Explicit formulas for biharmonic submanifolds in Sasakian space forms
aa r X i v : . [ m a t h . DG ] J un EXPLICIT FORMULAS FOR BIHARMONIC SUBMANIFOLDS INSASAKIAN SPACE FORMS
D. FETCU AND C. ONICIUC
Dedicated to Professor Neculai Papaghiuc on his 60-th birthday
Abstract.
We classify the biharmonic Legendre curves in a Sasakian space form,and obtain their explicit parametric equations in the (2 n + 1)-dimensional unitsphere endowed with the canonical and deformed Sasakian structures defined byTanno. Then, composing with the flow of the Reeb vector field, we transforma biharmonic integral submanifold into a biharmonic anti-invariant submanifold.Using this method we obtain new examples of biharmonic submanifolds in spheresand, in particular, in S . Introduction
Biharmonic maps between Riemannian manifolds φ : ( M, g ) → ( N, h ) are thecritical points of the bienergy functional E ( φ ) = R M | τ ( φ ) | v g . They represent anatural generalization of the well-known harmonic maps ([8]), the critical points ofthe energy functional E ( φ ) = R M | dφ | v g , and of the biharmonic submanifolds inEuclidean spaces defined by B.-Y. Chen ([7]).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 bienergyfunctional was derived by G. Y. Jiang in [14]: τ ( φ ) = − ∆ τ ( φ ) − trace R N ( dφ, τ ( φ )) dφ = 0 . Since any harmonic map is biharmonic, we are interested in non-harmonic bihar-monic maps, which are called proper-biharmonic .A special case of biharmonic maps is represented by the biharmonic Riemannianimmersions, or biharmonic submanifolds. There are several results of classificationor construction for such submanifolds in space forms ([15], [5]). Then, the nextstep would be the study of biharmonic submanifolds in Sasakian space forms. Inthis context J. Inoguchi classified in [13] the proper-biharmonic Legendre curves andHopf cylinders in a 3-dimensional Sasakian space form M ( c ), and in [11] the explicitparametric equations were obtained. Then, T. Sasahara and his collaborators stud-ied the biharmonic integral surfaces and 3-dimensional biharmonic anti-invariantsubmanifolds in S ([1], [17]).Recent results on biharmonic submanifolds in spaces of nonconstant sectionalcurvature were obtained by T. Ichiyama, J. Inoguchi and H. Urakawa in [12], byY.-L. Ou and Z.-P. Wang in [16], and by W. Zhang in [20]. Mathematics Subject Classification.
Key words and phrases.
Biharmonic submanifolds, Sasakian space forms, Legendre curves, inte-gral submanifolds.The authors were partially supported by the Grant CEEX, ET, 5871/2006, Romania.
Biharmonic submanifolds in pseudo-Euclidean spaces were also studied, and manyexamples and classification results were obtained (for example, see [2], [7]).The goals of our paper are to obtain new classification results for biharmonicLegendre curves in any dimensional Sasakian space form and to provide a methodfor constructing biharmonic submanifolds. In order to obtain explicit examples, weuse the (2 n + 1)-dimensional unit sphere S n +1 as a model of Sasakian space form.For a general account of biharmonic maps see [15] and The Bibliography of Bi-harmonic Maps [18].
Conventions.
We work in the C ∞ category, that means manifolds, metrics, con-nections and maps are smooth. The Lie algebra of the vector fields on M is denotedby C ( T M ). 2.
Preliminaries
Contact manifolds. A contact metric structure on a manifold N n +1 is givenby ( ϕ, ξ, η, g ), where ϕ is a tensor field of type (1 ,
1) on N , ξ is a vector field, η isan 1-form and g is a Riemannian metric such that ϕ = − I + η ⊗ ξ, η ( ξ ) = 1 ,g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y ) , g ( X, ϕY ) = dη ( X, Y ) , ∀ X, Y ∈ C ( T N ) . A contact metric structure ( ϕ, ξ, η, g ) is called normal if N ϕ + 2 dη ⊗ ξ = 0 , where N ϕ ( X, Y ) = [ ϕX, ϕY ] − ϕ [ ϕX, Y ] − ϕ [ X, ϕY ] + ϕ [ X, Y ] , ∀ X, Y ∈ C ( T N ) , is the Nijenhuis tensor field of ϕ .A contact metric manifold ( N, ϕ, ξ, η, g ) is a
Sasakian manifold if it is normal or,equivalently, if ( ∇ X ϕ )( Y ) = g ( X, Y ) ξ − η ( Y ) X, ∀ X, Y ∈ C ( T N ) . The contact distribution of a Sasakian manifold (
N, ϕ, ξ, η, g ) is defined by { X ∈ T N : η ( X ) = 0 } , and an integral curve of the contact distribution is called Legendrecurve . A submanifold M of N which is tangent to ξ is said to be anti-invariant if ϕ maps any vector tangent to M and normal to ξ to a vector normal to M .Let ( N, ϕ, ξ, η, g ) be a Sasakian manifold. The sectional curvature of a 2-planegenerated 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 -sectionalcurvature c is called a Sasakian space form and it is denoted by N ( c ).The curvature tensor field of a Sasakian space form N ( c ) is given by(2.1) R ( X, Y ) Z = c +34 { g ( Z, Y ) X − g ( Z, X ) Y } + c − { η ( Z ) η ( X ) Y −− 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 + 2 g ( X, ϕY ) ϕZ } . Let S n +1 = { z ∈ C n +1 : | z | = 1 } be the unit (2 n +1)-dimensional sphere endowedwith its standard metric field g . Consider the following structure tensor fields on S n +1 : ξ = −I z for each z ∈ S n +1 , where I is the usual almost complex structureon C n +1 defined by I z = ( − y , ..., − y n +1 , x , ..., x n +1 ) , IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 3 for z = ( x , ..., x n +1 , y , ..., y n +1 ), and ϕ = s ◦ I , where s : T z C n +1 → T z S n +1 denotes the orthogonal projection. Equipped with these tensors, S n +1 becomes aSasakian space form with ϕ -sectional curvature equal to 1 ([3]).Now, consider the deformed structure on S n +1 , introduced by Tanno in [13], η = aη , ξ = 1 a ξ , ϕ = ϕ , g = ag + a ( a − η ⊗ η , where a is a positive constant. The structure ( ϕ, ξ, η, g ) is still a Sasakian structureand ( S n +1 , ϕ, ξ, η, g ) is a Sasakian space form with constant ϕ -sectional curvature c = a − If a manifold N admits three Sasakian structures( ϕ a , ξ a , η a , g ), a = 1 , ,
3, satisfying ϕ c = − ϕ a ϕ b + η b ⊗ ξ a = ϕ b ϕ a − η a ⊗ ξ b ,ξ c = − ϕ a ξ b = ϕ b ξ a , η c = − η a ◦ ϕ b = η b ◦ ϕ a , for an even permutation ( a, b, c ) of (1 , , Sasakian 3-structure . The dimension of such a manifold is of the form 4 n + 3. Wenote that the maximum dimension of a submanifold of a 3-Sasakian manifold N n +3 which is an integral submanifold with respect to all three Sasakian structures is n .3. Biharmonic Legendre curves in Sasakian space forms
Definition 3.1.
Let ( N m , g ) be a Riemannian manifold and γ : I → N a curveparametrized by arc length, that is | γ ′ | = 1. Then γ is called a Frenet curve ofosculating order r , 1 ≤ r ≤ m , if there exists orthonormal vector fields E , E , ..., E r along γ such that E = γ ′ = T ∇ T E = κ E ∇ T E = − κ E + κ E ... ∇ T E r = − κ r − E r − , where κ , ..., κ r − are positive functions on I . Remark 3.2.
A geodesic is a Frenet curve of osculating order 1; a circle is a Frenetcurve of osculating order 2 with κ = constant; a helix of order r , r ≥
3, is a Frenetcurve of osculating order r with κ , ..., κ r − constants; a helix of order 3 is called,simply, helix.Now let ( N n +1 , ϕ, ξ, η, g ) be a Sasakian space form with constant ϕ -sectionalcurvature c and γ : I → N a Legendre Frenet curve of osculating order r . As(3.1) ∇ T T = ( − κ κ ′ ) E + ( κ ′′ − κ − κ κ ) E + (2 κ ′ κ + κ κ ′ ) E + κ κ κ E and(3.2) R ( T, ∇ T T ) T = − ( c + 3) κ E − c − κ g ( E , ϕT ) ϕT, D. FETCU AND C. ONICIUC we get(3.3) τ ( γ ) = ∇ T T − R ( T, ∇ T T ) T = ( − κ κ ′ ) E + (cid:16) κ ′′ − κ − κ κ + ( c +3) κ (cid:17) E +(2 κ ′ κ + κ κ ′ ) E + κ κ κ E + c − κ g ( E , ϕT ) ϕT. In the following we shall solve the biharmonic equation τ ( γ ) = 0. The problem isto find the relation between ϕT and the Frenet frame field. The simplest two casesare provided by c − κ g ( E , ϕT ) = 0. So, Case I: c = In this case γ is proper-biharmonic if and only if κ = constant > , κ = constant κ + κ = 1 κ κ = 0 . One obtains
Theorem 3.3. If c = 1 and n ≥ , then γ is proper-biharmonic if and only if either γ is a circle with κ = 1 , or γ is a helix with κ + κ = 1 . Remark 3.4. If n = 1 and γ is a non-geodesic Legendre curve we have ∇ T T = ± κ ϕT and then E = ± ϕT and ∇ T E = ±∇ T ϕT = ± ( ξ ∓ κ T ) = − κ T ± ξ .Therefore κ = 1 and γ cannot be biharmonic. Case II: c = , E ⊥ ϕ T. In this case γ is proper-biharmonic if and only if κ = constant > , κ = constant κ + κ = c +34 κ κ = 0 . Before stating the theorem we need the following
Lemma 3.5.
Let γ be a Legendre Frenet curve of osculating order 3 and E ⊥ ϕT .Then { T = E , E , E , ϕT, ξ, ∇ T ϕT } are linearly independent, in any point, andhence n ≥ .Proof. Since γ is a Frenet curve of osculating order 3, we have E = γ ′ = T ∇ T E = κ E ∇ T E = − κ E + κ E ∇ T E = − κ E . It is easy to see that, in an arbitrary point, the system S = { T = E , E , E , ϕT, ξ, ∇ T ϕT } IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 5 has only non-zero vectors and T ⊥ E , T ⊥ E , T ⊥ ϕT, T ⊥ ξ, T ⊥ ∇ T ϕT. Thus S is linearly independent if and only if S = { E , E , ϕT, ξ, ∇ T ϕT } is linearlyindependent. Further, as E ⊥ ξ, E ⊥ ∇ T ϕT, E ⊥ ξ, E ⊥ ∇ T ϕT, ϕT ⊥ ξ, ϕT ⊥ ∇ T ϕT, and E ⊥ E ⊥ ϕT, it follows that S is linearly independent if and only if S = { ξ, ∇ T ϕT } is linearlyindependent. But ∇ T ϕT = ξ + κ ϕE , κ = 0, and therefore S is linearly indepen-dent. (cid:3) Now we can state
Theorem 3.6.
Assume that c = 1 and ∇ T T ⊥ ϕT . We have1) If c ≤ − then γ is biharmonic if and only if it is a geodesic.2) If c > − then γ is proper-biharmonic if and only if eithera) n ≥ and γ is a circle with κ = c +34 . In this case { E , E , ϕT, ξ } are linearlyindependent,orb) n ≥ and γ is a helix with κ + κ = c +34 . In this case { E , E , E , ϕT,ξ, ∇ T ϕT } are linearly independent. Case III: c = , E k ϕ T. In this case γ is proper-biharmonic if and only if κ = constant > , κ = constant κ + κ = cκ κ = 0 . We can assume that E = ϕT . Then we have ∇ T T = κ E = κ ϕT , ∇ T E = ∇ T ϕT = ξ − κ T . That means E = ξ and κ = 1. Hence ∇ T E = ∇ T ξ = − ϕT = − E .Therefore Theorem 3.7. If c = 1 and ∇ T T k ϕT , then { T, ϕT, ξ } is the Frenet frame field of γ and we have1) If c ≤ then γ is biharmonic if and only if it is a geodesic.2) If c > then γ is proper-biharmonic if and only if it is a helix with κ = c − (and κ = 1 ). Remark 3.8. If n = 1 then ∇ T T k ϕT and we reobtain Inoguchi’s result [13]. Case IV: c = ( E , ϕ T ) is not constant 0 , − Assume that γ is a Legendre Frenet curve of osculating order r , 4 ≤ r ≤ n + 1, n ≥
2. If γ is biharmonic it follows that ϕT ∈ span { E , E , E } .Now, we denote f ( t ) = g ( E , ϕT ) and differentiating along γ we obtain f ′ ( t ) = g ( ∇ T E , ϕT ) + g ( E , ∇ T ϕT ) = g ( ∇ T E , ϕT ) + g ( E , ξ + κ ϕE )= g ( ∇ T E , ϕT ) = g ( − κ T + κ E , ϕT )= κ g ( E , ϕT ) . D. FETCU AND C. ONICIUC
Since ϕT = g ( ϕT, E ) E + g ( ϕT, E ) E + g ( ϕT, E ) E , the curve γ is proper-biharmonic if and only if κ = constant > κ + κ = c +34 + c − f κ ′ = − c − f g ( ϕT, E ) κ κ = − c − f g ( ϕT, E ) . Using the expression of f ′ ( t ) we see that the third equation of the above system isequivalent to κ = − c − f + ω , where ω = constant. Replacing in the second equation it follows κ = c + 34 − ω + 3( c − f , which implies f = constant. Thus κ = constant > g ( E , ϕT ) = 0 and then ϕT = f E + g ( ϕT, E ) E . It follows that there exists an unique constant α ∈ (0 , π ) \ { π , π, π } such that f = cos α and g ( ϕT, E ) = sin α .We can state Theorem 3.9.
Let c = 1 , n ≥ and γ a Legendre Frenet curve of osculating order r ≥ such that g ( E , ϕT ) is not constant , or − . We havea) If c ≤ − then γ is biharmonic if and only if it is a geodesic.b) If c > − then γ is proper-biharmonic if and only if ϕT = cos α E + sin α E and κ = constant > , κ = constant κ + κ = c +34 + c − cos α κ κ = − c − sin 2 α , where α ∈ (0 , π ) \ { π , π, π } is a constant such that c + 3 + 3( c −
1) cos α > and c −
1) sin 2 α < . Remark 3.10.
In this case we may obtain biharmonic curves which are not helices.
Remark 3.11.
We note that a preliminary version of the full classification of theproper-biharmonic Legendre curves in Sasakian space forms was obtained in [10].In the following, we shall choose the unit (2 n + 1)-dimensional sphere S n +1 withits canonical and modified Sasakian structures as a model for the complete, simplyconnected Sasakian space form with constant ϕ -sectional curvature c > −
3, and wewill find the explicit equations of biharmonic Legendre curves obtained in the firstthree cases, viewed as curves in R n +2 . Theorem 3.12.
Let γ : I → ( S n +1 , ϕ , ξ , η , g ) , n ≥ , be a proper-biharmonicLegendre curve parametrized by arc length. Then the equation of γ in the Euclideanspace E n +2 = ( R n +2 , h , i ) , is either γ ( s ) = 1 √ (cid:16) √ s (cid:17) e + 1 √ (cid:16) √ s (cid:17) e + 1 √ e where { e i , I e j } are constant unit vectors orthogonal to each other, or γ ( s ) = 1 √ As ) e + 1 √ As ) e + 1 √ Bs ) e + 1 √ Bs ) e , where (3.4) A = √ κ , B = √ − κ , κ ∈ (0 , , and { e i } are constant unit vectors orthogonal to each other, with h e , I e i = h e , I e i = h e , I e i = h e , I e i = 0 , A h e , I e i + B h e , I e i = 0 . Proof.
Let us denote by ˙ ∇ and by e ∇ the Levi-Civita connections on ( S n +1 , g ) and( R n +2 , h , i ), respectively.First, assume that γ is the biharmonic circle, that is κ = 1. From the Gauss andFrenet equations we get e ∇ T T = ˙ ∇ T T − h T, T i γ = κ E − γ and e ∇ T e ∇ T T = ( − κ − T = − T, which implies γ ′′′ + 2 γ ′ = 0 . The general solution of the above equation is γ ( s ) = cos (cid:16) √ s (cid:17) c + sin (cid:16) √ s (cid:17) c + c , where { c i } are constant vectors in E n +2 .Now, as γ satisfies h γ, γ i = 1 , h γ ′ , γ ′ i = 1 , h γ, γ ′ i = 0 , h γ ′ , γ ′′ i = 0 , h γ ′′ , γ ′′ i = 2 , h γ, γ ′′ i = − , and since in s = 0 we have γ = c + c , γ ′ = √ c , γ ′′ = − c , we obtain c + 2 c + c = 1 , c = 12 , c + c = 0 , c = 0 , c = 12 , c + c = 12 , where c ij = h c i , c j i . Further, we get that { c i } are orthogonal vectors in E n +2 with | c | = | c | = | c | = √ .Finally, using the fact that γ is a Legendre curve one obtains easily that h c i , I c j i = 0for any i, j = 1 , , γ is the biharmonic helix, that is κ + κ = 1, κ ∈ (0 , e ∇ T T = ˙ ∇ T T − h T, T i γ = κ E − γ, e ∇ T e ∇ T T = κ e ∇ T E − T = κ (cid:16) − κ T + κ E (cid:17) − T = − (cid:16) κ + 1 (cid:17) T + κ κ E and e ∇ T e ∇ T e ∇ T T = − (cid:16) κ +1 (cid:17) e ∇ T T + κ κ e ∇ T E = − (cid:16) κ +1 (cid:17) e ∇ T T − κ κ E = − γ ′′ − κ γ. Hence γ iv + 2 γ ′′ + κ γ = 0 , and its general solution is γ ( s ) = cos( As ) c + sin( As ) c + cos( Bs ) c + sin( Bs ) c , where A , B are given by (3.4) and { c i } are constant vectors in E n +2 . D. FETCU AND C. ONICIUC As γ satisfies h γ, γ i = 1 , h γ ′ , γ ′ i = 1 , h γ, γ ′ i = 0 , h γ ′ , γ ′′ i = 0 , h γ ′′ , γ ′′ i = 1 + κ , h γ, γ ′′ i = − , h γ ′ , γ ′′′ i = − (1 + κ ) , h γ ′′ , γ ′′′ i = 0 , h γ, γ ′′′ i = 0 , h γ ′′′ , γ ′′′ i = 3 κ + 1 , and since in s = 0 we have γ = c + c , γ ′ = Ac + Bc , γ ′′ = − A c − B c , γ ′′′ = − A c − B c , we obtain(3.5) c + 2 c + c = 1(3.6) A c + 2 ABc + B c = 1(3.7) Ac + Ac + Bc + Bc = 0(3.8) A c + AB c + A Bc + B c = 0(3.9) A c + 2 A B c + B c = 1 + κ (3.10) A c + ( A + B ) c + B c = 1(3.11) A c + ( AB + A B ) c + B c = 1 + κ (3.12) A c + A B c + A B c + B c = 0(3.13) A c + A c + B c + B c = 0(3.14) A c + 2 A B c + B c = 3 κ + 1where c ij = h c i , c j i . Since the determinant of the system given by (3.7), (3.8), (3.12)and (3.13) is − A B ( A − B ) = 0 it follows that c = c = c = c = 0 . The equations (3.5), (3.9) and (3.10) give c = 12 , c = 0 , c = 12 , and, from (3.6), (3.11) and (3.14) follows that c = 12 , c = 0 , c = 12 . Therefore, we obtain that { c i } are orthogonal vectors in E with | c | = | c | = | c | = | c | = √ .Finally, since γ is a Legendre curve one obtains the conclusion of the Theorem. (cid:3) Theorem 3.13.
Let γ : I → ( S n +1 , ϕ, ξ, η, g ) , n ≥ , a > , a = 1 , be a proper-biharmonic Legendre curve parametrized by arc length such that g ( ∇ γ ′ γ ′ , ϕγ ′ ) = 0 .Then the equation of γ in the Euclidean space E n +2 , is either γ ( s ) = 1 √ (cid:16)r a s (cid:17) e + 1 √ (cid:16)r a s (cid:17) e + 1 √ e for n ≥ or, for n ≥ , γ ( s ) = 1 √ As ) e + 1 √ As ) e + 1 √ Bs ) e + 1 √ Bs ) e , where (3.15) A = r κ √ aa , B = r − κ √ aa , κ ∈ (cid:16) , a (cid:17) , IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 9 and { e i , I e j } are constant unit vectors orthogonal to each other.Proof. Let us denote by ∇ , ˙ ∇ and by e ∇ the Levi-Civita connections on ( S n +1 , g ),( S n +1 , g ) and ( R n +2 , h , i ), respectively.First we consider the case when γ is the biharmonic circle, that is κ = c +34 . Let T = γ ′ be the unit tangent vector field (with respect to the metric g ) along γ . Usingthe two Sasakian structures on S n +1 we obtain ˙ ∇ T T = ∇ T T and ˙ ∇ T E = ∇ T E .From the Gauss and Frenet equations we get e ∇ T T = ˙ ∇ T T − h T, T i γ = κ E − a γ and e ∇ T e ∇ T T = ( − κ − a ) T = − a T. Hence aγ ′′′ + 2 γ ′ = 0 , with the general solution γ ( s ) = cos (cid:16)r a s (cid:17) c + sin (cid:16)r a s (cid:17) c + c , where { c i } are constant vectors in E n +2 .As γ verifies the following equations, h γ, γ i = 1 , h γ ′ , γ ′ i = 1 a , h γ, γ ′ i = 0 , h γ ′ , γ ′′ i = 0 , h γ ′′ , γ ′′ i = 2 a , h γ, γ ′′ i = − a , and in s = 0 we have γ = c + c , γ ′ = q a c , γ ′′ = − a c , one obtains c + 2 c + c = 1 , c = 12 , c + c = 0 , c = 0 , c = 12 , c + c = 12 , where c ij = h c i , c j i . Consequently, we obtain that { c i } are orthogonal vectors in E n +2 with | c | = | c | = | c | = √ .Finally, using the facts that γ is a Legendre curve and g ( ∇ γ ′ γ ′ , ϕγ ′ ) = 0 one obtainseasily that h c i , I c j i = 0 for any i, j = 1 , , γ is a biharmonic helix, that is κ + κ = c +34 , κ ∈ (cid:16) , c +34 (cid:17) .First we obtain ˙ ∇ T T = ∇ T T , ˙ ∇ T E = ∇ T E and ˙ ∇ T E = ∇ T E .From the Gauss and Frenet equations we get e ∇ T T = ˙ ∇ T T − h T, T i γ = κ E − a γ, e ∇ T e ∇ T T = κ e ∇ T E − a T = κ (cid:16) − κ T + κ E (cid:17) − a T = − (cid:16) κ + 1 a (cid:17) T + κ κ E , and e ∇ T e ∇ T e ∇ T T = − (cid:16) κ + a (cid:17) e ∇ T T + κ κ e ∇ T E = − (cid:16) κ + a (cid:17) e ∇ T T − κ κ E = − a γ ′′ − a κ γ. Therefore aγ iv + 2 γ ′′ + κ γ = 0 , and its general solution is γ ( s ) = cos( As ) c + sin( As ) c + cos( Bs ) c + sin( Bs ) c , where A , B are given by (3.15) and { c i } are constant vectors in E n +2 . The curve γ satisfies h γ, γ i = 1 , h γ ′ , γ ′ i = 1 a , h γ, γ ′ i = 0 , h γ ′ , γ ′′ i = 0 , h γ ′′ , γ ′′ i = 1 + aκ a , h γ, γ ′′ i = − a , h γ ′ , γ ′′′ i = − aκ a , h γ ′′ , γ ′′′ i = 0 , h γ, γ ′′′ i = 0 , h γ ′′′ , γ ′′′ i = 3 aκ + 1 a , and in s = 0 we have γ = c + c , γ ′ = Ac + Bc , γ ′′ = − A c − B c , γ ′′′ = − A c − B c . Then, it follows(3.16) c + 2 c + c = 1(3.17) A c + 2 ABc + B c = 1 a (3.18) Ac + Ac + Bc + Bc = 0(3.19) A c + AB c + A Bc + B c = 0(3.20) A c + 2 A B c + B c = 1 + aκ a (3.21) A c + ( A + B ) c + B c = 1 a (3.22) A c + ( AB + A B ) c + B c = 1 + aκ a (3.23) A c + A B c + A B c + B c = 0(3.24) A c + A c + B c + B c = 0(3.25) A c + 2 A B c + B c = 3 aκ + 1 a where c ij = h c i , c j i .The solution of the system given by (3.18), (3.19), (3.23) and (3.24) is c = c = c = c = 0 . From equations (3.16), (3.20) and (3.21) we get c = 12 , c = 0 , c = 12 , and, from (3.17), (3.22), (3.25), c = 12 , c = 0 , c = 12 . We obtain that { c i } are orthogonal vectors in E with | c | = | c | = | c | = | c | = √ .Finally, since γ is a Legendre curve and g ( ∇ γ ′ γ ′ , ϕγ ′ ) = 0, one obtains the conclusion. (cid:3) Just like for S (see [11]) one obtains for S n +1 endowed with the modified Sasakianstructure, with 0 < a < c > IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 11
Theorem 3.14.
Let γ : I → ( S n +1 , ϕ, ξ, η, g ) be a biharmonic Legendre curveparametrized by its arc length such that ∇ γ ′ γ ′ = √ c − ϕγ ′ . Then the equation of γ in the Euclidean space E n +2 is γ ( s ) = q BA + B cos( As ) e − q BA + B sin( As ) I e + q AA + B cos( Bs ) e + q AA + B sin( Bs ) I e = q BA + B exp( − i As ) e + q AA + B exp(i Bs ) e , where { e i } are constant unit orthogonal vectors in E n +2 and (3.26) A = s − a − p ( a − a − a , B = s − a + 2 p ( a − a − a . Remark 3.15.
The ODE satisfied by proper-biharmonic Legendre curves in the(2 n + 1)-sphere, in the fourth case, may be also obtained but the computations arerather complicated.4. Biharmonic submanifolds in Sasakian space forms
A method to obtain biharmonic submanifolds in a Sasakian space form is providedby the following Theorem.
Theorem 4.1.
Let ( N n +1 , ϕ, ξ, η, g ) be a strictly regular Sasakian space form withconstant ϕ -sectional curvature c and let i : M → N be an r -dimensional integralsubmanifold of N . Consider F : f M = I × M → N, F ( t, p ) = φ t ( p ) = φ p ( t ) , where I = S or I = R and { φ t } t ∈ R is the flow of the vector field ξ . Then F :( f M , e g = dt + i ∗ g ) → N is a Riemannian immersion and it is proper-biharmonic ifand only if M is a proper-biharmonic submanifold of N .Proof. From the definition of the flow of ξ we have dF ( t, p ) (cid:16) ∂∂t (cid:17) = dds | s = t { φ p ( s ) } = ˙ φ p ( t ) = ξ ( φ p ( t )) = ξ ( F ( t, p )) , i.e. ∂∂t is F -correlated to ξ and (cid:12)(cid:12)(cid:12) dF ( t, p ) (cid:16) ∂∂t (cid:17)(cid:12)(cid:12)(cid:12) = | ξ ( F ( t, p )) | = 1 = (cid:12)(cid:12)(cid:12) ∂∂t (cid:12)(cid:12)(cid:12) . The vector X p ∈ T p M can be identified to (0 , X p ) ∈ T ( t ′ ,p ) ( I × M ), for any t ′ ∈ I ,and we have dF ( t,p ) ( X p ) = ( dF ) ( t,p ) ( ˙ γ (0)) = dds | s =0 { φ t ( γ ( s )) } = ( dφ t ) p ( X p ) . Since φ t is an isometry, we have | dF ( t,p ) ( X p ) | = | ( dφ t ) p ( X p ) | = | X p | .Moreover, g (cid:16) dF ( t,p ) (cid:16) ∂∂t (cid:17) , dF ( t,p ) ( X p ) (cid:17) = g ( ξ ( φ p ( t )) , ( dφ t ) p ( X p ))= g (( dφ t ) p ( ξ p ) , ( dφ t ) p ( X p )) = g ( ξ p , X p )= 0 , so F : ( I × M, dt + i ∗ g ) → N is a Riemannian immersion.Let F − ( T N ) be the pull-back bundle over f M and ∇ F the pull-back connectiondetermined by the Levi-Civita connection on N . We shall prove that τ ( F ) ( t,p ) = ( dφ t ) p ( τ ( i )) and τ ( F ) ( t,p ) = ( dφ t ) p ( τ ( i )) , so, from the point of view of harmonicity and biharmonicity, f M and M have thesame behaviour.We start with two remarks. Let σ ∈ F − ( T N ) defined by σ ( t,p ) = ( dφ t ) p ( Z p ),where Z is a vector field along M , that is Z p ∈ T p N , ∀ p ∈ M . We have(4.1) ( ∇ FX σ ) ( t,p ) = ( dφ t ) p ( ∇ NX Z ) , ∀ X ∈ C ( T M ) . Then, if σ ∈ F − ( T N ), it follows that ϕσ ∈ F − ( T N ), ( ϕσ ) ( t,p ) = ϕ φ p ( t ) ( σ ( t,p ) ),and(4.2) ∇ F ∂∂t ϕσ = ϕ ∇ F ∂∂t σ. Now, we consider { X , ..., X r } a local orthonormal frame field on U , where U is anopen subset of M . The tension field of F is given by τ ( F ) = ∇ F ∂∂t dF (cid:16) ∂∂t (cid:17) − dF (cid:16) ∇ f M ∂∂t ∂∂t (cid:17) + r X a =1 {∇ FX a dF ( X a ) − dF ( ∇ f MX a X a ) } . As ∇ F ∂∂t dF (cid:16) ∂∂t (cid:17) = ∇ Nξ ξ = 0 , ∇ f M ∂∂t ∂∂t = ∇ I ∂∂t ∂∂t = 0 , ( ∇ FX a dF ( X a )) ( t,p ) = ( dφ t ) p ( ∇ NX a X a ) , dF ( t,p ) ( ∇ f MX a X a ) = ( dφ t ) p ( ∇ MX a X a )we obtain τ ( F ) ( t,p ) = ( dφ t ) p ( τ ( i )) . The next step is to prove that ∇ F ∂∂t τ ( F ) = − ϕ ( τ ( F )). Since [ ∂∂t , X a ] = 0, a = 1 , ..., r , ∇ F ∂∂t dF ( X a ) = ∇ FX a dF (cid:16) ∂∂t (cid:17) . But (cid:16) ∇ FX a dF (cid:16) ∂∂t (cid:17)(cid:17) ( t,p ) = ∇ NdF ( t,p ) X a ξ = ∇ N ( dφ t ) p X a ξ = − ϕ (( dφ t ) p ( X a ))= − ( dφ t ) p ( ϕX a ) , so(4.3) (cid:16) ∇ F ∂∂t dF ( X a ) (cid:17) ( t,p ) = − ( dφ t ) p ( ϕX a ) . We note that R F (cid:16) ∂∂t , X a (cid:17) dF ( X a ) = ∇ F ∂∂t ∇ FX a dF ( X a ) − ∇ FX a ∇ F ∂∂t dF ( X a )and, on the other hand (cid:16) R F (cid:16) ∂∂t , X a (cid:17) dF ( X a ) (cid:17) ( t,p ) = R Nφ t ( p ) ( ξ, ( dφ t ) p ( X a ))( dφ t ) p ( X a ) = ξ. Therefore(4.4) ∇ F ∂∂t ∇ FX a dF ( X a ) − ∇ FX a ∇ F ∂∂t dF ( X a ) = ξ IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 13
From (4.1) and (4.3) we get(4.5) (cid:16) ∇ FX a ∇ F ∂∂t dF ( X a ) (cid:17) ( t,p ) = − ( dφ t ) p ( ∇ NX a ϕX a )= − ( dφ t ) p ( ξ + ϕ ∇ NX a X a )and from (4.3)(4.6) (cid:16) ∇ F ∂∂t dF ( ∇ f MX a X a ) (cid:17) ( t,p ) = (cid:16) ∇ F ∂∂t dF ( ∇ MX a X a ) (cid:17) ( t,p ) = − ( dφ t ) p ( ϕ ∇ MX a X a ) . Replacing (4.5) and (4.6) in (4.4), we obtain ξ = ∇ F ∂∂t ∇ FX a dF ( X a ) − ∇ F ∂∂t dF ( ∇ f MX a X a ) + ∇ F ∂∂t dF ( ∇ f MX a X a ) − ∇ FX a ∇ F ∂∂t dF ( X a )= ∇ F ∂∂t ∇ dF ( X a , X a ) − ( dφ t ) p ( ϕ ∇ MX a X a ) + ( dφ t ) p ( ξ + ϕ ∇ NX a X a ) , so (cid:16) ∇ F ∂∂t ∇ FX a dF ( X a ) (cid:17) ( t,p ) = − ϕ ( dφ t ) p ( ∇ d i ( X a , X a )) . As ∇ dF ( X a , X a ) = 0, we obtain ∇ F ∂∂t τ ( F ) = − ϕ ( τ ( F )) . From (4.2) we have(4.7) ∇ F ∂∂t ∇ F ∂∂t τ ( F ) = −∇ F ∂∂t ϕ ( τ ( F )) = − ϕ ∇ F ∂∂t τ ( F ) = ϕ τ ( F )= − τ ( F ) , and from (4.1)(4.8) ( ∇ FX a ∇ FX a τ ( F )) ( t,p ) = ( dφ t ) p ( ∇ NX a ∇ NX a τ ( i )) , (4.9) (cid:16) ∇ F ∇ f MXa X a τ ( F ) (cid:17) ( t,p ) = ( dφ t ) p (cid:16) ∇ N ∇ MXa X a τ ( i ) (cid:17) . From (4.7), (4.8) and (4.9) we obtain(4.10) − (∆ F τ ( F )) ( t,p ) = − τ ( F ) ( t,p ) − ( dφ t ) p (∆ i τ ( i )) . After a straightforward computation we get(4.11) trace R F ( dF, τ ( F )) dF = − τ ( F ) + ( dφ t ) p (trace R Np ( d i , τ ( i )) d i ) . Finally, from (4.10) and (4.11) we obtain τ ( F ) ( t,p ) = ( dφ t ) p ( τ ( i )) . (cid:3) Remark 4.2.
The previous result was expected because of the following simpleremark. Assume that ( N n +1 , ϕ, ξ, η, g ) is a compact strictly regular Sasakian spaceform with constant ϕ -sectional curvature c and let G : M → N be an arbitrarysmooth map from a compact Riemannian manifold M . If F is biharmonic, then themap G is biharmonic, where F : f M = S × M → N , F ( t, p ) = φ t ( G ( p )). Indeed, an arbitrary variation { G s } s of G induces a variation { F s } s of F . We havethat τ ( p,t ) ( F s ) = ( dφ t ) G s ( p ) ( τ p ( G s )) and, from the biharmonicity of F and the FubiniTheorem, we get0 = dds | s =0 { E ( F s ) } = 12 dds | s =0 Z f M | τ ( F s ) | v e g = π dds | s =0 Z M | τ ( G s ) | v g = π dds | s =0 { E ( G s ) } . Next, consider the unit (2 n +1)-dimensional sphere S n +1 endowed with its canon-ical or modified Sasakian structure. The flow of ξ is φ t ( z ) = exp( − i ta ) z , where z = ( z , ..., z n +1 ) = ( x , ..., x n +1 , y , ..., y n +1 ) . From the above expression of the flow and from Theorems 3.14, 3.13, 3.12 and 4.1we obtain explicit examples of proper-biharmonic submanifolds in ( S n +1 , ϕ, ξ, η, g ), a >
0, of constant mean curvature. In particular, we reobtain a result in [1] and, for n = 1, the result in [11]. Proposition 4.3 ([1]) . Let F : f M → S ⊂ R be a proper-biharmonic anti-invariantimmersion. Then the position vector of f M in R is F ( t, u, v ) = exp( − i t ) √ u ) , i exp( − i u ) sin( √ v ) , i exp( − i u ) cos( √ v )) . Proof.
It was proved in [17] that the proper-biharmonic integral surface of ( S ,ϕ , ξ , η , g ) is given by f ( u, v ) = 1 √ u ) , i exp( − i u ) sin √ v, i exp( − i u ) cos √ v ) . Now, composing with the flow of ξ we reobtain the result in [1]. (cid:3) Proposition 4.4 ([11]) . Let M be a surface in ( S n +1 , ϕ, ξ, η, g ) with a ∈ (0 , ,with the position vector in the Euclidean space E n +2 given by F ( t, s ) = exp (cid:16) − i ta (cid:17)(cid:16)r BA + B exp( − i As ) e + r AA + B exp(i Bs ) e (cid:17) , where { e , e } is an orthonormal system of constant vectors in the Euclidean space ( R n +2 , h , i ) , with e orthogonal to I e and A , B are given by (3.26).Then M is a proper-biharmonic surface in S n +1 ( c ) . Proper-biharmonic submanifolds of ( S , g )We consider the Euclidean space E with three complex structures, I = (cid:18) − I I (cid:19) , J = I − I I − I , K = −IJ , where I n denotes the n × n identity matrix. We define three vector fields on S by ξ = −I z, ξ = −J z, ξ = −K z, z ∈ S , and consider their dual 1-forms η = η , η , η . Let ϕ a defined by ϕ = ϕ = s ◦ I , ϕ = s ◦ J , ϕ = s ◦ K . Then ( ϕ a , ξ a , η a , g ), a = 1 , ,
3, determine a Sasakian 3-structure on S . IHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS 15
Now, we shall indicate a method to construct proper-biharmonic submanifolds in( S , g ). We consider γ = γ ( s ) a proper-biharmonic curve in ( S , g ), parametrizedby arc-length, which is a Legendre curve for two of the three contact structures (itwas proved in [9] that there is no proper-biharmonic curve which is Legendre withrespect to all three contact structures on S ). For example, assume that γ is aLegendre curve for η and η . Composing with the flow of ξ (or ξ ) we obtain abiharmonic surface which is Legendre with respect to η (or η ). Then, composingwith the flow of ξ (or ξ ) we get a biharmonic 3-dimensional submanifold of ( S , g ).Using this method, from Theorems 3.12 and 4.1, we obtain 4 classes of proper-biharmonic surfaces in ( S , g ) and 4 classes of proper-biharmonic 3-dimensionalsubmanifolds of ( S , g ), all of constant mean curvature.For example, from Theorems 3.12 and 4.1, composing first with the flow of ξ andthen with that of ξ , we get the explicit parametric equations of proper-biharmonic3-dimensional submanifolds of ( S , g ). Proposition 5.1.
Let M be a -dimensional submanifold in S such that its positionvector field in E is either x = x ( u, t, s )= √ (cid:16) cos( u ) cos( √ s ) cos( t ) e + cos( u ) sin( √ s ) cos( t ) e + cos( u ) cos( t ) e − cos( u ) cos( √ s ) sin( t ) I e − cos( u ) sin( √ s ) sin( t ) I e − cos( u ) sin( t ) I e − sin( u ) cos( √ s ) cos( t ) J e − sin( u ) sin( √ s ) cos( t ) J e − sin( u ) cos( t ) J e − sin( u ) cos( √ s ) sin( t ) K e − sin( u ) sin( √ s ) sin( t ) K e − sin( u ) sin( t ) K e (cid:17) , where { e i , I e j } , { e i , J e j } are systems of constant orthonormal vectors in E , or x = x ( u, t, s )= √ (cid:16) cos( u ) cos( As ) cos( t ) e + cos( u ) sin( As ) cos( t ) e + cos( u ) cos( Bs ) cos( t ) e + cos( u ) sin( Bs ) cos( t ) e − cos( u ) cos( As ) sin( t ) I e − cos( u ) sin( As ) sin( t ) I e − cos( u ) cos( Bs ) sin( t ) I e − cos( u ) sin( Bs ) sin( t ) I e − sin( u ) cos( As ) cos( t ) J e − sin( u ) sin( As ) cos( t ) J e − sin( u ) cos( Bs ) cos( t ) J e − sin( u ) sin( Bs ) cos( t ) J e − sin( u ) cos( As ) sin( t ) K e − sin( u ) sin( As ) sin( t ) K e − sin( u ) cos( Bs ) sin( t ) K e − sin( u ) sin( Bs ) sin( t ) K e (cid:17) , where A = √ κ , B = √ − κ , κ = constant ∈ (0 , , and { e i } are constant orthonormal vectors in E such that h e , I e i = h e , I e i = h e , I e i = h e , I e i = 0 , h e , J e i = h e , J e i = h e , J e i = h e , J e i = 0 , and A h e , I e i + B h e , I e i = A h e , J e i + B h e , J e i = 0 . Then M is a proper-biharmonic submanifold of ( S , g ) .Proof. As the flows of ξ and ξ are given by φ t ( z ) = (cos t ) z − (sin t ) I z, φ t ( z ) = (cos t ) z − (sin t ) J z, the Proposition follows by a straightforward computation. (cid:3) Remark 5.2.
Note that there exist vectors { e i } which satisfy the hypotheses of theprevious Proposition. For example the first three or four vectors, respectively, fromthe canonical basis of E satisfy the required properties. References [1] K. Arslan, R. Ezentas, C. Murathan, T. Sasahara. , Beitr¨age Algebra Geom., to appear.[2] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, V.J. Papantoniou.
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Department of Mathematics, ”Gh. Asachi” Technical University of Iasi, Bd. CarolI no. 11, 700506 Iasi, Romania
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