Let f : (M, g) → (N , h) be a smooth map between two Riemannian manifolds and defineits first tension field by τ( f ) = trace∇d f. The map is called harmonic [10] if it is a criticalpoint of the energy functional
E( f ) =∫
M
|d f |2 υg,
where τ( f ) is the tension field of f . Biharmonic submanifolds in Euclidean spaces weredefined by Chen in [7]. The Euler–Lagrange equation for the energy functional is τ( f ) = 0.
Euler–Lagrange equation for the bienergy functional was derived by Jiang [15] by
τ2( f ) = −J f (τ ( f )) = −�τ( f ) − traceRN (d f, τ ( f ))d f = 0,
where J f is the Jacobi operator of f . It is trivial that any harmonic map is biharmonic. Ifthe map is non-harmonic biharmonic map, then we call it as proper biharmonic. Biharmonicsubmanifolds have been studied by many geometers. For example see [5–7,11–14,18,19]and references therein.
C. Özgür (B) · S. GüvençDepartment of Mathematics, Balikesir University, Çagıs, 10145 Balıkesir, Turkeye-mail: [email protected]
On the other hand, in [1], Blair, Carriazo and Alegre introduced the notion of a general-ized Sasakian space form and proved some of its basic properties. Many examples of thesemanifolds were presented by using some different geometric techniques such as Riemanniansubmersions, warped products or conformal and related transformations. New results ongeneralized complex space forms were also obtained. In [12,14], Fetcu and Oniciuc studiedbiharmonic Legendre curves in Sasakian space forms. In [20], the first author and M. M.Tripathi studied Legendre curves in α-Sasakian manifolds. In the present paper, we studybiharmonic Legendre curves in generalized Sasakian space forms with constant functions.We obtain curvature characterizations of these kinds of curves.
The paper is organized as follows: In Sect. 2, we give brief introduction about generalizedSasakian space forms. In Sect. 3, we give the main results of the study. In Sect. 4, someapplications are given for Sasakian, Kenmotsu and cosymplectic space forms.
2 Generalized Sasakian space forms
A (2n + 1)-dimensional Riemannian manifold M is said to be an almost contact metricmanifold [4], if there exist on M a (1, 1 ) tensor field ϕ, a vector field ξ , a 1-form η and aRiemannian metric g satisfying
for any vector fields X, Y on M . Such a manifold is said to be a contact metric manifold ifdη = , where (X, Y ) = g(X, ϕY ) is called the fundamental 2-form of M [4].
On the other hand, the almost contact metric structure of M is said to be normal if
[ϕ, ϕ](X, Y ) = −2dη(X, Y )ξ,
for any vector fields X, Y on M , where [ϕ, ϕ] denotes the Nijenhuis torsion of ϕ, given by
[ϕ, ϕ](X, Y ) = ϕ2[X, Y ] + [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ].A normal contact metric manifold is called a Sasakian manifold [4]. It can be proved that
an almost contact metric manifold is Sasakian if and only if
(∇Xϕ)Y = g(X, Y )ξ − η(Y )X.
In [21], Oubiña introduced the notion of a trans-Sasakian manifold. An almost contactmetric manifold M is called a trans-Sasakian manifold if there exist two functions α and β
on M such that
(∇Xϕ)Y = α[g(X, Y )ξ − η(Y )X ] + β[g(ϕX, Y )ξ − η(Y )ϕX ], (2.1)
for any vector fields X, Y on M . From (2.1), it is easy to see that
∇X ξ = −αϕX + β[X − η(X)ξ ]. (2.2)
If β = 0 (resp. α = 0), then M is said to be an α- Sasakian manifold (resp. β-Kenmotsumanifold). Sasakian manifolds (resp. Kenmotsu manifolds [16]) appear as examples of α-Sasakian manifolds (β-Kenmotsu manifolds), with α = 1 (resp. β = 1).
Another kind of trans-Sasakian manifolds is that of cosymplectic manifolds, obtained forα = β = 0. From (2.2), for a cosymplectic manifold it follows that
∇X ξ = 0,
which means that ξ is a Killing vector field for a cosymplectic manifold [3].
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On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Proposition 2.1 [17] A trans-Sasakian manifold of dimension greater than or equal to 5 iseither α-Sasakian, β-Kenmotsu or cosymplectic.
For an almost contact metric manifold M , a ϕ-section of M at p ∈ M is a section π ⊆ Tp Mspanned by a unit vector X p orthogonal to ξp and ϕX p . The ϕ- sectional curvature of π isdefined by K (X ∧ϕX) = R(X, ϕX, ϕX, X). A Sasakian manifold with constant ϕ -sectionalcurvature c is called a Sasakian space form. Similarly, a Kenmotsu manifold with constantϕ-sectional curvature c is called a Kenmotsu space form. A cosymplectic manifold withconstant ϕ-sectional curvature c is called a cosymplectic space form.
Given an almost contact metric manifold M with an almost contact metric structure(ϕ, ξ, η, g), M is called a generalized Sasakian space form [1] if there exist three functionsf1, f2 and f3 on M such that
R(X, Y )Z = f1 {g(Y, Z)X − g(X, Z)Y } + f2{g(X, ϕZ)ϕY
for any vector fields X, Y, Z on M , where R denotes the curvature tensor of M . If M is aSasakian space form then f1 = c+3
4 , f2 = f3 = c−14 , if M is a Kenmotsu space form then
f1 = c−34 , f2 = f3 = c+1
4 , if M is a cosymplectic space form then f1 = f2 = f3 = c4 .
Now, we give the following results from [1] and [2] to use in the next chapter:
Proposition 2.2 [2] Let M( f1, f2, f3) be an α -Sasakian generalized Sasakian-space-form.Then α does not depend on the direction of ξ and the following equation holds:
f1 − f3 = α2.
Moreover, if M is connected, then α is a constant.
Theorem 2.1 [2] Let M( f1, f2, f3) be a connected α-Sasakian generalized Sasakian-space-form with dimension greater than or equal to 5. Then f1, f2 and f3 are constant functions,related as follows:
i If α = 0, then f1 = f2 = f3 and M is a cosymplectic manifold of constant ϕ-sectionalcurvature.
ii If α = 0, then f1 − α2 = f2 = f3.
Proposition 2.3 [2] Let M( f1, f2, f3) be a β-Kenmotsu generalized Sasakian-space-form.Then, β depends only on the direction of ξ and the functions f1, f3 satisfy the equation:
f1 − f3 + ξ (β) + β2 = 0.
Theorem 2.2 [1] Let M( f1, f2, f3) be a β -Kenmotsu generalized Sasakian-space-formwith dimension greater than or equal to 5. Then f1, f2 and f3 depend only on the directionof ξ and the following equations hold:
ξ ( f1) + 2β f3 = 0,
ξ ( f2) + 2β f2 = 0.
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Theorem 2.3 [2] Let M be a 3-dimensional (α, β) trans-Sasakian manifold such that α, β
depend only on the direction of ξ . Then M is a generalized Sasakian space form M( f1, f2, f3)
with functions:
f1 = 3τ − 2(α2 − ξ (β) − β2
), f2 = 0, f3 = 3τ − 3
(α2 − ξ (β) − β2
),
where τ denotes the scalar curvature of M.
The contact distribution of an almost contact metric manifold M with an almost contactmetric structure (ϕ, ξ, η, g) is defined by
{X ∈ T M : η(X) = 0}and an integral curve of the contact distribution is called a Legendre curve [4].
3 Biharmonic Legendre curves in generalized Sasakian space forms
Let (M, g) be an m-dimensional Riemannian manifold and γ : I → M a curve parametrizedby arc length. Then γ is called a Frenet curve of osculating order r , 1 ≤ r ≤ m, if thereexists orthonormal vector fields E1, E2, . . . , Er along γ such that
E1 = γ ′ = T,
∇T E1 = κ1 E2,
∇T E2 = −κ1 E1 + κ2 E3, (3.1)
...
∇T Er = −κr−1 Er−1,
where κ1, . . . , κr−1 are positive functions on I .A geodesic is a Frenet curve of osculating order 1; a circle is a Frenet curve of osculating
order 2 such that κ1 is a non-zero positive constant; a helix of order r , r ≥ 3, is a Frenetcurve of osculating order r such that κ1, . . . , κr−1 are non-zero positive constants; a helix oforder 3 is called simply a helix.
Now let (M2n+1, ϕ, ξ, η, g) be a generalized Sasakian space form with functions f1, f2,
f3 and γ : I → M a Legendre Frenet curve of osculating order r . By the use of (2.3) and(3.1), it can be easily seen that
∇T ∇T T = −κ21 E1 + κ ′
1 E2 + κ1κ2 E3,
∇T ∇T ∇T T = −3κ1κ′1 E1 + (
κ ′′1 − κ3
1 − κ1κ22
)E2
+ (2κ ′
1κ2 + κ1κ′2
)E3 + κ1κ2κ3 E4,
R(T,∇T T )T = −κ1 f1 E2 − 3κ1 f2g(ϕT, E2)ϕT + f3κ1η(E2)ξ.
So we have
τ(γ ) = ∇T ∇T ∇T T − R(T,∇T T )T
= −3κ1κ′1 E1
+ (κ ′′
1 − κ31 − κ1κ
22 + κ1 f1
)E2 (3.2)
+(2κ ′1κ2 + κ1κ
′2)E3 + κ1κ2κ3 E4
+ 3κ1 f2g(ϕT, E2)ϕT − f3κ1η(E2)ξ.
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On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Let m = min {r, 4}. From (3.2), the curve γ is proper-biharmonic if and only if κ1 > 0and
1. f2 = 0 or ϕT ⊥ E2 or ϕT ∈ span {E2, . . . , Em}; and2. f3 = 0 or E2 ⊥ ξ or ξ ∈ span {E2, . . . , Em}; and3. g(τ (γ ), Ei ) = 0, for any i = 1, m.
So we can state the following theorem:
Theorem 3.1 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) and m = min {r, 4}. Then γ is proper biharmonicif and only if
1. f2 = 0 or ϕT ⊥ E2 or ϕT ∈ span {E2, . . . , Em}; and2. f3 = 0 or ξ ⊥ E2 or ξ ∈ span {E2, . . . , Em}; and3. the first m of the following equations are satisfied (replacing κm = 0):
κ1 = constant > 0,
κ21 + κ2
2 = f1 + 3 f2 [g(ϕT, E2)]2 − f3 [η(E2)]2 ,
κ ′2 + 3 f2g(ϕT, E2)g(ϕT, E3) − f3η(E2)η(E3) = 0,
κ2κ3 + 3 f2g(ϕT, E2)g(ϕT, E4) − f3η(E2)η(E4) = 0.
Now we give the interpretations of Theorem 3.1.
Case I If f2 = f3 = 0 then M2n+1 is a Riemannian space of constant curvature. Biharmoniccurves in a Riemannian space of constant curvature was studied in [9]. The following resultwas given:
Theorem 3.2 [9] Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = f3 = 0. Then γ is proper biharmonic ifand only if either
1. γ is a circle with κ1 = √f1, where f1 > 0 is a constant; or
2. γ is a helix with κ21 + κ2
2 = f1, where f1 > 0 is a constant. In both cases, if f1 is not apositive constant, then such a proper biharmonic Legendre curve does not exist.
From Theorem 3.2, for a Riemannian space of constant curvature, it is clear that there isno this kind of proper biharmonic Legendre Frenet curve of osculating order r > 3.
Case II f2 = 0, f3 = 0 and ξ ⊥ E2.In this case, η(E2) = 0. From Theorem 3.1, we obtain
κ1 = constant > 0,
κ21 + κ2
2 = f1,
κ ′2 = 0,
κ2κ3 = 0.
(3.3)
Hence we can state the following Theorem:
Theorem 3.3 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = 0, f3 = 0 and ξ ⊥ E2. Then γ is properbiharmonic if and only if either
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1. γ is a Legendre circle with κ1 = √f1, where f1 > 0 is a constant; or
2. γ is a Legendre helix with κ21 + κ2
2 = f1, where f1 > 0 is a constant.
If r > 3 or f1 is not a positive constant, then such a proper biharmonic Legendre curvedoes not exist.
Proof Assume that γ is proper biharmonic. We use Eqs. (3.3). If r > 3, then from (3.3),κ2 = 0 or κ3 = 0, a contradiction. If r = 2, then γ is a Legendre circle, since κ1 = √
f1,where f1 is a positive constant. Finally, if r = 3, then κ2 is a non-zero constant. Hence, γ isa helix with κ2
1 + κ22 = f1, where f1 is a positive constant.
Conversely, let γ be a Legendre circle with κ1 = √f1 or a Legendre helix with κ2
1 +κ22 =
f1, where f1 > 0 is a constant. It is clear that γ satisfies the first two/three equations of(3.3), respectively. Thus, Theorem 3.1 gives us γ is a proper biharmonic curve. ��
Case III f2 = 0, f3 = 0, ξ ∈span{E2, . . . , Em} and η(E2) = 0.
Let m = min {r, 4} = 4, that is, r ≥ 4. In this case, we can write
ξ = cos u1 E2 + sin u1 cos u2 E3 + sin u1 sin u2 E4, (3.4)
where u1 : I → R is the angle function between ξ and E2; u2 : I → R is the angle functionbetween E3 and the orthogonal projection of ξ onto span{E3, E4}. Equation (3.4) gives us
η(E2) = cos u1,
η(E3) = sin u1 cos u2,
η(E3) = sin u1 sin u2.
(3.5)
Let r = 3. Then E4 does not exist, which gives us ξ ∈span{E2, E3}. Hence we can write
ξ = cos u1 E2 + sin u1 E3, (3.6)
where u1 : I → R is the angle function between ξ and E2. We can also obtain (3.6) from(3.4), taking u2 = 0.
Finally, let r = 2. Then E3 and E4 do not exist, which gives us ξ ∈span{E2}. Thus ξ ‖ E2,that is,
ξ = ±E2. (3.7)
(3.7) can also be found taking u1 = 0, π and u2 = 0 in (3.4).Thus by the use of Theorem 3.1, Eqs. (3.4), (3.6) and (3.7), we can state the following
Theorem:
Theorem 3.4 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = 0, f3 = 0, ξ ∈span{E2, . . . , Em} andη(E2) = 0. If r ≥ 4, then γ is proper biharmonic if and only if
κ1 = constant > 0,
κ21 + κ2
2 = f1 − f3 cos2 u1,
κ ′2 − f3 cos u1 sin u1 cos u2 = 0,
κ2κ3 − f3 cos u1 sin u1 sin u2 = 0.
If r = 3, the first three of the above equations are satisfied, taking u2 = 0. If r = 2, the firsttwo of the above equations are satisfied, taking u1 = 0, π .
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On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Now, let (M2n+1, ϕ, ξ, η, g) be a trans-Sasakian generalized Sasakian space form andγ : I → M a Legendre curve of osculating order r . Since γ is a Legendre curve, η(T ) = 0.Then
∇T ξ = −αϕT + βT (3.8)
which gives us
g(∇T ξ, T ) = β. (3.9)
Differentiating η(T ) = 0 along γ , if we use (3.1) and (3.9), we find
κ1η(E2) = −β. (3.10)
Corollary 3.1 Let γ be a Legendre Frenet curve of osculating order r in a trans-Sasakiangeneralized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f1 =constant, f2 = 0, f3 and β
are non-zero constants, ξ ∈span{E2, . . . , Em}, η(E2) = 0. Then γ is proper biharmonic ifand only if γ is a helix with κ1 = ±β > 0, κ2 = ±α = √
f1 − f3 − β2, where f1 − f3 −β2
is a positive constant, ξ ‖ E2, α is a non-zero constant, ϕT ‖ E3 and dimM = 3.
Proof Since M is a trans-Sasakian generalized Sasakian space form, using (2.2) and (3.1),we obtain
∇T η(E1) = κ1η(E2) + β = 0 (3.11)
∇T η(E2) = κ2η(E3) − αg(ϕT, E2), (3.12)
∇T η(E3) = −κ2η(E2) + κ3η(E4) − αg(ϕT, E3),
∇T η(E4) = −κ3η(E3) + κ4η(E5) − αg(ϕT, E4).
Assume that γ is proper biharmonic.
1. If r = 2, from Theorem 3.1, we have
κ1 = constant > 0,
κ21 = f1 − f3 [η(E2)]
2 ,
and ξ ∈span{E2}, that is, ξ ‖ E2. Thus we get η(E2) = ±1. So γ is a circle withκ1 = √
f1 − f3 , where f1 − f3 > 0 is a constant and ξ ‖ E2. Differentiating ξ = ±E2
along γ , we find α = 0 and κ1 = ±β. Since α = 0, M is a β-Kenmotsu generalizedSasakian space form. From Proposition 2.3, we have
f1 − f3 + β2 = 0,
which contradicts f1 − f3 > 0.2. If r = 3, from Theorem 3.1, we have
κ1 = constant > 0, (3.13)
κ21 + κ2
2 = f1 − f3 [η(E2)]2 , (3.14)
κ ′2 − f3η(E2)η(E3) = 0, (3.15)
and ξ ∈span{E2, E3}. Differentiating (3.14) and using (3.12), (3.15), we find
2κ2η(E3) = αg(ϕT, E2). (3.16)
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C. Özgür, S. Güvenç
(3.11) gives us η(E2) is a constant, since β is a non-zero constant. Hence using (3.12)we get
κ2η(E3) = αg(ϕT, E2). (3.17)
From (3.16) and (3.17), we have η(E3) = 0. Since ξ ∈span{E2, E3} and η(E3) = 0, wehave ξ ‖ E2, that is, η(E2) = ±1. Finally, from Eqs. (3.11), (3.13), (3.14) and (3.15),we obtain γ is a helix with κ1 = ±β > 0, κ2 = √
f1 − f3 − β2, where f1 − f3 − β2 isa positive constant and ξ ‖ E2. Differentiating ξ = ±E2 along γ , we find α = 0. In thiscase, using Proposition 2.1, we find dimM = 3.
3. If r ≥ 4, then dimM ≥ 5. Since β is a non-zero constant, using Proposition 2.1, wefind α = 0. Hence M is a β-Kenmotsu generalized Sasakian space form with non-zeroconstant β and dimM ≥ 5. Using Theorem 2.2, we find f3 = 0, a contradiction.
Conversely, let γ be the given curve. It is clear that the first three of the equations inTheorem 3.1 are satisfied (replacing κm = 0). So γ is proper biharmonic. ��Case IV f2 = 0, f3 = 0 and ϕT ⊥ E2.
In this case, g(ϕT, E2) = 0. Hence, from Theorem 3.1, we have the following theorem:
Theorem 3.5 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = 0, f3 = 0 and ϕT ⊥ E2. Then γ isproper biharmonic if and only if
1. γ is a Legendre circle with κ1 = √f1, where f1 > 0 is a constant; or
2. γ is a Legendre helix with κ21 + κ2
2 = f1, where f1 > 0 is a constant.
In both cases, unless f1 is a positive constant, then there is no this kind of proper bihar-monic Legendre curve.
Proof The proof is similar to the proof of Theorem 3.3. ��Case V f2 = 0, f3 = 0, ϕT ∈span{E2, E3, E4} and g(ϕT, E2) = 0.
Let m = min {r, 4} = 4, that is, r ≥ 4. In this case, ϕT can be written as
ϕT = cos ω1 E2 + sin ω1 cos ω2 E3 + sin ω1 sin ω2 E4, (3.18)
where ω1 : I → R is the angle function between ϕT and E2; ω2 : I → R is the anglefunction between E3 and the orthogonal projection of ϕT onto span{E3, E4}. From (3.18),we have
g(ϕT, E2) = cos ω1,
g(ϕT, E3) = sin ω1 cos ω2,
g(ϕT, E4) = sin ω1 sin ω2.
(3.19)
Let r = 3. Then E4 does not exist, which gives us ϕT ∈span{E2, E3}. Hence we can write
ϕT = cos ω1 E2 + sin ω1 E3, (3.20)
where u1 : I → R is the angle function between ξ and E2. We can also obtain (3.20) from(3.18), taking ω2 = 0.
Finally, let r = 2. Then E3 and E4 do not exist, which gives us ϕT ∈span{E2}. ThusϕT ‖ E2, that is,
ϕT = ±E2. (3.21)
(3.21) can also be found taking ω1 = 0, π and ω2 = 0 in (3.18).In view of Theorem 3.1, Eqs. (3.18), (3.20) and (3.21), we obtain the following result:
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Theorem 3.6 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = 0, f3 = 0, ϕT ∈span{E2, . . . , Em} andg(ϕT, E2) = 0. If r ≥ 4, then γ is proper biharmonic if and only if
κ1 = constant > 0,
κ21 + κ2
2 = f1 + 3 f2 cos2 ω1,
κ ′2 + 3 f2 cos ω1 sin ω1 cos ω2 = 0,
κ2κ3 + 3 f2 cos ω1 sin ω1 sin ω2 = 0.
If r = 3, the first three of the above equations are satisfied, taking ω2 = 0. If r = 2, the firsttwo of the above equations are satisfied, taking ω1 = 0, π .
Corollary 3.2 Let γ be a Legendre Frenet curve of osculating order r in a connected trans-Sasakian generalized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f1 =constant, f2 is anon-zero constant, f3 = 0, ϕT ∈span{E2, . . . , Em} and g(ϕT, E2) = 0. Then γ is properbiharmonic if and only if γ is a Frenet curve of order r ≥ 4 with
κ1 = −β
η(E2)= constant > 0,
κ2 = √δ > 0,
κ3 = −3g(ϕT, E2)g(ϕT, E4)√δ
> 0,
κ4 = −βg(ϕE2, E5)
η(E2)g(ϕT, E4)> 0, (if r ≥ 5),
where
δ = f1 + 3 f2 [g(ϕT, E2)]2 − β2
[η(E2)]2 > 0
is a constant, g(ϕT, E3) = 0, α = 0, g(ϕT, E2) = 0 and g(ϕT, E4) = 0 are constants,β = 0 and η(E2) = 0.
Proof Since M is a trans-Sasakian manifold, it is easy to show that
Let α = 0. Then (3.30) gives us g(ϕT, E3) = 0, that is, ϕT = ±E2. Then (3.24)gives us κ2 = 0, which is a contradiction. Thus α = 0. In this case, (3.32) gives us[η(E2)]2 + [η(E3)]2 = 1. So ξ ∈span{E2, E3} and ϕT = ±E2. Hence ξ = ±E3. From(3.11), we have β = 0. Differentiating ξ = ±E3 and using (3.1), (3.8), (3.27), (3.28)and (3.29), we obtain κ1 = √
f1 + 3 f2 − α2, κ2 = ±α > 0, where f1 + 3 f2 − α2 > 0,α = 0 is a constant. In this case, M is a connected α-Sasakian generalized Sasakian spaceform. If dimM ≥ 5, using Theorem 2.1, we find f2 = f3, a contradiction. If dimM = 3,using Theorem 2.3, we have f2 = 0, which is also a contradiction.Let r ≥ 4. In this case, Theorem 3.1 gives us
κ1 = constant > 0, (3.33)
κ21 + κ2
2 = f1 + 3 f2 [g(ϕT, E2)]2 , (3.34)
κ ′2 + 3 f2g(ϕT, E2)g(ϕT, E3) = 0, (3.35)
κ2κ3 + 3 f2g(ϕT, E2)g(ϕT, E4) = 0 (3.36)
and ϕT ∈span{E2, E3, E4}. Differentiating (3.34) and using (3.23), (3.35), we find
− 2κ2g(ϕT, E3) = αη(E2). (3.37)
3. Let r ≥ 4 and g(ϕT, E3) = 0. We find α = 0. Since g(ϕT, E3) = 0,we get ϕT ∈span{E2, E4}. From (3.23), g(ϕT, E2) = 0 is a constant. So usingϕT ∈span{E2, E4} and equation (3.36), g(ϕT, E4) is a non-zero constant. Using (3.11),(3.33), (3.34), 3.35) and (3.36), we obtain κ1 = −β
η(E2)= constant> 0, κ2 = √
δ,
κ3 = −3g(ϕT,E2)g(ϕT,E4)√δ
= constant> 0, where
δ = f1 + 3 f2 [g(ϕT, E2)]2 − β2
[η(E2)]2 = constant > 0.
If r ≥ 5, differentiating g(ϕT, E5) = 0 and using (3.22), we find κ4 = −βg(ϕE2,E5)η(E2)g(ϕT,E4)
.4. If r ≥ 4 and g(ϕT, E3) = 0, then α = 0 and η(E2) = 0. Since dimM ≥ 5 and α = 0,
we have β = 0. This contradicts η(E2) = 0.
Conversely, let γ be the given curve. It is obvious that all four of the equations in Theorem3.1 are satisfied. So γ is proper biharmonic. ��
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On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Case VI f2 = 0, f3 = 0, ϕT ⊥ E2 and ξ ⊥ E2.
In this case, g(ϕT, E2) = 0 and η(E2) = 0. Using Theorem 3.1, we have the followingresult:
Theorem 3.7 Let γ be a Legendre curve in a generalized Sasakian space form (M2n+1, ϕ, ξ,
η, g) with f2 = 0, f3 = 0, ϕT ⊥ E2 and ξ ⊥ E2. Then γ is proper biharmonic if and onlyif
1. γ is a circle with κ1 = √f1, where f1 > 0 is a constant; or
2. γ is a helix with κ21 + κ2
2 = f1, where f1 > 0 is a constant.
There is no this kind of proper biharmonic Legendre curve unless f1 is a positive constant.
Proof The proof is similar to the proof of Theorem 3.3. ��Case VII f2 = 0, f3 = 0, ϕT ⊥ E2, ξ ∈span{E2, . . . , Em} and η(E2) = 0.
Since g(ϕT, E2) = 0, using Theorem 3.1, Eqs. (3.4) and (3.5), we have the followingtheorem:
Theorem 3.8 Let γ be a Legendre Frenet curve of osculating order r in a general-ized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f2 = 0, f3 = 0, ϕT ⊥ E2,ξ ∈span{E2, . . . , Em} and η(E2) = 0. If r≥ 4, then γ is proper biharmonic if and onlyif
κ1 = constant > 0,
κ21 + κ2
2 = f1 − f3 cos2 u1,
κ ′2 − f3 cos u1 sin u1 cos u2 = 0,
κ2κ3 − f3 cos u1 sin u1 sin u2 = 0.
(3.38)
If r = 3, the first three of the above equations are satisfied, taking u2 = 0. If r = 2, the firsttwo of the above equations are satisfied, taking u1 = 0, π .
Corollary 3.3 Let γ be a Legendre Frenet curve of osculating order r in a trans-Sasakiangeneralized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f1 =constant, f2 and f3 are non-zero constants, ϕT ⊥ E2, ξ ∈span{E2, . . . , Em}, η(E2) = 0. Then γ is proper biharmonicif and only if γ is a helix of order 4 with
κ1 = −β
η(E2)= constant > 0,
κ2 = √δ > 0,
κ3 = f3η(E2)η(E4)√δ
= constant > 0,
where
δ = f1 − f3 [η(E2)]2 − β2
[η(E2)]2
is a positive constant, η(E3) = 0, α = 0.
Proof The proof is similar to the proof of Corollary 3.1. ��Case VIII f2 = 0, f3 = 0, ϕT ∈span{E2, . . . , Em}, g(ϕT, E2) = 0 and ξ ⊥ E2.
In this case, η(E2) = 0. Then (3.18) and (3.19) are satisfied. Hence we have:
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C. Özgür, S. Güvenç
Theorem 3.9 Let γ be a Legendre Frenet curve of osculating order r in a generalizedSasakian space form (M2n+1, ϕ, ξ, η, g) with ϕT ∈span{E2, . . . , Em}, g(ϕT, E2) = 0 andξ ⊥ E2. If r ≥ 4, then γ is proper biharmonic if and only if
κ1 = constant > 0,
κ21 + κ2
2 = f1 + 3 f2 cos2 ω1,
κ ′2 + 3 f2 cos ω1 sin ω1 cos ω2 = 0,
κ2κ3 + 3 f2 cos ω1 sin ω1 sin ω2 = 0.
(3.39)
If r = 3, the first three of the above equations are satisfied, taking ω2 = 0. If r = 2, the firsttwo of the above equations are satisfied, taking ω1 = 0, π .
Corollary 3.4 Let γ be a Legendre Frenet curve of osculating order r in a trans-Sasakiangeneralized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f1 =constant, f2 and f3 are non-zero constants, ϕT ∈span{E2, . . . , Em} , g(ϕT, E2) = 0 and ξ ⊥ E2. Then γ is properbiharmonic if and only if
1. γ is a circle with κ1 = √f1 + 3 f2, where α = β = 0, ϕT ‖ E2 and f1 + 3 f2 is a
positive constant; or2. γ is a helix with κ1 = √
f1 + 3 f2 − α2, κ2 = ±α > 0, where f1 +3 f2 −α2 > 0, α = 0is a constant, β = 0, ϕT ‖ E2 and ξ ‖ E3; or
3. γ is a Frenet curve of order r ≥ 4 with
κ1 = δ > 0,
κ2 =√
f1 + 3 f2 [g(ϕT, E2)]2 − δ2 > 0,
κ3 = −3 f2g(ϕT,E2)g(ϕT,E4)√f1+3 f2[g(ϕT,E2)]2−δ2
> 0,
κ4 = δg(ϕE2,E5)g(ϕT,E4)
> 0, (if r ≥ 5);where g(ϕT, E3) = 0, g(ϕT, E2) = 0 and g(ϕT, E4) = 0 are constants, f1 +3 f2 [g(ϕT, E2)]2 − δ2 > 0 and δ > 0 are constants.
Proof The proof is similar to the proof of Corollary 3.2. ��Case IX f2 = 0, f3 = 0, ϕT ∈span{E2, . . . , Em}, g(ϕT, E2) = 0, ξ ∈span{E2, . . . , Em}and η(E2) = 0.
In this case, by the use of Theorem 3.1, Eqs. (3.4), (3.5), (3.18) and (3.19), we can statethe following theorem:
Theorem 3.10 Let γ be a Legendre Frenet curve of osculating order r in a gen-eralized Sasakian space form (M2n+1, ϕ, ξ, η, g) with ϕT ∈span{E2, . . . , Em} andξ ∈span{E2, . . . , Em} . If r ≥ 4, then γ is proper biharmonic if and only if
κ1 = constant > 0,
κ21 + κ2
2 = f1 + 3 f2 cos2 ω1 − f3 cos2 u1,
κ ′2 + 3 f2 cos ω1 sin ω1 cos ω2 − f3 cos u1 sin u1 cos u2 = 0,
κ2κ3 + 3 f2 cos ω1 sin ω1 sin ω2 − f3 cos u1 sin u1 sin u2 = 0.
If r = 3, the first three of the above equations are satisfied, taking ω2 = 0 and u2 = 0. Ifr = 2, the first two of the above equations are satisfied, taking u1 = 0, π and ω1 = 0, π .
Corollary 3.5 Let γ be a Legendre Frenet curve of osculating order r ≥ 4 in a trans-Sasakian generalized Sasakian space form (M2n+1, ϕ, ξ, η, g) with f1 =constant, f2 andf3 are non-zero constants, ϕT ∈span{E2, . . . , Em} , g(ϕT, E2) = 0, ξ ∈span{E2, . . . , Em}and η(E2) = 0. Then γ is proper biharmonic if and only if
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On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Since M is a trans-Sasakian manifold, if we write (3.12),(3.23) and (3.41) in (3.43), we obtain
2κ2μ = λ, (3.44)
where we denote
λ = (3 f2 + f3)αg(ϕT, E2)η(E2),
μ = f3η(E2)η(E3) − 3 f2g(ϕT, E2)g(ϕT, E3).
If λ = 0 and μ = 0, then (3.44) gives us κ2 = λ2μ
= 0. So κ3 is found from equation(3.42).
If μ = 0, it is obvious from (3.41) that κ2 is a constant. From (3.40), we find κ2 = √δ,
where
δ = f1 + 3 f2 [g(ϕT, E2)]2 − f3 [η(E2)]
2 − β2
[η(E2)]2 > 0
123
C. Özgür, S. Güvenç
is a constant. So (3.42) gives us
κ3 = f3η(E2)η(E4) − 3 f2g(ϕT, E2)g(ϕT, E4)√δ
> 0.
��
4 Applications
Let (M2n+1, ϕ, ξ, η, g) be a Sasakian space form. So α = 1, β = 0, f1 = c+34 , f2 = f3 =
c−14 . In this case, equation (3.10) becomes
κ1η(E2) = 0,
which gives us η(E2) = 0, since κ1 > 0. Hence, all possible cases are Case I, Case VI andCase VIII.
For Case I and Case VI, if we use Theorem 3.2 and Theorem 3.7, we can state the followingtheorem:
Theorem 4.1 [14] Let γ be a Legendre curve in a Sasakian space form (M2n+1, ϕ, ξ, η, g)
with c = 1 or ϕT ⊥ E2. Then γ is proper biharmonic if and only if
1. γ is a circle with κ1 = 12
√c + 3, where c > −3; or
2. γ is a helix with κ21 + κ2
2 = c+34 , where c > −3.
If c ≤ −3, such a proper biharmonic Legendre curve does not exist.
For the Case VIII, using Corollary 3.4, we give the following theorem:
Theorem 4.2 [14] Let γ be a Legendre curve in a Sasakian space form (M2n+1, ϕ, ξ, η, g)
with c = 1, ϕT ∈span{E2, . . . , Em}, g(ϕT, E2) = 0. Then γ is proper biharmonic if andonly if
1. γ is a helix with κ1 = √c − 1 and κ2 = 1, where c > 1, ϕT ‖ E2 and ξ ‖ E3; or
2. γ is a Frenet curve of order r ≥ 4 with
κ1 = δ,
κ2 = 1
2
√c + 3 + 3(c − 1) [g(ϕT, E2)]2 − 4δ2,
κ3 = −3(c − 1)g(ϕT, E2)g(ϕT, E4)
2√
c + 3 + 3(c − 1) [g(ϕT, E2)]2 − 4δ2> 0,
κ4 = δg(ϕE2, E5)
g(ϕT, E4)> 0 (if r ≥ 5);
where g(ϕT, E3)= 0, g(ϕT, E2) = 0 and g(ϕT, E4) = 0 are constants, c+3+3(c−1)
[g(ϕT, E2)]2 − 4δ2 > 0 and δ > 0 are constants.
Proof If we use α = 1, β = 0, f1 = c+34 , f2 = f3 = c−1
4 in Corollary 3.4, the proof isclear. ��Remark 4.1 κ4 does not need to be constant. So, there exists proper biharmonic curves whichare not helices in a Sasakian space form with dimM ≥ 5.
123
On some classes of biharmonic Legendre curves in generalized Sasakian space forms
Now, let (M2n+1, ϕ, ξ, η, g) be a Kenmotsu space form. So α = 0, β = 1, f1 = c−34 ,
f2 = f3 = c+14 . In this case, by the use of Theorems 2.2 and 2.3, we find f2 = c+1
4 = 0,that is, c = −1. Thus, the only possible case is Case I. From Theorem 3.2, we can state thefollowing theorem:
Theorem 4.3 There does not exist a proper biharmonic Legendre Frenet curve in a Kenmotsuspace form.
Proof Let (M2n+1, ϕ, ξ, η, g) be a Kenmotsu space form. Since c = −1, we have f1 =c−3
4 = −1 < 0. Hence, from Theorem 3.2, a proper biharmonic Legendre curve does notexist in M . ��
Let (M2n+1, ϕ, ξ, η, g) be a cosymplectic space form. So α = β = 0, f1 = f2 = f3 = c4 .
In this case, Eq. (3.10) becomes
κ1η(E2) = 0,
which gives us η(E2) = 0, since κ1 > 0. Hence, all possible cases are Case I, Case VI andCase VIII. In Case I, since f1 = f2 = f3 = c
4 = 0, it contradicts f1 > 0. For the Case VI,from Theorem 3.7, we can state the following theorem:
Theorem 4.4 Let γ be a Legendre Frenet curve in a cosymplectic space form (M2n+1, ϕ, ξ,
η, g) with c = 0, ϕT ⊥ E2 and ξ ⊥ E2. Then γ is proper biharmonic if and only if
1. γ is a circle with κ1 =√
c2 , where c > 0; or
2. γ is a helix with κ21 + κ2
2 = c4 , where c > 0.
For the Case VIII, using Corollary 3.4, we give the following theorem:
Theorem 4.5 Let γ be a Legendre curve in a cosymplectic space form (M2n+1, ϕ, ξ, η, g)
with c = 0, ϕT ∈span{E2, E3, E4}, g(ϕT, E2) = 0 and ξ ⊥ E2. Then γ is proper bihar-monic if and only if
1. γ is a circle with κ1 = √c, where ϕT ‖ E2 and c > 0; or,
2. γ is a Frenet curve of order r ≥ 4 with
κ1 = δ > 0,
κ2 = 1
2
√c + 3c [g(ϕT, E2)]2 − 4δ2 > 0,
κ3 = −3cg(ϕT, E2)g(ϕT, E4)
2√
c + 3c [g(ϕT, E2)]2 − 4δ2> 0,
κ4 = δg(ϕE2, E5)
g(ϕT, E4)(if r ≥ 5);
where g(ϕT, E3) = 0, g(ϕT, E2) = 0 and g(ϕT, E4) = 0 are constants, c +3c [g(ϕT, E2)]2 − 4δ2 > 0 and δ > 0 are constants.
Proof The proof is easily done if we use α = β = 0, f1 = f2 = f3 = c4 in Corollary 3.4. ��
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C. Özgür, S. Güvenç
References
1. Alegre, P., Blair, D.E., Carriazo, A.: Generalized Sasakian space-forms. Israel J. of Math. 141, 157–183(2004)
3. Blair, D.E.: The theory of quasi-Sasakian structures. J. Differ. Geom. 1, 331–345 (1967)4. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhauser, Boston (2002)5. Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of S3. Internat. J. Math. 12, 867–876
(2001)6. Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres. Israel J. Math. 130, 109–123
(2002)7. Chen, B.-Y.: A report on submanifolds of finite type. Soochow J. Math. 22, 117–337 (1996)8. Chen, B.-Y., Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces.
Kyushu J. Math. 52, 167–185 (1998)9. Cho, J.T., Inoguchi, J., Lee, J.-E.: On slant curves in Sasakian 3-manifolds. Bull. Austral. Math. Soc.
74(3), 359–367 (2006)10. Eells Jr, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160
(1964)11. Fetcu, D.: Biharmonic curves in the generalized Heisenberg group. Beitrage zur Algebra und Geometrie
46, 513–521 (2005)12. Fetcu, D.: Biharmonic Legendre curves in Sasakian space forms. J. Korean Math. Soc. 45, 393–404
(2008)13. Fetcu, D., Oniciuc, C.: Biharmonic hypersurfaces in Sasakian space forms. Differ. Geom. Appl. 27,
713–722 (2009)14. Fetcu, D., Oniciuc, C.: Explicit formulas for biharmonic submanifolds in Sasakian space forms. Pac. J.
Math. 240, 85–107 (2009)15. Jiang, G.Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A
7, 389–402 (1986)16. Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tôhoku Math. J. 24, 93–103 (1972)17. Marrero, J.C.: The local structure of trans-Sasakian manifolds. Ann. Mat. Pura Appl. 162, 77–86 (1992)18. Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un.
Mat. Argent. 47, 1–22 (2006)19. Ou, Y.-L.: p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps. J. Geom.
Phys. 56, 358–374 (2006)20. Özgür, C., Tripathi, M.: On Legendre curves in α-Sasakian manifolds. Bull. Malays. Math. Sci. Soc.
31(1), 91–96 (2008)21. Oubiña, J.A.: New classes of almost contact metric structures. Publ. Math. Debrecen 32(3–4), 187–193
