijθi ∂j, ∂k M = i,j θigij gjk = i θiδik =θk = θ(∂k). MEpomènwc, gia kˆje X = i Xi∂i ìpou X ∈ T M èqoume i Xi∂i = V, X M = V, i Xi∂i M = i Xi V, ∂i M = i Xiθ(∂i) = θθ(X ).Sth sunèqeia ja deÐxoume ìti to dianusmatikì pedÐo V ∈ T M tètoio ¸steθ(X) = V, X gia kˆje X ∈ T M eÐnai monadikì.Prˆgmati, jewroÔme èna ˆllo dianusmatikì pedÐo W ∈ T M tètoio ¸steθ(X) = W, X gia kˆje X ∈ T M . Tìte, èqoume V, X = W, X giakˆje X ∈ T M . Autì shmaÐnei ìti V = W .Epistrèfoume sthn apìdeixh tou jewr matoc 4.3.1. kai èqoume mtrace∇dH = − ⊥H − traceB(−, AH−) − A∇e⊥i H (ei) + ∇ei AH (ei) i=1'Omwc m m m 2 A∇e⊥i H (ei) + ∇ei AH (ei) =2 A∇e⊥i H (ei) + (d|H |2) = i=1 i=1 2traceA∇(⊥−)H (−) + m grad(|H |)2) 2

54 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESìpou d(|H|)2 → (d(|H|)) ≡ grad(|H|2) ∈ T MAnakefalai¸noume lègontac ìti h apeikìnish i eÐnai diarmonik  an kai mìno anτ2(i) = 0. Jètoume th tim  tou trace∇dH sth sqèsh τ2(i) = m trace∇dH +mH = 0 kai èqoume−∇⊥H − traceB(−, AH −) + mH = 2traceA∇⊥(−)H (−) + m grad(|H |2) 2Efìson to aristerì mèloc thc sqèshc an kei sto kˆjeto q¸ro thc M kai todexiì mèloc thc sqèshc ston efaptìmeno q¸ro thc M , èqoume −∇⊥H − traceB(−, AH−) + mH = 0 2traceA∇(⊥−)H (−) + m grad(|H |2) = 0 2kai to je¸rhma èqei apodeiqjeÐ.Pìrisma 4.3.1. 'Estw M mia upopollaplìthta thc Sn me ∇⊥H = 0.Tìte h apeikìnish ègklishc i : M → Sn eÐnai diarmonik  an kai mìno anmH = traceB(−, AH−).ApìdeixhApì thn upìjesh gnwrÐzw ìti ∇⊥H = 0, dhlad  h sunˆrthsh ègklishci : M → Sn èqei parˆllhlo dianusmatikì pedÐo mèshc kampulìthtac kai katˆsunèpeia to |H| eÐnai stajerì katˆ m koc thc M . Sto prohgoÔmeno je¸rhmaapodeÐxame pwc h i eÐnai diarmonik  an kai mìno an isqÔoun ta ex c:(1) −∇⊥H − traceB(−, AH−) + mH = 0(2) 2traceA∇⊥(−)H (−) + m grad(|H |2 ) = 0 2Epeid  ∇⊥H = 0 h pr¸th sqèsh gÐnetai traceB(−, AH−) = mH

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 55kai to pìrisma apedeÐqjhke.Prìtash 4.3.1. 'Estw M mia uperepifˆneia thc Sn. Tìte h apeikìnishègklishc i : M → Sn eÐnai diarmonik  an kai mìno an(1) ∇⊥H = (m − |B|2)H(2) 2traceA∇⊥(−)H (−) + m grad(|H |2 ) =0 2Apìdeixh'Eqoume traceB(−, AH−) = 1 (traceA)η|B|2 = |B|2H mìpou H = 1 (traceA)ηa = 1 (traceA)η m mkai h ηa m eÐnai mia orjokanonik  bˆsh tou T M⊥. a=1Sto Je¸rhma 4.3.1. apodeÐxame ìti h i eÐnai diarmonik  an kai mìno an(1) −∇⊥H − traceB(−, AH−) + mH = 0(2) 2traceA∇(−)H(−) + m grad(|H |2 ) = 0 2H pr¸th sqèsh gÐnetai −∇⊥H − |B|2H + mH = 0 ⇔ ∇⊥H = (m − |B|2)HEpomènwc, h i eÐnai diarmonik  an kai mìno an(1) ∇⊥H = (m − |B|2)H(2) 2traceA∇(−)H (−) + m grad(|H |2 ) =0 2

56 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESPrìtash 4.3.2. 'Estw M = Sm(a) × b = p = (x1, ..., xm+1, b); x12 + ... + xm2 +1 = a2, a2 + b2 = 1, 0 < a < 1 miaparˆllhlh upersfaÐra thc Sm+1.H M eÐnai diarmonik  upopollaplìthta thc S m+1 an kai mìno an a = √1 kai 2b = + √1   b = − √1 2 2ApìdeixhJewroÔme to sÔnolo Γ(T M ) = X = (X1, ...Xm, 0) ∈ Rm+2; x1X1 + ... +xm+1Xm+1 = 0 twn tom¸n (section) thc efaptìmenhc dèsmhc thc M kaiξ = (x1, ..., xm+1, − a2 ) èna dianusmatikì pedÐo thc M. b'Eqoume ξ, X = x1 X 1 + ... + xm+1X m+1 + (− a2 )0 = 0 bkai ξ, p = (x1)2 + ... + (xm+1)2 − a2 b = a2 − a2 = 0 b ξ, ξ = (x1)2 + ... + (xm+1)2 + (− a2 )2 = a2 + a4 = c2 b b2ìpou c > 0. Apì tic duo pr¸tec sqèseic sumperaÐnoume ìti to ξ eÐnai tom (section) thc orjog¸niac dèsmhc thc M , dhlad  ξ ∈ Γ(T M ⊥).Prìkeitai dhlad  gia mia C∞−apeikìnish ξ : M → TM⊥p → ξ(p) tètoia ¸ste π ◦ ξ = id, ìpou π ◦ ξ : M → M me tim  (π ◦ ξ)(p) = pgia kˆje p ∈ M kai π : T M ⊥ → M h dianusmatik  dèsmh pˆnw sth M .Jètoume η = 1 ξ kai sumbolÐzoume me −Aη X to efaptìmeno dianusmatikì pedÐo cthc Sm+1, dhlad  −AηX = (∇XSm+1 η)

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 57ìpou h apeikìnish Aη : C(T M ) → C(T M ) X → AηXeÐnai digrammik , autosuzug c kai kaleÐtai telest c sq matoc   deÔterh jemeli¸dhcmorf  sth kˆjeth dieÔjunsh ξ.Apì to tÔpo tou Weingarten èqoume ìti ∇Sm+1 η = ∇⊥X η − Aη X Xìpou to dianusmatikì pedÐo ∇X⊥ η orÐzei mia sunoq  pou eÐnai sumbat  stosÔnolo twn tom¸n thc orjog¸niac dèsmhc T M ⊥.Jètw η = 1 ξ kai h sqèsh grˆfetai c ∇Sm+1 1 ξ = ∇⊥X 1 ξ − Aη X ⇔ c c X 1 ∇Sm+1 ξ = 1 (∇⊥X ξ − Aξ X ) = 1 (∇XRm+1 ξ − AξX) c c c X = 1 (∇RXm+1 ξ + ξ, X p) = 1 ∇(X 1,...,X m+1 ,0) (x1, ..., xm+1, − a2 ) = 1 X c c b cEpomènwc, ∇X⊥ η − Aη X = 1 X ⇔ ∇⊥(−)η − Aη(−) = 1 (−) c cApì th teleutaÐa sqèsh, èqoume ∇⊥η = 0 kai Aη = 1 I kai to diˆnusma mèshc ckampulìthtac gÐnetai H = 1 (traceA)η = − 1 η m c AH = A− 1 η = − 1 Aη = − 1 (− 1 )I = 1 I c c c c c2ApodeÐxame sto pìrisma 4.3.1 ìti h apeikìnish ègklishc miac upopollaplìth-tac M thc Sn me ∇⊥H = 0 eÐnai diarmonik  an kai mìno an mH = traceB(−, AH−) = |B|2H

58 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTESMe efarmog  tou porÐsmatoc autoÔ, h teleutaÐa sqèsh mac dÐnei c2 = 1 ⇔ a2 + a4 = 1 b2ìpou a2 +b2 = 1 kai 0 < a < 1 . Oi lÔseic tou sust matoc twn duo exis¸sewneÐnai (a = √1 , b = + √1 ) kai (a = √1 , b = − √1 ). 2 2 2 2'Ara h upopollaplìthta M eÐnai diarmonik  thc Sm+1 an kai mìno an a = √1 kai b = √1   b = − √1 . 2 2 2EÐdame ìti oi mh armonikèc diarmonikèc kampÔlec thc S3 èqoun stajer  gew-daisiak  kampulìthta. Oi B.Y. Chen kai S. Ishikawa sthn ergasÐa touc [5],apèdeixan ìti to mètro tou dianÔsmatoc mèshc kampulìthtac twn mh armonik¸ndiarmonik¸n epifanei¸n thc S3 eÐnai stajerì.DiatÔpwsan kai apèdeixan to parakˆtw je¸rhma :Je¸rhma 4.3.2. 'Estw M mia epifˆneia thc S3. H M eÐnai mh armonik  di-armonik  upopollaplìthta an kai mìno an to |H| eÐnai stajerì kai to |B|2 = 2.Prokeimènou na taxinom soume tic diarmonikèc epifˆneiec thc S3 parajètoumeto apotèlesma thc ergasÐac [13] tou Z.H. Hou.Je¸rhma 4.3.3. 'Estw M mia uperepifˆneia thc S3 me stajer  mèsh kam-pulìthta.(1) An |B|2 = 2, tìte h M eÐnai eÐte topikˆ isometrik  me èna tm ma thc uper-sfaÐrac S 2( √1 ) sthn S3 eÐte eÐnai topikˆ isometrik  me èna tm ma tou tìrou 2S1( √1 ) × S1( √1 ) 2 2(2) An h M eÐnai sumpag c kai prosanatolismènh kai |B|2 = 2, tìte h MeÐnai eÐte isometrik  mia mikr  upersfaÐra S 2( √1 ) eÐte isometrik  me ton tìro 2S1( √1 ) × S1( √1 ). 2 2

4.3. DIARMONIKŸES EPIFŸANEIES STHN S3 59 An lˆboume upìyh mac ìti o tìroc tou Clifford S 1 ( √1 ) × S 1( √1 ) eÐnai ar- 2 2monik  epifˆneia thc S3 tìte sundiˆzontac to je¸rhma 4.3.2. kai to je¸rhma4.3.3., èqoume :Je¸rhma 4.3.4. 'Estw M mia mh armonik  diarmonik  epifˆneia thc S3.(1) An h M eÐnai mh sumpag c, tìte aut  eÐnai topikˆ isometrik  me èna tm mathc sfaÐrac S 2( √1 ) sthn S3. 2(2) An h M eÐnai sumpag c kai prosanatolismènh, tìte eÐnai isometrik  me thsfaÐra S 2( √1 ) aktÐnac √1 . 2 2Anakefalai¸nontac, ta apotelèsmata pou katal goume eÐnai ta ex c :'Estw M m mia diarmonik  upopollaplìthta thc tridiˆstathc sfaÐrac S3.Tìte,(1) An m = 1, dhlad  h M eÐnai mia kampÔlh thc S3, tìte h M eÐnai isometrik eÐte(i) me ènan kÔklo aktÐnac √1 , ìtan h gewdaisiak  kampulìthta eÐnai Ðsh me th 2monˆda, dhlad  kg = 1, eÐte(ii) me mia gewdaisiak  kampÔlh tou tìrou tou CliffordS 1( √1 ) × S1( √1 ), ìtan h gewdaisiak  kampulìthta ikanopoieÐ th sqèsh 2 20 < kg < 1.(2) An m = 2, dhlad  h M eÐnai mia uperepifˆneia tìte:(i) an h M eÐnai mh sumpag c tìte aut  eÐnai topikˆ isometrik  me èna tm mathc sfaÐrac S 2( √1 ) sthn S3, kai 2(ii) an h M eÐnai sumpag c kai prosanatolismènh tìte eÐnai isometrik  me thsfaÐra S 2( √1 ) aktÐnac √1 . 2 2

60 KEFŸALAIO 4. DIARMONIKŸES UPOPOLLAPLŸOTHTES

BibliografÐa [1] William M. Boothby: An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, Inc., 1986. [2] Manfredo Do Carmo: Riemannian Geometry, theory and applications, 1992. [3] B. Y. Chen: Total MeanCurvature and Submanifolds of Finite Type, Series in Pure Mathematics -Volume 1, World Scientific, 1984. [4] B. Y. Chen: Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), 169-188. [5] B. Y. Chen, S. Ishikawa: Biharmonic pseudo-Riemannian submanifplds in pseudo-Euclidean spaces, Kyushu J. Math. 52, 1998, pp. 167-185. [6] R. Caddeo, S. Montaldo and C. Oniciuc: Biharmonic submanifolds of S3, Intern.J. Math., 12 (2001), 867-876. [7] R. Caddeo, S. Montaldo and Paola piu: On Biharmonic maps, Amer. Math. Soc. 288 (2001), 286-290. 61

62 BIBLIOGRAFŸIA [8] R. Caddeo, S. Montaldo and C. Oniciuc: Biharmonic submanifolds in spheres, Israel. J. Math., 130 (2002), 109-123. [9] Krishan L. Duggal and Aurel Bejancu: Lightlike Submanifolds of Semie-Riemannian Manifolds and Applications, Kluwer Academic Pub- lishers, 1996. [10] M. Dajczer: Submanifolds and Isometric Immersions, Mathematics Lecture series 13, Publish or Perish, 1990. [11] J. Eells, L. Lemaire: A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. [12] J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. [13] Z.H. Hou: Hypersurfaces in a sphere with constant mean curvature, Proc. Amer. Math. Soc. 125 (1997), 1193-1196. [14] G. Y. Jiang: 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A 7no 2 (1986), 130-144. [15] G. Y. Jiang: 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7no 4 (1986), 389-402. [16] Serge Lang: Differential and Riemannian manifolds, Graduate Texts in Mathematics 160, Springer-Verlag, 1995.

BIBLIOGRAFŸIA 63[17] John M. Lee: Riemannian Geometry: An introduction to curvature, Graduate Texts in Mathematics, Springer-Verlag, 1997.[18] John M. Lee: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer-Verlag, 2003.[19] E. Loubeau and Oniciuc: The index of biharmonic maps in spheres, Compositio Math. 141 (2005) 729-745.[20] B. O’Neil: Semie-Riemannian Geometry with Applications to Relativ- ity, Academic Press, 1983.[21] C. Oniciuc: Biharmonic maps between Riemannian manifolds, An. Stiint. Umv. Al. I. Cuza Iasi Mat. (N.S) 48 (2002) 237-248.[22] Peter Petersen: Riemannian Geometry, Graduate Texts in Mathemat- ics, Springer-Verlag, 1998.[23] BasÐleioc PapantwnÐou: DiaforÐsimec Pollaplìthtec, Pan/mio Pa- tr¸n, 1993.[24] BasÐleioc PapantwnÐou: Tanustik  Anˆlush kai GewmetrÐa Riemann, Tìmoi I kai II, Pan/mio Patr¸n, 1995.[25] H. Urakawa, Calculus of Variations and Harmonic Maps, Amer. Math. Soc., Providance, 1993.[26] T.J. Willmore: Riemannian Geometry, Oxford Science Publications, 1993.