The contact Whitney sphere

No e di Ma ema ica 20, n. 2, 2000/2001, 125–133. The con ac Whi ney sphe e Da id E. Blai Depa men o Ma hema ics, Michigan S a e Uni e si y, Eas Lansing, Michigan 48824-1027, USA [email p o ec ed] Al onso Ca iazo∗ Depa amen o de Geome ´ıa y Topolog´ıa, Facul ad de Ma em´a icas, Uni e sidad de Se illa, Apdo. Co eos 1160, 41080–Se illa, Spain [email p o ec ed] Recei ed: 2 Ma ch 2001; accep ed: 27 Ma ch 2001. Abs ac . In his pape , we in oduce he con ac Whi ney sphe e as an imbedding o he n–dimensional uni sphe e as an in eg al submani old o he s anda d con ac s uc u e on R2n+1. We ob ain a gene al inequali y o in eg al submani olds in R2n+1, in ol ing bo h he scala cu a u e and he mean cu a u e, and we use he equali y case in o de o cha ac e ize he con ac Whi ney sphe e. We also s udy a simila p oblem o an i-in a ian submani olds o R2n+1, angen o he s uc u e ec o field. Keywo ds: In eg al submani olds, Whi ney sphe es, scala cu a u e, mean cu a u e MSC 2000 classi ica ion: 53C15, 53C40, 53D10. In oduc ion One o he mos in e es ing opics in con ac geome y is he s udy o in eg al submani olds, i.e., submani olds imme sed in a con ac mani old, such ha he con ac o m es ic ed o he submani old anishes. In pa icula , 1–dimensional in eg al submani olds a e called Legend e cu es. A well-known p ope y o Legend e cu es o he s anda d con ac s uc u e dz−ydxon R3is ha he p ojec ion ¯γo a closed Legend e cu e γin R3 o he xy-plane mus ha e sel -in e sec ions and algeb aic (signed) a ea ze o. On he o he hand, we can hink o he pai o γand i s p ojec ion ¯γin he ollowing e ms. Suppose ha γi sel does no ha e sel -in e sec ions and ega d ¯γas a Lag angian submani old in C∼ =R2wi h sel -in e sec ions; hen hink o going om ¯γ o γas a way o emo ing he singula i y bu p ese ing he “Lag angian-Legend e” p ope y. A gene aliza ion o his can be done wi h he Whi ney sphe es. The Whi ney sphe es a e usually defined as a amily o Lag angian imme sions o he uni ∗Pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain 1999) 126 D. E. Blai , A. Ca iazo sphe e Sn, cen e ed a he o igin o Rn+1,in oCn∼ =R2ngi en by (u0,u 1,...,u n)→ 1+u2 0 (u1,...,u n,u 0u1,...,u 0un)+B, (1) whe e is a posi i e numbe and Bis a ec o o Cn. The numbe and he ec o Ba e called he adius and he cen e o he Whi ney sphe e, espec i ely. F om a opological poin o iew, i is well-known ha he sphe e can no be imbedded in Cnas a Lag angian submani old. The Whi ney sphe es ha e he bes possible beha iou , because hey ha e only one double poin a he poles o Sn. In he con ac mani old R2n+1 wi h i s usual con ac me ic s uc u e, we ha e he ollowing p esen a ion o he Whi ney sphe es as a amily o imbedded sphe es and in eg al submani olds o he con ac s uc u e (u0,u 1,...,u n)→ 1+u2 0 (u0u1,...,u 0un,u 1,...,u n, u0 1+u2 0 +C(1 + u2 0)) + B, whe e is a posi i e numbe , Bis a ec o o R2n+1 and Cis a eal cons an . We e e o hese sphe es as he con ac Whi ney sphe es. The e o e, wi h his p esen a ion we ha e emo ed he p e ious singula i y a he poles o Sn. On he o he hand, he Whi ney sphe es in Cnha e an in e es ing geome - ic p ope y. I was p o en by Bo elli, Chen and Mo an [3] and independen ly by Ros and U bano [6] ha i Mnis a Lag angian submani old o Cn,wi h mean cu a u e ec o Hand scala cu a u e τ, hen |H|2≥2(n+2) n2(n−1)τ,wi h equali y i and only i Mis ei he o ally geodesic o a (piece o a) Whi ney sphe e. Mo eo e , in [4], Cas o cons uc s a one-pa ame e amily o Lag angian sphe es including he Whi ney sphe e, such ha hey sa is y a geome ic equal- i y o ype τ=µ|H|2,wi hµ>0. In his pape , we es ablish an analogue o he abo e esul o in eg al sub- mani olds in R2n+1 wi h i s s anda d Sasakian s uc u e, such ha he equali y case holds o he con ac Whi ney sphe es. We also gi e a cha ac e iza ion by he second undamen al o m, simila o ha o Ros and U bano in [6]. Finally, we s udy he same p oblem o an i-in a ian submani olds angen o he s uc u e ec o field on R2n+1 and ob ain he co esponding esul s. 1 P elimina ies. Le (R2n+1,φ,ξ,η,g) deno e he mani old R2n+1 wi h i s usual Sasakian s uc u e gi en by η=1 2(dz − n  i=1 yidxi),ξ=2 ∂ ∂z, The con ac Whi ney sphe e 127 g=η⊗η+1 4 n  i=1 (dxi⊗dxi+dyi⊗dyi), φ( n  i=1 (Xi ∂ ∂xi +Yi ∂ ∂yi )+Z∂ ∂z)= n  i=1 (Yi ∂ ∂xi −Xi ∂ ∂yi )+ n  i=1 Yiyi ∂ ∂z, whe e (xi,y i,z),i=1...n a e he ca esian coo dina es. I is well-known ha (R2n+1,φ,ξ,η,g) is a Sasakian-space- o m, wi h con- s an φ–sec ional cu a u e equal o −3. Hence, i s cu a u e enso  Ris gi en by  R(X,Y )Z=−η(X)η(Z)Y+η(Y)η(Z)X−g(X,Z)η(Y)ξ+ +g(Y,Z)η(X)ξ−g(Z, φY )φX +g(Z, φX)φY −2g(X,φY )φZ, (2) o any ec o fields X,Y, Z. Fo mo e de ails and backg ound, we e e o [2]. Le Mbe an n-dimensional Riemannian mani old isome ically imme sed in he Sasakian-space- o m R2n+1. We also deno e by g he me ic on M. Le ∇( esp.  ∇) be he Le i–Ci i a connec ion o M( esp. R2n+1). Then, he Gauss–Weinga en o mulas a e gi en by  ∇XY=∇XY+σ(X,Y ), ∇XV=−AVX+DXV, o any angen ec o fields Xand Yand any no mal ec o field V, whe e Dis he connec ion in he no mal bundle, σis he second undamen al o m o Mand Ais he shape ope a o . The mean cu a u e ec o His defined by H=(1/n) ace σ. A submani old Mis called an in eg al submani old i η es ic ed o M anishes. I is well-known ha he con ac subbundle {η=0}admi s in eg al submani olds up o and including dimension n, bu o no highe dimension, see e.g. [2]. A di ec consequence o his defini ion is ha φX is a no mal ec o field, o any angen ec o field X, i.e., Mis an an i-in a ian submani old o R2n+1. Hence, in a neighbo hood o e e y poin p∈M,wecanconside alocal o hono mal ame {e1,...,e n,e 1∗,...,e n∗,ξ}, such ha e1,...,e na e angen o Mand ei∗=φei, o any i=1,...,n. Such a ame is called a Legend e ame. I we pu σk ij =g(σ(ei,e j),e k∗), i can be p o ed ha σi jk =σk ji =σj ik,(3) o any i, j, k =1,...,n. On he o he hand, we ha e [2, p. 128] Aξ=0,(4) equi alen ly σ(X, Y ) is pe pendicula o ξ, o any angen ec o fields Xand Y. 128 D. E. Blai , A. Ca iazo 2 Cha ac e izing he Con ac Whi ney Sphe e Le Mbe an n–dimensional submani old o R2n+1. Gi en a angen o hono - mal ame {e1,...,e n}, he scala cu a u e τo Mis defined by τ= i<j K(ei∧ej), whe e K(ei∧ej) is he sec ional cu a u e o he plane sec ion spanned by ei and ej. By ollowing he same s eps as in he p oo o Lemma 1 o [3], and by i ue o (2), (3) and (4), we ob ain he ollowing inequali y o in eg al submani olds, in ol ing bo h cu a u es |H|and τ: P oposi ion 1. Le Mnbe an in eg al submani old o R2n+1. Then, he squa e o i s mean cu a u e |H|2and i s scala cu a u e τsa is y a each poin he ollowing inequali y: |H|2≥2(n+2) n2(n−1)τ. (5) The equali y holds i and only i he e exis s a eal unc ion λdefined on M, such ha he second undamen al o m σo Msa isfies σ(e1,e 1)=3λe1∗,σ(e2,e 2)=... =σ(en,e n)=λe1∗, σ(e1,e j)=λej∗,σ(ej,e k)=0,2≤j=k≤n, (6) whe e {e1,...,e n,e 1∗,...,e n∗,ξ}is a Legend e ame such ha e1∗is pa allel o H. On he o he hand, we can also cha ac e ize he equali y case o he abo e inequali y by he beha io o he second undamen al o m o he submani old. We ob ain a o mula simila o ha o [6]: P oposi ion 2. Le Mnbe an in eg al submani old o R2n+1. Then, M sa isfies he equali y case o (5) a e e y poin , i and only i σ(X,Y )= n n+2{g(X, Y )H+g(φX, H)φY +g(φY, H)φX},(7) o any angen ec o fields Xand Y. P oo . I he equali y case o (5) holds o e e y p∈M, hen, in a neigh- bo hood o e e y poin , we can find a Legend e ame wi h e1∗pa allel o Hand such ha i sa isfies (6), wi h λ=n n+2|H|. Hence, i ollows ha (7) holds o any angen ec o fields Xand Y. The con e se can be e ified di ec ly. QED The con ac Whi ney sphe e 129 We now p oceed o show ha he only non- i ial example o an in eg al sub- mani old sa is ying he equali y in (5), is he con ac Whi ney sphe e. Fi s , we s a e he ollowing wo lemmas. The fi s one can be easily p o ed by s aigh - o wa d compu a ion. Lemma 1. Le π:R2n+1 →Cnbe he diffe en ial map gi en by: π(x1,...,x n,y 1,...,y n,z)=1 2(y1,...,y n,x 1,...,x n).(8) Then, π:(R2n+1,φ,ξ,η,g)→(Cn,J,G)is a Riemannian subme sion, whe e we deno e by (Cn,J,G) he usual Kaehle ian s uc u e on Cn. Mo eo e , i sa isfies he ollowing condi ions: i) The e ical subspace Vpo he subme sion a p∈R2n+1 is equal o he span o ξp; ii) g=π∗G+η⊗η; iii) φX =(Jπ∗X)∗, o any ec o field Xon R2n+1, whe e ∗deno es he ho izon al li wi h espec o η. Lemma 2 (Uniqueness o he li o he Whi ney sphe e). Le π: R2n+1 →Cnbe he Riemannian subme sion gi en by (8),andw:Sn→Cna Whi ney imme sion gi en by (1).I ψ:Sn→R2n+1 is an in eg al imme sion, such ha π◦ψ=w, hen ψis he con ac Whi ney imme sion gi en by ψ(u0,u 1,...,u n)= 2 1+u2 0 (u0u1,...,u 0un,u 1,...,u n,2 u0 1+u2 0 +C(1+u2 0))+ ˜ B, whe e Cis a cons an and ˜ Bisa ec o o R2n+1 such ha π(˜ B)=B. P oo . Since π◦ψ=wand n i=0 u2 i= 1, we can w i e ψas ψ(u0,u 1,...,u n)= 2 1+u2 0 (u0u1,...,u 0un,u 1,...,u n, (u1,...,u n))+ ˜ B, (9) o any cons an ec o ˜ Bsuch ha π(˜ B)=B. Then, diffe en ia ing (9), we ob ain, o any i=1,...,n: ∂ ∂ui =2  u2 0(1 + u2 0)−u2 i(1 −u2 0) u0(1 + u2 0)2 ∂ ∂xi − j=i uiuj(1 −u2 0) u0(1 + u2 0)2 ∂ ∂xj + +1+u2 0+2u2 i (1 + u2 0)2 ∂ ∂yi +2 j=i uiuj (1 + u2 0)2 ∂ ∂yj + 2ui +(1+u2 0)∂ ∂ui (1 + u2 0)2 ∂ ∂z    .(10) 130 D. E. Blai , A. Ca iazo Since ψis an in eg al imme sion in R2n+1,weha eη(∂ ∂ui)=0, o any i=1,...,n, whe e ηis he con ac o m on R2n+1. Hence, i ollows om (10) ha he unc ion mus sa is y he sys em o pa ial diffe en ial equa ions ∂ ∂ui ( 1+u2 0 )=2 ui(3u2 0−1) u0(1 + u2 0)3,i=1,...,n, which implies ha =2 u0 1+u2 0 +C(1 + u2 0), whe e Cis a eal cons an . QED We can now s a e he main heo em: Theo em 1. Le Mnbe an in eg al submani old o he s anda d con ac s uc u e on R2n+1. Then, he equali y case o (5) holds a e e y poin p∈M, i and only i ei he Mis a o ally geodesic submani old o i is a po ion o a con ac Whi ney sphe e. P oo . Le ψ:M→R2n+1 be an in eg al imme sion and pu  ψ=π◦ψ. Then, i ollows om Lemma 1 ha  ψ:M→Cnis a Lag angian imme sion, and ha he me ics induced on Mby bo h ψand  ψag ee. Deno e by σand ∇ he second undamen al o m o  ψand he Le i–Ci i a connec ion o Cn, espec i ely. Then, i ollows om he well-known O’Neill equa ions [5], ha  ∇X∗Y∗=(∇XY)∗+1 2η([X∗,Y∗])ξ o any ec o fields X,Y on Cn angen o M.SinceMis no mal o ξ,weha e η([X∗,Y∗]) = 0, and he e o e σ(X,Y )=(σ(X,Y ))∗and  H∗=H, (11) whe e  His he mean cu a u e ec o o  ψ. Suppose ha he equali y case o (5) holds o any p∈M. Then, P oposi- ion 2 implies ha σsa isfies (7). Hence, by i ue o (11), we ha e ha σ(X,Y )= n n+2G(X, Y ) H+G(JX,  H)JY +G(JY,  H)JX o any ec o fields X, Y on Cn angen o M. The e o e, i ollows om [6, Theo em 2], ha ei he  ψis o ally geodesic o  ψ(M) is an open po ion o a Whi ney sphe e. I he fi s case holds, (11) implies ha ψis also o ally geodesic. On he o he hand, i  ψ(M) is a po ion o a Whi ney sphe e, Lemma 2 implies ha ψ(M) mus be a po ion o a co esponding con ac Whi ney sphe e. The con e se can be p o ed by s aigh o wa d compu a ion. QED The con ac Whi ney sphe e 131 3 An i-in a ian Submani olds. We now conside an (n+ 1)–dimensional an i-in a ian submani old Mn+1 o R2n+1; such a submani old is angen o he s uc u e ec o field ξ. In pa - icula , om he Sasakian condi ion on R2n+1, ∇Xξ=−φX, and he e o e σ(X,ξ)=−φX, (12) o any ec o field X angen o M. In his case, i is possible o s a e an inequali y simila o (5): P oposi ion 3. Le Mn+1 be an (n+1)–dimensional an i-in a ian sub- mani old o R2n+1. Then, he squa e o i s mean cu a u e |H|2and i s scala cu a u e τsa is y a each poin he ollowing inequali y: |H|2≥2(n+2) (n+1) 2(n−1)τ. (13) The equali y holds i and only i he e exis s a eal unc ion λdefined on M, such ha he second undamen al o m σo Msa isfies σ(e1,e 1)=3λe1∗,σ(e2,e 2)=... =σ(en,e n)=λe1∗, σ(e1,e j)=λej∗,σ(ej,e k)=0,2≤j=k≤n, whe e {e1,...,e n,ξ,e 1∗,...,e n∗}is a local o hono mal ame such ha e1,..., en,ξa e angen o M,ei∗=φei, o any i=1,...,n,ande1∗is pa allel o H. We can also desc ibe he second undamen al o m o he submani olds sa - is ying he equali y case o (13) a e e y poin : P oposi ion 4. Le Mn+1 be an (n+1)–dimensional an i-in a ian sub- mani old o R2n+1. Then, Msa isfies he equali y case o (13) a e e y poin , i andonlyi σ(X,Y )=n+1 n+2{(g(X, Y )−η(X)η(Y))H+ +(g(φX, H)−n+2 n+1η(X))φY +(g(φY, H)−n+2 n+1η(Y))φX},(14) o any angen ec o fields Xand Y. A submani old Mo R2n+1, angen o ξ,issaid obe o ally con ac geodesic [1, p.110] i σ(X,Y )=η(X)σ(Y,ξ)+η(Y)σ(X,ξ),(15) o any angen ec o fields Xand Y. I is clea ha e e y o ally con ac geodesic submani old is minimal. Then, i ollows om (12), (14) and (15) ha 132 D. E. Blai , A. Ca iazo e e y an i-in a ian o ally con ac geodesic submani old sa isfies he equali y case o (13). Wi h an addi ional condi ion, we can cha ac e ize he non- i ial an i-in a ian submani olds sa is ying ha equali y. Le π:R2n+1 →Cnbe he Riemannian subme sion gi en by (8), and suppose ha he e exis s a Lag angian submani old No Cn, such ha he ollowing diag am commu es M−→ R2n+1 ↓↓π N−→ Cn, whe e Mis he se o fib es o e N. Now, we s a e he ollowing heo em: Theo em 2. Unde he abo e condi ions, he equali y case o (13) holds o any p∈M, i and only i ei he Mis a o ally con ac geodesic submani old o is locally isome ic o he Riemannian p oduc o a po ion o a Whi ney sphe e and R. P oo . As in he p oo o Theo em 1, i ollows om he O’Neill equa ions o he Riemannian subme sion π ha σ(X∗,Y∗)=(σ(X, Y ))∗and  H∗=n+1 nH, (16) whe e σand  Ha e he second undamen al o m and he mean cu a u e ec o o N, espec i ely. On he o he hand, we also know ha η(∇X∗Y∗)=−g(∇X∗ξ,Y ∗)=g(φX∗,Y∗)=0, since Mis an i-in a ian . This implies ha Mis locally isome ic o he Rie- mannian p oduc o Nand R. Suppose now ha he equali y case o (13) holds o any p∈M. Then, P oposi ion 4 implies ha σsa isfies (14). Hence, i ollows om (11) ha σ sa isfies he equa ion o [6, Theo em 2], and so, ei he Nis o ally geodesic o is a po ion o a Whi ney sphe e. Finally, i only emains o ema k ha , i Nis o ally geodesic, hen, we ob ain om (16) ha σ(X,Y ) = 0, o any angen ec o fields Xand Y, pe pendicula o ξ, and he e o e, Mis o ally con ac geodesic. The con e se can be p o ed by s aigh o wa d compu a ion. QED Acknowledgemen s. This a icle was w i en while he second au ho was isi ing Michigan S a e Uni e si y in 1999. He would like o exp ess his deep app ecia ion o p o esso s D.E. Blai and B.-Y. Chen o he wa m hospi ali y he ecei ed du ing his isi . The con ac Whi ney sphe e 133 Re e ences [1] A. 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