Riemannian submersions and slant submanifolds

Publ. Ma h. Deb ecen 61 / 3-4 (2002), 523–532 Riemannian subme sions and slan submani olds By JOS´ E L. CABRERIZO (Se illa), ALFONSO CARRIAZO (Se illa), LUIS M. FERN´ ANDEZ (Se illa) and MANUEL FERN´ ANDEZ (Se illa) Abs ac . We s udy he ela ionship be ween slan submani olds in bo h Com- plex and Con ac Geome y h ough Riemannian subme sions. We p esen some con- s uc ion p ocedu es o ob ain slan submani olds in he uni sphe e and in a S ie el mani old. We also gene alize hem by means o he Boo hby–Wang ib a ion. Finally, we show some cha ac e iza ion heo ems o h ee-dimensional slan submani olds. 0. In oduc ion The geome y o slan submani olds has been inc easingly s udied since B.-Y. Chen de ined slan imme sions in complex mani olds as a na u al gene aliza ion o bo h holomo phic and o ally eal imme sions (see [7]). La e , a simila no ion o slan submani old was in oduced in Con ac Geome y, which is specially impo an o submani olds angen o he s uc u e ec o ield o a con ac me ic mani old. The pu pose o he p esen pape is o s udy he close ela ionship be ween bo h heo ies h ough Riemannian subme sions. In pa icula , we p o e ha , in some condi ions, a submani old o an almos He mi ian mani old is slan i and only i i s li by a Riemannian subme sion is a slan submani old o an almos con ac me ic mani old. Ma hema ics Subjec Classi ica ion: 53C15, 53C40. Key wo ds and ph ases: Riemannian subme sion, Kaehle ian mani old, Sasakian man- i old, slan submani old. The au ho s wish o hank P o . Manuel Ba os o his aluable sugges ions and help ul ema ks. The au ho s a e pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain, 2001). 524 Jos´e L. Cab e izo e al. We use his esul as a me hod o ind in e es ing examples. Examples o p ope slan submani olds o a Sasakian-space- o m o cons an φ-sec ional cu a u e cha e been gi en in [4], [10] (c=−3) and [6] (c < −3), bu , un il now, he e we e no examples in a Sasakian-space- o m wi h c > −3. In ac , in his pape we exhibi a cons uc ion p ocedu e o ob ain examples o slan submani olds in he uni sphe e wi h i s usual Sasakian s uc u e (c= 1). A e wa ds, we ex end i in o de o ge ample examples in Sasakian-space- o ms wi h cons an φ-sec ional cu a u e c, o any c > −3. Mo eo e , we also cons uc examples o slan imme sions in o a S ie el mani old and we gene alize bo h p ocedu es by using he Boo hby–Wang ib a ion. Finally, we p esen some classi ica ions o h ee-dimensional slan submani olds o R5, by a ending o hei second undamen al o m. 1. P elimina ies In his sec ion, we ecall some basic o mulas and de ini ions abou slan submani olds in bo h Complex and Con ac Geome y, which we shall use la e . Fo de ails and backg ound on complex and con ac mani olds, we e e o he s anda d e e ences [1], [13]. A submani old No an almos He mi ian mani old ( e N, g, J) is said o be slan [7] i o each nonze o ec o X angen o Na p, he angle θ(X), 0 ≤θ(X)≤π/2, be ween JX and TpNis a cons an , called he slan angle o he submani old. In pa icula , holomo phic and o ally eal submani olds appea as slan submani olds wi h slan angle 0 and π/2, espec i ely. A slan submani old is called p ope slan i i is nei he holomo phic no o ally eal. In he case whe e Nis a Riemann su ace and e Nis a Kaehle mani old, S. S. Che n and J. G. Wol son in oduced he no ion o Kaehle angle, de ined o be he angle be ween J∂/∂x and ∂/∂y, whe e z=x+√−1yis a local complex coo dina e on N[9]. I is clea ha i Nis a su ace wi h cons an Kaehle angle α, hen i is a slan submani old wi h slan angle θsa is ying θ=α( esp. θ=π−α) when α∈[0, π/2] ( esp. α∈(π/2, π]). Pu JX =PX +FX, o any angen ec o ield X, whe e PX ( esp. FX) deno es he angen ial ( esp. no mal) componen o JX. Then, θ- slan submani olds a e cha ac e ized by he o mula: P2=−cos2θId . Riemannian subme sions and slan submani olds 525 A special ype o p ope slan submani old is ha o Kaehle ian slan submani old, i.e., a p ope slan submani old sa is ying ∇0P= 0, whe e ∇0deno es he Le i–Ci i a connec ion on N. I is easy o show ha a Kaehle ian slan submani old is a Kaehle ian mani old wi h espec o he induced me ic and wi h he almos complex s uc u e gi en by (sec θ)P. In a simila way, gi en a submani old M angen o he s uc u e ec o ield ξo an almos con ac me ic mani old ( M, φ, ξ, η, G), i is said o be slan [4] i he angle θ(X) be ween φX and TpMis a cons an , which is independen o he choice o p∈Mand X∈Tp(M) Span(ξp). In pa icula , o θ= 0 ( esp. θ=π/2) we ob ain he in a ian ( esp. an i-in a ian ) submani olds. Now, i we deno e by TX ( esp. NX) he angen ial ( esp. no mal) componen o φX, he e is an equa ion which cha ac e izes θ-slan submani olds: T2=−cos2θ(Id −η⊗ξ). In con ac geome y, he simila no ion o Kaehle ian slan submani- olds is gi en by p ope θ-slan submani olds sa is ying (∇XT)Y= cos2θ(g(X, Y )ξ−η(Y)X), o any angen ec o ields X, Y , whe e ∇deno es he Le i–Ci i a con- nec ion on M. This non- i ial ac is shown in [4]. The e o e, by ol- lowing he complex case no a ion, we call such a submani old a Sasakian slan submani old. On he o he hand, he possibili y o ob aining an in- duced con ac me ic s uc u e on a slan submani old o a con ac me ic mani old is s udied in [5]. 2. Main esul s Le Mbe a (2m+ 1)-dimensional almos con ac me ic mani old wi h s uc u e enso s (φ, ξ, η, G) and e Nbe a eal 2m-dimensional almos He mi ian mani old wi h s uc u e (J, g). Le suppose ha he e exis s a Riemannian subme sion π: M→e Nsa is ying he condi ions: i) The e ical subspace Vpo he subme sion a p∈ Mis equal o he span o ξp, ii) φX∗= (JX)∗, 526 Jos´e L. Cab e izo e al. o any ec o ield Xon e N, whe e ∗deno es he ho izon al li wi h espec o π. In ac , since πis a Riemannian subme sion, we also ha e iii) G(X∗, Y ∗) = g(X, Y ), o any ec o ields X, Y on e N. Now, le Mbe an (n+ 1)-dimensional submani old angen o he s uc u e ec o ield ξo Mand Nbe an n-dimensional submani old o e N. Th oughou in his sec ion we assume ha he ollowing diag am commu es (2.1) M−−−−→ M   y  yπ N−−−−→ e N whe e Mis he se o ib es o e N. Then, we s a e he ollowing heo em: Theo em 2.1. In he abo e condi ions, we ha e: (a) Mis θ-slan in Mi and only i Nis θ-slan in e N. Mo eo e , i Mis a Sasakian mani old, we also ha e: (b) Mis Sasakian θ-slan in Mi and only i Nis Kaehle ian θ-slan in e N. P oo . S a emen (a) ollows di ec ly om i)–iii). In ac , in any almos con ac me ic mani old, φξ = 0, and hen, he condi ion o M being a slan submani old is eally ela ed o i s con ac dis ibu ion, which is he ho izon al subspace o he subme sion a any poin . Now, suppose ha Mis a Sasakian mani old and deno e by ∇( esp. ∇0) he Le i–Ci i a connec ion on M( esp. N). I ollows om he well- known O’Neill equa ions o he subme sion ha ∇X∗Y∗= (∇0 XY)∗+η(∇X∗Y∗)ξ, η(∇X∗Y∗) = −G(X∗, TY ∗), o any ec o ields X, Y on e N angen o N. Then, we ha e (∇X∗T)Y∗= ((∇0 XP)Y)∗−G(X∗, T2Y∗)ξ, (∇ξT)Y∗= 0, which imply (b). ¤ Riemannian subme sions and slan submani olds 527 No ice ha , in pa icula , s a emen (a) o Theo em 2.1 implies s a e- men s (3) and (4) o [13, P oposi ion 3.2, p. 459]. By using Theo em 2.1, we can show he ollowing cons uc ion p oce- du e o gi ing examples o p ope slan submani olds in he uni sphe e. Le π:S2m+1 →CPm(4) be he well-known Hop ib a ion, whe e S2m+1 ( esp. CPm(4)) is endowed wi h i s usual Sasakian ( esp. Kaehle- ian) s uc u e. Gi en any isome ic imme sion :N→CPm(4), hen M=π−1(N) is a p incipal ci cle bundle o e Nwi h o ally geodesic i- b es and he li ˆ :M→S2m+1 o is an isome ic imme sion such ha he ollowing diag am commu es: Mˆ −−−−→ S2m+1   y  yπ N −−−−→ CPm(4). I ollows om Theo em 2.1 ha , in o de o ob ain a θ-slan subman- i old o S2m+1, i is enough o conside a θ-slan submani old o CPm(4). Fo example, we could ake he examples gi en in [11]. The abo e p oce- du e was i s poin ed ou by B.-Y. Chen and Y. Tazawa in [8]. We can also conside he li o he Ve onese sequence in o de o ge new examples o p ope slan imme sions in o S2m+1. We ecall ha he Ve onese sequence ψ0, . . . , ψmis de ined, o any p= 0, . . . , m, by ψp:S2→CPm:ψp[z0, z1] = [gp,0(z0/z1), . . . , gp,m(z0/z1)], whe e [z0, z1]∈CP1=S2, and gp,j(z) = p! (1 + zz)psµm j¶zj−pX k (−1)kµj p−k¶µm−j k¶(zz)k, o any j= 0, . . . , m. I was shown in [2] ha e e y ψpis a con o mal minimal imme sion wi h cons an cu a u e and cons an Kaehle angle αpsuch ha an2αp 2=p(m−p+ 1) (p+ 1)(m−p). By combining his p ocedu e and a D-homo he ic de o ma ion, we may also ob ain he ollowing heo em, simila o [6, Theo em 3.5]: 528 Jos´e L. Cab e izo e al. Theo em 2.2. Le cbe a cons an wi h c > −3. Then, he e ex- is p ope slan submani olds in a Sasakian-space- o m wi h cons an φ- sec ional cu a u e c. P oo . Fi s , we can choose a p ope slan submani old o S2m+1, gi en by he abo e cons uc ion p ocedu e. We deno e he usual Sasakian s uc u e on S2m+1 by (φ, ξ, η, G). Then, o any c > −3, we conside he cons an a= 4/(c+ 3) >0 and he D-homo he ic de o ma ion: e φ=φ, e ξ=1 aξ, eη=aη, e G=aG +a(a−1)η⊗η. I was shown in [1] ha S2m+1 wi h his s uc u e is a Sasakian-space- o m wi h cons an φ-sec ional cu a u e (4/a)−3 = c. Finally, i is easy o p o e ha a D-homo he ic de o ma ion maps slan submani olds in o slan submani olds. ¤ A mo e elabo a e cons uc ion p ocedu e o ob aining slan sub- mani olds in a ce ain almos con ac me ic mani old can be shown as ollows. Le Hbe he closed connec ed subg oup in S3×S3gi en by H={(z, z) : z∈S1}and conside he homogeneous space (S3×S3)/H. Since his is a compac simply connec ed 5-dimensional spin mani old wi h H2((S3×S3)/H;Z) = Z, i ollows om a classic esul o Smale [12] ha i is di eomo phic o S2×S3. On he o he hand, i we deno e by V(2,4) he S ie el mani old o o hono mal 2- ames in 4-space, i is known ha V(2,4) is di eomo phic o (S3×S3)/H and hen, he e is a di eomo - phism :V(2,4) →S2×S3. Le eπ:S3→S2be he Hop ib a ion and pu F= (id ×eπ)◦ . Hence, F:V(2,4) →Q2is a subme sion, whe e Q2deno es he complex quad ic S2×S2. Now, pu S2 ∗=S2 {(0,0,1)} and le E:S2 ∗→Cbe he co esponding s e eog aphic p ojec ion, which p ese es he complex s uc u e o C. Then, V(2,4)∗→Q2∗→C2is a subme sion, whe e Q2∗( esp. V(2,4)∗) deno es he mani old S2 ∗×S2 ∗( esp. F−1(Q2∗)). I is clea ha , i we conside on C2i s usual Kaehle s uc u e, V(2,4)∗can be endowed wi h a na u al almos con ac me ic s uc u e such ha (E, E)◦F|V(2,4)∗ is a Riemannian subme sion sa is ying he abo e s a ed condi ions i)–ii). Hence, we ob ain ample examples o slan su aces in V(2,4)∗by consid- e ing he li s o slan su aces in C2(see, o ins ance, [7]). Mo eo e , we can gi e a gene aliza ion o he p e ious cons uc ion p ocedu es. Le Mbe a (2m+ 1)-dimensional compac egula con ac Riemannian subme sions and slan submani olds 529 mani old. Acco ding wi h a classical esul o Boo hby–Wang [3], one can see Mas a ci cle bundle o e a 2m-dimensional compac symplec ic mani old e N: π: M−→ e N. Since e Nca ies a global symplec ic o m Ω, he e exis a Riemannian me ic gand a enso ield Jo ype (1,1) such ha (g, J) is an almos Kaehle s uc u e on e Nwi h Ω as i s undamen al 2- o m. Deno e by η he con ac o m on Mwi h d η=π∗Ω and ξi s cha ac e is ic ec o ield and de ine a enso ield φand a Riemannian me ic Gon Mby φX = (Jπ∗X)∗ and G=π∗(g)+η⊗η, espec i ely. Then, i can be p o ed ha (φ, ξ, η, G) is a K-con ac s uc u e on Mand π: ( M, G)→(e N, g) is a Riemannian subme sion. Now, we ha e: Theo em 2.3. In he abo e condi ions, le Mbe a submani old o M. Then, Mis a S1-in a ian θ-slan submani old i and only i M=π−1(N), whe e Nis a θ-slan submani old o e N. P oo . Le Nbe a submani old o e Nand deno e by M=π−1(N). Then, Mis a submani old o Mand he cha ac e is ic ec o ield ξis angen o M, in pa icula , Mis S1-in a ian . The con e se o he abo e s a ed ac also holds, ha is, i Mis a S1-in a ian submani old o M, hen ξis angen o Mand he e exis s a submani old Nin e Nwi h M=π−1(N). Hence, he p oo concludes by applying Theo em 2.1. ¤ 3. Some applica ions We now p oceed o show some applica ions o he abo e s a ed e- la ionship be ween slan submani olds and Riemannian subme sions, by conside ing he di e en ial map gi en by π:R5−→ C2; (x1, x2, y1, y2, z)7−→ 1 2(y1, y2, x1, x2). I is easy o see ha , i we ha e on R5( esp. C2) i s usual Sasakian ( esp. Kaehle ian) s uc u e, hen πis a Riemannian subme sion sa is ying con- di ions i)–ii). The e o e, by using his subme sion, we can ob ain examples o slan submani olds in R5by aking he li s o Examples 2.1, 2.3, 2.4 530 Jos´e L. Cab e izo e al. and 2.5 o [7]. No ice ha hose examples will be simila o Examples 3.7– 3.10 o [4]. Now, suppose ha we ha e a 3-dimensional submani old M angen o he s uc u e ec o ield on R5and a su ace Nin C2sa is ying dia- g am (2.1). Then, we ha e he ollowing classi ica ion heo em: Theo em 3.1. In he abo e condi ions, Mis a 3-dimensional slan submani old o R5wi h pa allel mean cu a u e ec o i and only i Mis one o he ollowing submani olds: (a) a submani old locally isome ic o an open po ion o he p oduc o a plane ci cle and a ci cula cylinde . (b) a submani old locally isome ic o an open po ion o he p oduc o a ci cula cylinde and R. (c) a minimal slan submani old in R5. Mo eo e , i ei he case (a) o case (b) occu s, hen Mis an an i-in a ian submani old. P oo . Fi s , i is known ha i he mean cu a u e ec o o Mis pa allel hen he mean cu a u e ec o o Nis also pa allel, and ha M is minimal i and only i Nis minimal (see, o ins ance, [13, p. 462–463]). Hence, he p oo o his heo em ollows om Theo em 1.1 o [7, p. 50] and by aking in o accoun ha , i Mis an an i-in a ian submani old, hen η(∇X∗Y∗) = 0, o any X,Y angen o N, which means ha , in his case, Mis locally isome ic o he Riemannian p oduc o Nand R.¤ In he same condi ions, we can also classi y he submani old Ma - ending o a pa icula beha iou o i s second undamen al o m σ: Theo em 3.2. Mis a 3-dimensional slan submani old o R5sa is y- ing (3.1) (∇Xσ)(Y, Z) = G(Y, TX)NZ +G(Z, TX)NY o any angen ec o ields X, Y, Z o hogonal o ξ, i and only i Mis one o he ollowing submani olds: (a) a submani old locally isome ic o an open po ion o he p oduc o a plane ci cle and a ci cula cylinde . (b) a submani old locally isome ic o an open po ion o he p oduc o a ci cula cylinde and R. Riemannian subme sions and slan submani olds 531 (c) a li by πo an open po ion o a plane in C2. Mo eo e , i ei he case (a)o case (b)occu s, hen Mis an an i-in a ian submani old. P oo . I ollows om he O’Neill equa ions ha (∇X∗σ)(Y∗, Z∗) = ((∇Xσ0)(Y, Z))∗+G(Y∗, TX∗)NZ∗+G(Z∗, TX∗)NY ∗, o any X, Y, Z angen o N, whe e σ0deno es he second undamen al o m o N, and so, Msa is ies (3.1) i and only i σ0is pa allel. The e o e, his p oo wo ks as ha o Theo em 3.1, by applying now Theo em 1.2 o [7, p. 51]. ¤ A su icien condi ion o a submani old M, in he abo e condi ions, o sa is y equa ion (3.1) is o be o ally con ac geodesic, i.e., such ha σ(X, Y ) = η(X)σ(Y, ξ) + η(Y)σ(X, ξ), o any angen ec o ields Xand Y. In ac he e a e examples o o ally con ac geodesic slan submani olds in R5(see, o ins ance, Example 3.7 o [4]). Re e ences [1] D. E. Blai , Con ac Mani olds in Riemannian Geome y, Lec u e No es in Ma h- ema ics, 509, Sp inge -Ve lag,New Yo k, 1976. [2] J. Bol on, G. R. Jensen, M. Rigoli and L. M. Woodwa d , On con o mal minimal imme sions o S 2in o CP n,Ma h. Ann. 279 (1988), 599–620. [3] W. M. Boo hby and H. C. Wang , On con ac mani olds, Ann. o Ma h. 68 (1958), 721–734. [4] J. L. Cab e izo, A. Ca iazo, L. M. Fe n andez and M. Fe n andez , Slan submani olds in Sasakian mani olds, Glasgow Ma h. J. 42 (2000), 125–138. [5] J. L. Cab e izo, A. Ca iazo, L. M. Fe n andez and M. Fe n andez , S uc- u e on a slan submani old o a con ac mani old, Indian J. Pu e Appl. Ma h. 31 (2000), 857–864. [6] J. L. Cab e izo, A. Ca iazo, L. M. Fe n andez and M. Fe n andez , Exis- ence and uniqueness heo em o slan imme sions in Sasakian-space- o ms, Publ. Ma h. Deb ecen 58 no. 4 (2001), 559–574. [7] B. Y. Chen , Geome y o Slan Submani olds, Ka holieke Uni e si ei Leu en, Leu en, 1990. [8] B. Y. Chen and Y. Tazawa , Slan submani olds o complex p ojec i e and com- plex hype bolic spaces, Glasgow Ma h. J. 42 (2000), 439–454.