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7 Explicit formulas for biharmonic submanifolds in 0 0 non-Euclidean 3-spheres 2 n D. FETCU and C. ONICIUC a ∗ J 5 Dedicated to Professor Vasile Oproiu on his 65-th birthday ] G D Abstract . h We obtain the parametric equations of all biharmonic Legendre curves t a andHopfcylindersinthe 3-dimensionalunitsphereendowedwiththe mod- m ified Sasakian structure defined by Tanno. [ 2000 MSC: 53C42, 53B25. Key words: Biharmonic submanifolds, Sasakian space forms, Legendre 1 curves, Hopf cylinders. v 6 5 1 Introduction 1 1 0 Biharmonic maps φ : (M,g) (N,h) between Riemannian manifolds are the 7 → 0 critical points of the bienergy functional E2(φ) = 21RM |τ(φ)|2 νg and represent / a natural generalization of the well-known harmonic maps, the critical points of h at the energy functional E(φ) = 21RM |dφ|2 νg. The bienergy functional has been m suggested since 1964 by Eells and Sampson in their famous paper [13]. The bienergy functional can be viewed as the analogue for maps of the Willmore : v functional for Riemannian immersions. In a different setting, Chen defined the i X biharmonic submanifolds in an Euclidean space. If we apply the characterization r formula of biharmonic maps to Riemannian immersions into Euclidean spaces, a we recover Chen’s notion of biharmonic submanifold. The Euler-Lagrange equation for the energy functional is τ(φ) = 0, where τ(φ) = trace dφ is the tension field, and the Euler-Lagrange equation for the ∇ bienergy functional was derived by Jiang in [15]: τ (φ) = ∆τ(φ) trace RN(dφ,τ(φ))τ(φ) = 2 − − = 0. ∗Theauthors were partially supported by theGrant CEEX, ET, 5871/2006, Romania. 1 Since any harmonic map is biharmonic, we are interested in non-harmonic biharmonic maps, which are called proper-biharmonic. Thereareseveralclassificationresultsofproper-biharmonicsubmanifoldsin3- dimensional spaces. The proper-biharmonic submanifolds in R3, S3 and N3( 1), − the space of constant negative sectional curvature 1, were completely classified − in [11], [5] and [6], respectively. Then, the next step was to classify proper- biharmonic submanifolds in a 3-dimensional space of non-constant sectional curvature. In [8], [7] and [12] the authors classified the proper-biharmonic curves in the Heisenberg group and Cartan-Vranceanu 3-dimensional spaces, while, in [14], Inoguchi classified the proper-biharmonic Legendre curves and Hopf cylinders in a 3-dimensional Sasakian space form M3(c). For a general account of biharmonic maps see [16] and The bibliography of biharmonic maps [18]. The goal of this paper is to obtain explicitly the parametric equations of proper-biharmonic Legendre curves and Hopf cylinders in S3 with the modified Sasakian structure definedby Tanno, by usingthe results in [14]. The techniques used in our paper are introduced in [1] and [2]. Conventions. We work in the C category, that means manifolds, metrics, ∞ connections and maps are smooth. The Lie algebra of the vector fields on M is denoted by C(TM). 2 Preliminaries 2.1 Contact manifolds A contact metric structure on a manifold M2n+1 is given by (ϕ,ξ,η,g), where ϕ is a tensor field of type (1,1) on M, ξ is a vector field on M, η is an 1-form on M and g is a Riemannian metric, such that ϕ2 = I +η ξ, η(ξ) = 1,  − ⊗    g(ϕX,ϕY) = g(X,Y) η(X)η(Y), X,Y C(TM) .  − ∀ ∈  g(X,ϕY) = dη(X,Y), X,Y C(TM)   ∀ ∈ A contact metric structure (ϕ,ξ,η,g) is Sasakian if it is normal, that is N +2dη ξ = 0, ϕ ⊗ where N (X,Y)= [ϕX,ϕY] ϕ[ϕX,Y] ϕ[X,ϕY]+ϕ2[X,Y], X,Y C(TM), ϕ − − ∀ ∈ 2 is the Nijenhuis tensor field of ϕ, or, equivalently, if ( ϕ)(Y)= g(X,Y)ξ η(Y)X, X,Y C(TM). X ∇ − ∀ ∈ Let us consider a Sasakian manifold (M,ϕ,ξ,η,g). The sectional curvature of a 2-plane generated by X and ϕX, where X is an unit vector orthogonal to ξ, is called ϕ-sectional curvature determined by X. A Sasakian manifold with constant ϕ-sectional curvature c is called a Sasakian space form and it is denoted by M(c). The curvature tensor field of a Sasakian space form M(c) is given by R(X,Y)Z = c+3 g(Z,Y)X g(Z,X)Y + c 1 η(Z)η(X)Y 4 { − } −4 { − η(Z)η(Y)X +g(Z,X)η(Y)ξ g(Z,Y)η(X)ξ+ − − +g(Z,ϕY)ϕX g(Z,ϕX)ϕY +2g(X,ϕY)ϕZ . − } A contact metric manifold (M,ϕ,ξ,η,g) is called regular if for any point p M there exists a cubic neighborhood of p such that any integral curve of ξ ∈ passes through the neighborhood at most once, and strictly regular if all integral curves are homeomorphic to each other. Let (M,ϕ,ξ,η,g) be a regular contact metric manifold. Then the orbit space M¯ can benaturally organized as a manifold and, moreover, if M is compact then M is a principal circle bundle over M¯ (the Boothby-Wang Theorem). In this case the fibration π : M M¯ is called the Boothby-Wang fibration. A very known → example of a Boothby-Wang fibration is the Hopf fibration π : S2n+1 CPn. → We recall the following result obtained by Ogiue Theorem 2.1 ([17]) Let (M,ϕ,ξ,η,g) be a strictly regular Sasakian manifold. Then M¯ can be organized as a Ka¨hler manifold. Moreover, if (M,ϕ,ξ, η,g) is a Sasakian space form M(c), then M¯ has constant sectional holomorphic curvature c+3. 2.2 General results In [5] all biharmonic curves and surfaces in the 3-dimensional Euclidean unit sphere (S3,g ), where g is the usual metric, were classified: 0 0 Theorem 2.2 ([5]) LetMm beaproper-biharmonic submanifold ofS3. Wehave a) if m = 1, then M is either the circle of radius 1 or a geodesic of the torus √2 S1 1 S1 1 S3 with the slope different from 1; (cid:16)√2(cid:17)× (cid:16)√2(cid:17) ⊂ ± b) if m = 2, then M is the hypersphere S2 1 S3. (cid:16)√2(cid:17) ⊂ 3 We can think (S3,g ) as a Sasakian space form with constant ϕ -sectional 0 0 curvature 1. We know that in the geometry of Sasakian manifolds an important roleisplayedbytheintegralsubmanifoldsandHopfcylinders. Anaturalquestion is whether biharmonic Legendre curves and Hopf cylinders in (S3,ϕ ,ξ ,η ,g ) 0 0 0 0 exist. Since the torsion of a Legendre curve in (S3,ϕ ,ξ ,η ,g ) is 1 and a Hopf 0 0 0 0 cylinderisflat, itfollows thatin(S3,ϕ ,ξ ,η ,g )donotexistproper-biharmonic 0 0 0 0 Legendre curves and proper-biharmonic Hopf cylinders. Thenextstep is thestudyof theexistence ofbiharmonicLegendrecurves and of biharmonic Hopf cylinders in Sasakian space forms with constant ϕ-sectional curvature c = 1. Inoguchi gave the following classification: 6 Theorem 2.3 ([14]) LetM3(c)beaSasakianspace formofconstant ϕ-sectional curvature c and let γ : I M be a Legendre curve parametrized by arc length. → We have a) if c6 1, then γ is biharmonic if and only if it is geodesic; b) if c > 1, then γ is proper-biharmonic if and only if it is a helix with the curvature κ¯2 = c 1. − Theorem 2.4 ([14]) LetM3(c)beaSasakianspace formofconstant ϕ-sectional curvature candS aHopf cylinder, where γ¯ isacurveinthe orbitspace of M3(c), γ¯ parametrized by arc length. We have a) if c6 1, then S is biharmonic if and only if it is minimal; γ¯ b) if c > 1, then S is proper-biharmonic if and only if the curvature κ¯ of γ¯ γ¯ is constant κ¯2 = c 1. − LetM3(c)beaSasakianspaceformwithconstantϕ-sectionalcurvaturec > 1, and let γ¯ : I M¯ be a curve parametrized by arc length. We denote f = γ¯ 1 ′ → and f = J¯f the Frenet frame field along γ¯, where J¯ is the almost complex 2 1 structure on M¯. Consider S = π 1(γ¯) the Hopf cylinder correspondingto γ¯ and γ¯ − assume that it is biharmonic, that is k¯2 = c 1. Since [ξ,fH] = 0, g(ξ,ξ) = 1, − 1 g(fH,fH) = 1, we can choose a local chart x = x(u,v) such that ξ = x and 1 1 v fH = x . The parametric curves u x(u,v ) are proper-biharmonic Legendre 1 u → 0 curves parametrized by arc length and geodesics in S . Moreover, u x(u,v ) γ¯ 0 → are the only geodesics of S which are proper-biharmonic in M3(c). γ¯ Proposition 2.5 Let M3(c) be a Sasakian space form with constant ϕ-sectional curvature c > 1 and S a biharmonic Hopf cylinder. Let γ : I S be a geodesic γ¯ γ¯ → parametrized by arc length and i : S M3(c) the canonical inclusion. Then γ¯ → i γ :I M3(c) is proper-biharmonic if and only if γ = fH. ◦ → ′ ± 1 4 Proof. Let γ : I S be a geodesic parametrized by arc length. Then γ¯ → γ = c ξ+c fH, where c and c are real constants such that c2+c2 = 1 and fH ′ 1 2 1 1 2 1 2 1 is the horizontal lift of f . After a straightforward computation one obtains 1 τ(i γ) = c (c κ¯ 2c )fH and τ (i γ)= 2c c2(c κ¯ 2c )κ¯fH, ◦ 2 2 − 1 2 2 ◦ 1 2 2 − 1 2 where κ¯2 = c 1. Hence τ(i γ) = 0 and τ (i γ) = 0 if and only if c = 0, that 2 1 − ◦ 6 ◦ is γ = fH. ′ ± 1 (cid:3) It can beeasily proved that, if γ : I M3(c) is a biharmonicLegendre curve, → then it is a geodesic in the biharmonic Hopf cylinder x(s,t) = φ (t), where γ(s) φ is the uniparametric group of ξ. From now on we will use the parameters t { } (u,v) instead of (s,t). In the following we will choose the 3-sphere S3 with modified Sasakian structure as a model for the complete, simply connected Sasakian space form with constant ϕ-sectional curvature c > 1, and we will find the explicit equations of biharmonic Legendre curves and of biharmonic Hopf cylinders, viewed as submanifolds of R4. These equations are given by Theorem 3.1 and Theorem 4.1, respectively. 3 Proper-biharmonic Legendre curves Let S3 = z C2 : z = 1 be the unit 3-dimensional sphere endowed with { ∈ k k } its standard metric field g . Consider the following structure tensor fields on S3: 0 ξ = Jz for each z S3, where J is the usual almost complex structure on C2 0 − ∈ defined by Jz = ( y , y ,x ,x ) for z = (x ,x ,y ,y ), and ϕ = s J, where 1 2 1 2 1 2 1 2 0 − − ◦ s : T C2 T S3 denotes the orthogonal projection. Endowed with these tensors, z z → S3 becomes a Sasakian space form with ϕ -sectional curvature 1 (see [4]). Now 0 we consider the deformed structure 1 η =aη , ξ = ξ , ϕ = ϕ , g = ag +a(a 1)η η , 0 0 0 0 0 0 a − ⊗ where a is a positive constant. Such a deformation is called a -homothetic D deformation, since the two metrics restricted to the contact subbundle are D homothetic, and it was introduced in [19]. The deformed structure (ϕ,ξ,η,g) is stillaSasakianstructureand(S3,ϕ,ξ,η,g)isaSasakianspaceformwithconstant ϕ-sectional curvature c = 4 3 (see [2]). From now on we assume that a (0,1), a− ∈ that is c> 1. 5 Theorem 3.1 Let γ : I (S3,ϕ,ξ,η,g) be a Legendre curve parametrized by → arc length. Then it is proper-biharmonic if and only if, as a curve in R4 γ(s) = B cos(As)e B sin(As)Je + qA+B 1 −qA+B 1 (3.1) + A cos(Bs)e + A sin(Bs)Je , qA+B 3 qA+B 3 where e ,e isan orthonormal system of constant vectors inthe Euclidianspace 1 3 { } (R4, , ), with e orthogonal to Je , and 3 1 h i 3 2a 2√(a 1)(a 2)  A= q − − − − a (3.2)  .  3 2a+2√(a 1)(a 2)  B = q − a − −  Proof. Let us denote by , ˙ and the Levi-Civita connections on (S3,g), ∇ ∇ ∇ (S3,g ) and (R4, , ), respectively. e 0 h i Let γ : I (S3,g) be a proper-biharmonic Legendre curve parametrized by → arc length and let T = γ be the unit tangent vector field along the curve. Using ′ Theorem 2.3 we have g¯(T,T) = 1, g(T,ξ) = 0, T = κϕT, and κ2 = c 1. T ∇ − In order to find the explicit parametrization of γ as a curve in R4 we shall characterize γ as the solution of a certain fourth order ODE with constant coef- ficients. For this purpose the expressions of T, ϕT and ξ are needed. T T T ∇ ∇ ∇ First, we observe that g( XX,Z) = ag0(e˙XX,eZ), for anyeZ C(TS3) and ∇ ∇ ∈ for any X C(TS3) orthogonal to ξ. Then, using the properties of the Sasakian ∈ structures (ϕ,ξ,η,g) and (ϕ ,ξ ,η ,g ), we get 0 0 0 0 1 ˙ T = κϕT, ˙ ϕT = κT +ξ, ˙ ξ = ϕT. T T T ∇ ∇ − ∇ −a Further, using the Gauss equation of (S3,g ) in (R4, , ), we obtain 0 h i T = ˙ T T,T γ = κϕT 1γ  ∇T ∇T −h i − a e .  ϕT = κT +ξ, ξ = 1ϕT  ∇T − ∇T −a e e Now it follows 1 1 T = κ ϕT T = +κ2 T +κξ ∇T∇T ∇T − a −(cid:16)a (cid:17) e e e 6 and 2 1 1 1 T = +κ2 T + γ + +κ2 γ. ∇T∇T∇T −(cid:16)a (cid:17)(cid:16)∇T a (cid:17) a(cid:16)a (cid:17) e e e e As T = γ and κ2 = c 1= 4 4, we obtain the equation of γ (thought as a ∇T ′′ − a − curvee in R4) (3.3) a2γiv +a(6 4a)γ +γ = 0. ′′ − The general solution is γ(s) = cos(As)c +sin(As)c +cos(Bs)c +sin(Bs)c , 1 2 3 4 where A, B are given by (3.2) and c are constant vectors in R4. i { } It is easy to see that γ must verify the following relations 1 5 4a γ,γ = 1, γ′,γ′ = , γ,γ′ = 0, γ′,γ′′ = 0, γ′′,γ′′ = − , h i h i a h i h i h i a2 1 5 4a γ,γ′′ = , γ′,γ′′′ = − , γ′′,γ′′′ = 0, γ,γ′′′ = 0, h i −a h i − a2 h i h i 16a2 44a+29 γ′′′,γ′′′ = − , h i a3 Denoting c = c ,c , in s = 0 we have ij i j h i (3.4) c +2c +c = 1 11 13 33 1 (3.5) A2c +2ABc +B2c = 22 24 44 a (3.6) Ac +Ac +Bc +Bc = 0 12 23 14 34 (3.7) A3c +AB2c +A2Bc +B3c = 0 12 23 14 34 5 4a (3.8) A4c +2A2B2c +B4c = − 11 13 33 a2 1 (3.9) A2c +(A2+B2)c +B2c = 11 13 33 a 5 4a (3.10) A4c +(AB3+A3B)c +B4c = − 22 24 44 a2 (3.11) A5c +A3B2c +A2B3c +B5c = 0 12 23 14 34 (3.12) A3c +A3c +B3c +B3c = 0 12 23 14 34 16a2 44a+29 (3.13) A6c +2A3B3c +B6c = − . 22 24 44 a3 7 Since the determinant of the system given by (3.6), (3.7), (3.11) and (3.12) is A2B2(A2 B2)4 = 0 it follows that − − 6 c = c = c = c = 0. 12 23 14 34 The equations (3.4), (3.8) and (3.9) give B A c = , c = 0, c = , 11 13 33 A+B A+B and, from (3.5), (3.10) and (3.13) it results that B A c = , c = 0, c = . 22 24 44 A+B A+B We obtained that c are orthogonal vectors in R4 with c = c = B { i} k 1k k 2k qA+B and c = c = A . Let us consider c = c e , where e are mutually k 3k k 4k qA+B i k ik i { i} orthogonal unit constant vectors in R4. Now, if γ is a Legendre curve, one obtains T,ξ = 0. From the expression of h i γ and since, by definition, ξ = Jγ(s), we get e ,Je = e ,Je = 1 and 1 2 3 4 − h i −h i ± e ,Je = e ,Je = e ,Je = e ,Je = 0, from which arise the forms of 1 3 1 4 2 3 2 4 h i h i h i h i the vectors e . Therefore e = Je and e = Je . Finally, we obtain i 2 1 4 3 ∓ ± γ(s) = B cos(As)e B sin(As)Je + qA+B 1 −qA+B 1 + A cos(Bs)e + A sin(Bs)Je , qA+B 3 qA+B 3 and γ (s) = B cos(As)e + B sin(As)Je + 1 qA+B 1 qA+B 1 + A cos(Bs)e A sin(Bs)Je , qA+B 3−qA+B 3 but γ and γ parametrize the same curve since γ (s)= γ( s). 1 1 − (cid:3) Remark 3.2 The geometric interpretation of equation (3.3) can be obtained as follows. Denote by j the canonical inclusion of γ(R) in (S3,g ). Since T,T = 1, 0 h i a we obtain that τ(j)= a˙ T = aγ +γ and τ (j)= a2γiv +2aγ +(4a 3)γ. T ′′ 2 ′′ ∇ − 8 Therefore γ is a solution of (3.3) if and only if τ (j)+4(1 a)τ(j)= 0. 2 − We notethatRiemannianimmersionsφinspaceformswhichsatisfy theequation τ (φ) = λτ(φ), or equivalently, their mean curvature vectors are eigenvectors of 2 the rough Laplacian, were studied (for example, see [10]). Remark 3.3 In [3] the author studied the biharmonic curves in the Berger spheres S3. It was proved that biharmonic curves parametrized by arc length ǫ are helices and then their parametric equations were derived. We note that the Berger metric is homothetic to g. Therefore, by changing the parameter of the biharmonic curves of S3 we obtain the biharmonic curves of (S3,g), and then we ǫ may select the Legendre ones. However, our method is completely different and the constant vectors are precisely determined. 4 Proper-biharmonic Hopf cylinders Although proper-biharmonic curves in 3-dimensional spaces of non-constant sectional curvature were found, various attempts to find proper-biharmonicsurfaces failed. Now, in the case of Hopf cylinders in (S3,ϕ,ξ,η,g) we can state: Theorem 4.1 The parametric equation of the proper-biharmonic Hopf cylinder S in (S3,ϕ,ξ,η,g), thought as a surface in (R4, , ), is γ¯ h i x= x(u,v) = B cos(Au+ 1v)e B sin(Au+ 1v)Je + qA+B a 1 −qA+B a 1 (4.1) + A cos(Bu 1v)e + A sin(Bu 1v)Je , qA+B − a 3 qA+B − a 3 where e ,e isan orthonormal system of constant vectors inthe Euclidianspace 1 3 { } (R4, , ) with e orthogonal to Je , and 3 1 h i 3 2a 2√(a 1)(a 2)  A= q − − − − a (4.2)  .  3 2a+2√(a 1)(a 2)  B = q − a − −  Proof. We consider the Boothby-Wang fibration π : (S3,g) CP1, where → CP1 is the complex projective space with constant sectional holomorphic curvature 4. Let us denote by ¯, , ˙ and the Levi-Civita connections on CP1, a ∇ ∇ ∇ ∇ (S3,g), (S3,g ) and (R4, , ), respectivelye. 0 h i 9 Let S be a proper-biharmonic Hopf cylinder in (S3,g), where γ¯ : I CP1 is γ¯ → the base curve parametrized by arc length. Using Theorem 2.4, γ¯ has constant curvature κ¯ = √c 1. ± − We denote f = γ¯ and f = J¯f . The horizontal lift fH of f and ξ form a 1 ′ 2 1 1 1 global orthonormal frame field on S , and fH =ϕfH is normal to S . γ¯ 2 1 γ¯ In order to find the explicit parametrization of S as a surface in R4, the γ¯ expressions of ∇f1Hf1H, ∇f1Hξ, ∇f1Hf2H and ∇ξf2H are needed. First, we haeve that e e e ∇f1Hf1H = (∇¯f1f1)H −g(f1H,ϕf1H)= κ¯f2H. Usingagaing( X,Z) = ag ( ˙ X,Z),foranyZ C(TS3)andforanyX X 0 X ∇ ∇ ∈ ∈ C(TS3) orthogonal to ξ, and the properties of the Sasakian structures (ϕ,ξ,η,g) and (ϕ ,ξ ,η ,g ), we get 0 0 0 0 1 1 ∇˙ f1Hf1H = κ¯f2H, ∇˙ f1Hξ = −af2H, ∇˙ f1Hf2H = −κ¯f1H +ξ, ∇˙ ξf2H = af1H. Then 1 1 1 ∇f1Hf1H = κ¯f2H − ax, ∇f1Hξ =−af2H, ∇f1Hf2H = −κ¯f1H +ξ, ∇ξf2H = af1H. e e e e We recall that [ξ,fH] = 0, therefore we can choose a local chart x = x(u,v) 1 such that fH = x and ξ = x . 1 u v After a straightforward computation, assuming that κ¯ = √c 1, we obtain − a2x +a(6 4a)x +x= 0 uuuu uu  − (4.3) .  ax √c 1x +x = 0 uuv u v  − − Thus x = x(u,v) = cos(Au+ 1v)c +sin(Au+ 1v)c + a 1 a 2 (4.4) +cos(Bu 1v)c +sin(Bu 1v)c , − a 3 − a 4 where A, B are given by (4.2) and c are constant vectors in R4. Since i { } x,x = 1, x,x =0, x,x = 0, x ,x = 0, u v u v h i h i h i h i 1 1 x ,x = , x ,x = , x ,x = 0, x ,x = 0, h u ui a h v vi a2 h v uvi h u uvi one obtains, in (u,v) = (0,0), (4.5) c +2c +c = 1 11 13 33 10