Abstract
This paper is devoted to examine necessary and sufficient conditions for a Frenet curve to be f-harmonic, f-biharmonic, bi-f-harmonic and f-biminimal in three-dimensional \(\beta \)-Kenmotsu manifolds. In addition, such conditions are investigated for slant curves.
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1 Introduction
The concept of f-Kenmotsu manifold was defined for the first time in [9] by Jannsens and Vanhecke, where f is a real constant. Subsequently, Olszak and Rosca [15] investigated normal locally conformal almost cosymplectic manifolds and gave a differential geometric interpretation of such manifolds which are called f-Kenmotsu manifolds, where f is a function on M, [15].
On the other hand, in [7], Eells and Sampson defined harmonic maps between Riemannian manifolds, and in [6], Lemaire and Eells studied various topics in harmonic maps. On the other hand, Mangione published a paper which he considered harmonic maps in f-Kenmotsu manifold, in [13]. These maps are widely studied as they have an comprehensive field of study due to their wide applications.
In [7], Eells and Sampson studied not only harmonic maps, but also biharmonic maps between the Riemannian manifolds by generalizing harmonic maps. Besides, in [20], Perktaş et al. studied biharmonic curves in three-dimensional f-Kenmotsu manifold for the first time.
f-Harmonic maps between Riemannian manifolds were introduced by Lichnerowicz in 1970 and then examined by Eells and Lemaire, in [6]. f-Harmonic maps, as the solution of inhomogeneous Heisenberg spin systems and continuous spin systems, are of interest not only for mathematicians but also for physicists [2].
In [12], Lu defined f-biharmonic maps, which are the generalization of biharmonic maps. He also studied f-biharmonic maps between Riemannian manifolds, in [5]. Besides, Ou [16] gave a complete classification of f-biharmonic curves in three-dimensional Euclidean space and characterization of f-biharmonic curves in n-dimensional space forms.
Bi-f-harmonic maps as a generalization of biharmonic and f-harmonic maps were introduced by Ouakkas et al. [17]. In addition, Roth defined a non-f-harmonic, f-biharmonic map called as a proper f-biharmonic map [21]. In [19], Perktaş et al. obtained bi-f -harmonicity conditions for curves in Riemannian manifolds and discussed the particular cases of the Euclidean space, unit sphere and hyperbolic space.
Finally, Loubeau and Montaldo [11] studied biminimal curves in a Riemannian manifold. Moreover, Perktaş et al. handled these types of curves in f-Kenmotsu manifolds in [20]. On the other hand, Karaca and Özgür defined f-biminimal immersions and they handled f -biminimal curves in a Riemannian manifold, in [8].
This paper, which we prepared with the inspiration got from these studies, is organized as follows. In Sects. 2 and 3, we give basic definitions and properties of Frenet curves in three-dimensional \(\beta \)-Kenmotsu manifolds which will be needed in other sections, respectively. In Sect. 4, we prove that there is no proper f-harmonic Frenet curve in three-dimensional \(\beta \) -Kenmotsu manifold. In Sect. 5, we derive the f-biharmonicity conditions for a Frenet curve in a three-dimensional \(\beta \)-Kenmotsu manifold and give a nonexistence theorem. In Sect. 6, we get bi-f-harmonicity conditions not only for a Frenet curve but also a slant and a Legendre curve in three-dimensional \(\beta \)-Kenmotsu manifolds. Finally, in the last section, we investigate f-biminimality conditions.
2 Preliminaries
In this section, we remind some definitions and propositions which will be needed throughout the paper.
A differentiable manifold \(M^{2n+1}\) is called an almost contact metric manifold with the almost contact metric structure \((\varphi ,\xi ,\eta ,g) \) if it admits a tensor field \(\varphi \) of type (1, 1), a vector field \(\xi \) , a 1-form \(\eta \), and a Riemannian metric tensor field g satisfying the following conditions [3]:
where I denotes the identity transformation and \( X,Y \in \Gamma (TM) \).
An almost contact metric manifold is said to be an f-Kenmotsu manifold if the Levi–Civita connection \( \nabla \) of g satisfies
where f is a strictly positive differentiable function on M and \( X,Y \in \Gamma (TM) \), [9]. Here, if f is equal to a nonzero constant \( \beta \), then the manifold is called a \( \beta \)-Kenmotsu manifold [9, 14]. In particular, if \( \beta =1\), then the manifold is known as a Kenmotsu manifold [10].
For an f-Kenmotsu manifold, the curvature tensor field equation is given as
where \(X,Y,Z\in TM\) and r is the scalar curvature of M. [13].
Definition 2.1
\(\gamma :I \subset {\mathbb {R}}\longrightarrow M\) is called a slant curve if the contact angle \( \theta :I\rightarrow [0, 2\pi )\) of given by \(\cos \theta (s) = g(T(s), \xi ) \) is a constant function.
In particular, if \(\theta =\dfrac{\pi }{2}\) (or \(\dfrac{3\pi }{2})\), then is called a Legendre curve, [4].
Remark 2.2
For a slant curve in a \(\beta \)-Kenmotsu manifold, we have [4]
where \(\left| \sin \theta \right| \le \min {\dfrac{k_{1}}{\beta }}\) and
Remark 2.3
For a Legendre curve in a \(\beta \)-Kenmotsu manifold, we have
In particular, a Legendre curve in a \(\beta \)-Kenmotsu manifold is a circle [4, 20].
Definition 2.4
Let (M, g) and \(({\bar{M}},{\bar{g}})\) be Riemannian manifolds. Then, a harmonic map \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is defined as the critical point of the energy functional
where \(v_{g}\) is the volume element of (M, g). Using Euler–Lagrange equation of the energy functional \(E(\psi )\), where \(\tau (\psi )\) is the tension field of map \(\psi \), a map is called as harmonic if
Here, \(\nabla \) is the connection induced from the Levi–Civita connection \(\nabla ^{{\bar{M}}}\) of \({\bar{M}}\) and the pull-back connection \(\nabla ^{\psi }\) [7, 8].
As a natural generalization of harmonic maps, biharmonic maps are defined as below.
Definition 2.5
A map \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is defined as biharmonic if it is a critical point, for all variations, of the bienergy functional
Namely, \(\psi \) is a biharmonic map if \(\tau _{2} (\psi )\) which is the bitension field of \(\psi \) equals to
Here, \( R^{{\bar{M}}} \), the curvature tensor field of \({\bar{M}}\), is defined as
for any \(X,Y,Z \in \Gamma (T{\bar{M}})\) and \(\nabla ^{\psi }\) is the pull-back connection [7, 8].
Note that harmonic maps are always biharmonic and biharmonic maps which are not harmonic are called proper biharmonic maps [18].
Definition 2.6
A map \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is said to be an f-harmonic if it is critical point of f-energy functional
where \(f \in {C^{\infty }}({M,\mathrm I\!R})\) is a positive smooth function. Then, the f-harmonic map equation obtained using Euler–Lagrange equation as follows:
where \(\tau _{f}(\psi )\) is the f-tension field of the map \(\psi \).
Note that f-harmonic maps are generalizations of harmonic maps [1].
Definition 2.7
A map \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is said to be an f-biharmonic if it is critical point of the f-bienergy functional
The Euler–Lagrange equation for the f-biharmonic map is given by
where \(\tau _{2,f}(\psi )\) is the f-bitension field of the map \(\psi \) [5, 8].
Remark 2.8
An f-biharmonic map turns into a biharmonic map if f is a constant.
Definition 2.9
A map \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is said to be a bi-f-harmonic if it is a critical point of the bi-f-energy functional
The Euler–Lagrange equation for the bi-f-harmonic map is given by
where \(\tau _{f,2}(\psi ) \) is the bi-f-tension field of the map \(\psi \) [19].
Definition 2.10
An immersion \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is called biminimal if it is critical point of the bienergy functional \(E_{2}(\psi )\) for variations normal to the image \(\psi (M)\subset {\bar{M}}\), with fixed energy. Equivalently, there exists a constant \(\lambda \in \mathrm{I\!R}\), such that \(\psi \) is a critical point of the \(\lambda \)-bienergy
for any smooth variation of the map \(\psi _{t}:]-\epsilon ,+\epsilon [, \) \(\psi _{0}=\psi \), such that \(V=\textrm{d}\psi _{t}/\textrm{d}t|_{t_{0}}\) is normal to \(\psi (M)\) [11]. The Euler–Lagrange equation for a \(\lambda \)-biminimal immersion is
for some value of \(\lambda \in \mathrm{I\!R},\) where \([ . ]^{\perp } \) denotes the normal component of [. ].
An immersion is called free biminimal if it is biminimal for \(\lambda =0\) [8, 11].
Definition 2.11
An immersion \(\psi :(M,g)\rightarrow ({\bar{M}},{\bar{g}})\) is called f-biminimal if it is a critical point of the f-bienergy functional \(E_{2,f}(\psi )\) for variations normal to the image \(\psi (M)\subset {\bar{M}}\), with fixed energy. Equivalently, there exists a constant \(\lambda \in \mathrm{I\!R}\), such that \(\psi \) is a critical point of the \(\lambda \)-f-bienergy
for any smooth variation of the map \(\psi _{t}:]-\epsilon ,+\epsilon [, \) \(\psi _{0}=\psi \). Using the Euler–Lagrange equations for f-harmonic and f-biharmonic maps, an immersion is f-biminimal if
for some value of \(\lambda \in \mathrm{I\!R}.\)
An immersion is called free f-biminimal if it is f-biminimal for \(\lambda =0.\) If f is a constant, then the immersion is biminimal [8].
3 Frenet curves in three-dimensional \(\beta \)-Kenmotsu manifold
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s. The Serret–Frenet frame defined on \(\gamma \) denoted by \({T=\gamma ^{'}(s), N, B}\) which are the tangent, the principal normal, and the binormal vector fields, respectively. Here, Serret–Frenet formulas are given as
where \(k_{1}\) and \(k_{2}\) are the curvature and the torsion of the curve, respectively.
Using these Serret–Frenet formulas, we get
and by substituting (3.2) to the curvature tensor formula (2.4), we have
With the help of these calculations, we shall present f-harmonicity, f-biharmonicity, bi-f-harmonicity, and f-biminimality conditions of a Frenet curve in a three-dimensional \(\beta \)-Kenmotsu manifold as in the following sections.
4 f-Harmonic curves in three-dimensional \(\beta \)-Kenmotsu manifold
In this section, we investigate the f-harmonicity condition for a curve in a three-dimensional \(\beta \)-Kenmotsu manifold. Let \(\gamma :I\subset \mathrm{I\!R}\rightarrow {M}\) be a curve in a three-dimensional \(\beta \)-Kenmotsu manifold. Then, via definition (2.6), the f-harmonicity condition given as below
From (4.1), we get following nonexistence theorem.
Theorem 4.1
There is no proper f-harmonic Frenet curve in a three-dimensional \(\beta \)-Kenmotsu manifold.
Proof
Using the condition given in (4.1), it is easy to see that \(f^{'}=0\), so f is a constant. This situation contradicts the definition of a proper f-harmonic curve. \(\square \)
5 f-Biharmonic curves in three-dimensional \(\beta \)-Kenmotsu manifold
Here, we derive the f-biharmonicity condition for a curve in a three-dimensional \(\beta \)-Kenmotsu manifold. By substituting (3.2)–(3.5) in the equation of f-bitension field \(\tau _{2,f}(\gamma )\), f-biharmonicity condition is obtained as below
Taking the scalar product of (5.1) with T, N and B, respectively, we can state the following theorem.
Theorem 5.1
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\rightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s. Then, \(\gamma \) is an f-biharmonic curve if and only if
From Theorem 5.1, we obtain the following nonexistence theorems about f-biharmonic curves in three-dimensional \(\beta \)-Kenmotsu manifolds.
Theorem 5.2
There does not exist an f-biharmonic Frenet curve with constant curvature \(k_{1}\) in a three-dimensional \(\beta \)-Kenmotsu manifold.
Proof
Let \(k_{1}\) be a constant. Then, the first equation of (5.2) reduces to \(2k_{1}f^{'}=0\). Here, it is easy to see that f becomes a constant. This situation contradicts the definition of an f-biharmonic curve. \(\square \)
Theorem 5.3
There does not exist an f-biharmonic Legendre curve in a three-dimensional \(\beta \)-Kenmotsu manifold.
Proof
For a Legendre curve in a \(\beta \)-Kenmotsu manifold, it is well known that \(k_{1}=\beta \) where \(\beta \) is a constant, [4]. Therefore, the assumption \(k_{1} \ne \textrm{constant}\) contradicts the definition of a \(\beta \)-Kenmotsu manifold. \(\square \)
Theorem 5.4
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s. Then, \(\gamma \) is an f-biharmonic curve if and only if \(r,k_{1}\) and \(k_{2}\) satisfies the following conditions:
where \(f=ck_{1}^{-\frac{3}{2}}\) and c is the integration constant.
Proof
From the first equation of (5.2), it is easy to see that f equals to \(ck_{1}^{-\frac{3}{2}}\). Then, by substituting f and its derivatives into the second and third equation of (5.2), the proof is completed. \(\square \)
Next, we shall examine some special cases for an f-biharmonic curve in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 5-I: If \(k_{2}=0\), then (5.3) reduces to
Here, if we assume that \(\eta (N)=0\), then we obtain that \(\gamma \) is a Legendre curve. However, it is well known that for a Legendre curve in a three-dimensional \(\beta \)-Kenmotsu manifold \(\eta (N)=-1\), which contradicts our assumption. Therefore, in the second equation of (5.4), \(\eta (N)\) cannot be zero. In this case, we have following two subcases:
Subcase 5-I-1: If \((\dfrac{r}{2}+3\beta ^{2})=0\), then (5.4) reduces to
Then, we conclude the following theorem.
Theorem 5.5
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold of constant scalar curvature \(r= -6\beta ^{2}\) and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s with \(k_{1} \ne \textrm{constant}\) and \(k_{2}=0\). Then, \(\gamma \) is an f-biharmonic curve if and only if
where \(f=ck_{1}^{-\frac{3}{2}}\) and c is the integration constant.
Subcase 5-I-2: If \(\eta (B)=0\), then (5.4) reduces to
Since \(\xi =\eta (T)T+\eta (N)N\) and \((\eta (T))^2+(\eta (N))^2=1,\) we give the following theorem.
Theorem 5.6
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s with \(k_{1} \ne \textrm{constant},\) \(k_{2}=0\) and \(\eta (B)=0\). Then, \(\gamma \) is an f-biharmonic curve if and only if \(k_{1}\) satisfy the following differential equation:
where \(f=ck_{1}^{-\frac{3}{2}}\) and c is the integration constant.
Case 5-II: If \(k_{2}=\textrm{constant} >0\), then (5.3) reduces to
Hence, we have the following theorem.
Theorem 5.7
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s with \(k_{1} \ne \textrm{constant}\) and \(k_{2}=\textrm{constant} \). Then, \(\gamma \) is an f-biharmonic curve if and only if
and \(k_{1}\), \(k_{2},r\) satisfy the following differential equation:
Proof
From second equation of (5.5), we obtain that
Then, by substituting this result to the first equation of (5.5) and the formula \( f=ck_{1}^{-\frac{3}{2}},\) the proof is completed.
\(\square \)
Now, assume that \(\gamma :I\longrightarrow M\) is a slant curve such that N is non-parallel to \(\xi \). By means of Definition 2.1, Remark 2.2 and Theorem 5.1, the following theorem and corollary are obtained.
Theorem 5.8
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, \(\gamma \) is an f-biharmonic curve if and only if
where \(k_{1} \ne \textrm{constant}\).
Corollary 5.9
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, \(\gamma \) is an f-biharmonic curve if and only if
where \(k_{1} \ne \textrm{constant},\) \(f=ck_{1}^{-\frac{3}{2}}\), and c is the integration constant.
Now, we discuss some special cases for a slant f-biharmonic curve in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 5-III: If \(k_{1} \ne \textrm{constant}\) and \(k_{2}=0\), then (5.6) reduces to
Then, we get the following theorem:
Theorem 5.10
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, for \(k_{1} \ne \textrm{constant}\) and \(k_{2}=0,\) \(\gamma \) is an f-biharmonic curve if and only if M is of constant scalar curvature \(r= -6\beta ^{2}\) and
Case 5-IV: If \(k_{1} \ne \textrm{constant}\) and \(k_{2}=\textrm{constant}>0\), then (5.6) reduces to
Using first equation of (5.9), we get \(f=ck_{1}^{-\frac{3}{2}}\). Then, by substituting this result to the second and third equation of (5.9), we conclude the following.
Theorem 5.11
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve with \(k_{1} \ne \textrm{constant}\) and \(k_{2}=\textrm{constant}>0\). Then, \(\gamma \) is an f-biharmonic curve if and only if
where \(f=ck_{1}^{-\frac{3}{2}}\) and c is the integration constant.
6 Bi-f-harmonic curves in three-dimensional \(\beta \)-Kenmotsu manifold
In this section, we derive the bi-f-harmonicity condition for a Frenet curve in a three-dimensional \(\beta \)-Kenmotsu manifold. Using Eqs. (3.2)–(3.5) in the equation of bi-f-tension field \(\tau _{f,2}(\gamma )\), see [19], we obtain bi-f-harmonicity condition as below
Therefore, we can state the following theorem:
Theorem 6.1
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s. Then, \(\gamma \) is a bi-f-harmonic curve if and only if
Now, we shall examine some special cases for the bi-f-harmonic curves in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 6-I: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), then (6.2) reduces to
Since any of \((\dfrac{r}{2}+3\beta ^{2})\) or \(\eta (B)\) in the third equation of (6.3) can be equal to zero, we examine Case 6-I in two subcases.
Subcase 6-I-1: If \((\dfrac{r}{2}+3\beta ^{2})=0\), then (6.3) reduces to
Then, we have the following theorem.
Theorem 6.2
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve with \(k_{1}=\textrm{constant}>0,\) \(k_{2}=0\) and \(r=-6\beta ^{2}\). Then, \(\gamma \) is a bi-f-harmonic curve if and only if \(k_{1}, f, \beta \) satisfy the following differential equation:
Corollary 6.3
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve with \(k_{1}=\textrm{constant}>0,\) \(k_{2}=0\) and \(r=-6\beta ^{2}\). Then, \(\gamma \) is a bi-f-harmonic curve if and only if either
where \( \beta ^{2}-5k_{1}^{2} <0,\) or
where \( \beta ^{2}-5k_{1}^{2} >0,\) \(c_i\) \( (1 \le i \le 4)\) are real constants.
Subcase 6-I-2: If \(\eta (B)=0\), then (6.3) reduces to
Since \(\xi =\eta (T)T+\eta (N)N\) and \((\eta (T))^2+(\eta (N))^2=1,\) we give the following theorem.
Theorem 6.4
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s with \(k_{1} =\textrm{constant}>0,\) \(k_{2}=0\) and \(\eta (B)=0\). Then, \(\gamma \) is a bi-f-harmonic curve if and only if
Case 6-II: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\), then (6.2) reduces to
Then, we have the following.
Theorem 6.5
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve with \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\). Then, \(\gamma \) is a bi-f-harmonic curve if and only if
Now, assume that \(\gamma :I\longrightarrow M\) is a slant curve, such that N is non-parallel to \(\xi \). By means of Definition 2.1, Remark 2.2 and Theorem 6.1, the following theorem is obtained.
Theorem 6.6
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, \(\gamma \) is a bi-f-harmonic curve if and only if
We shall consider some special cases for bi-f-harmonic slant curves in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 6-III: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), then (6.5) reduces to
Hence, we give the following.
Theorem 6.7
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a slant curve. Then, for \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), \(\gamma \) is a bi-f-harmonic curve if and only if M is of constant scalar curvature \(r= -6\beta ^{2}\) and
Case 6-IV: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\), then (6.5) reduces to
We have the following theorem.
Theorem 6.8
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve with \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\). Then, \(\gamma \) is a bi-f-harmonic if and only if
Now, assume that \(\gamma :I\longrightarrow M\) is a Legendre curve. By means of Definition 2.1, Remark 2.3, and Theorem 6.1, the following theorem is obtained.
Theorem 6.9
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Legendre curve. Then, \(\gamma \) is a bi-f-harmonic curve if and only if the function f satisfies the following differential equation:
7 f-Biminimal curves in three-dimensional \(\beta \)-Kenmotsu manifold
Finally, in this section, we derive the f-biminimality condition for a Frenet curve in a three-dimensional \(\beta \)-Kenmotsu manifold. The f-biminimality condition, see [8], obtained as below using normal components of f-tension and f-bitension field with the help of \(\lambda \)-f-bienergy functional
Using (7.1) we obtain the following.
Theorem 7.1
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve parametrized by arclength s. Then, \(\gamma \) is an f-biminimal curve if and only if
Now, we discuss some special cases for a f-biminimal curve in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 7-I: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), then (7.2) reduces to
In the third equation of (7.3), \((\dfrac{r}{2}+3\beta ^{2})\) or \(\eta (B)\) can be equal to zero, so we consider Case 7-I in two subcases.
Subcase 7-I-1: If \((\dfrac{r}{2}+3\beta ^{2})=0\), then (7.3) reduces to
Subcase 7-I-2: If \(\eta (B)=0\), we know that \(\eta (T)^{2}+\eta (N)^{2}=1,\) which reduces (7.3) to the following:
Since in Subcase 7-I-1, \(r=-6\beta ^{2}\), then (7.4) and (7.5) overlap.
Thus, we get the following theorem.
Theorem 7.2
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve with \(k_{1}=\textrm{constant}>0\), \(k_{2}=0.\) Then, \(\gamma \) is an f-biminimal curve if and only if either \(r=-6\beta ^2\) or \(\eta (B)=0\) and, in both cases, f satisfies
where \( k_{1}^2+\beta ^2+\lambda <0,\) and
where \( k_{1}^2+\beta ^2+\lambda >0,\) \(c_i\) \( (1 \le i \le 4)\) are real constants.
Case 7-II: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\), then (7.2) reduces to
Using second equation of (7.6) into the first equation, we get the following theorem.
Theorem 7.3
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Frenet curve with \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\). Then, \(\gamma \) is an f-biminimal curve if and only if
Now, assume that \(\gamma :I\longrightarrow M\) is a slant curve, such that N is non-parallel to \(\xi \). By means of Definition 2.1, Remark 2.2, and Theorem 7.1, the following theorem is obtained.
Theorem 7.4
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, \(\gamma \) is an f-biminimal curve if and only if
Here, we examine some cases for the f-biminimal slant curves in a three-dimensional \(\beta \)-Kenmotsu manifold.
Case 7-III: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), then (7.7) reduces to
Then, we have the following.
Theorem 7.5
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve. Then, for \(k_{1}=\textrm{constant}>0\) and \(k_{2}=0\), \(\gamma \) is an f-biminimal curve if and only if M is of constant curvature \(r=-6\beta ^{2}\) and either
where \( k_{1}^2+\beta ^2+\lambda <0,\) or
where \( k_{1}^2+\beta ^2+\lambda >0,\) \(c_i\) \( (1 \le i \le 4)\) are real constants.
Case 7-IV: If \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\), then (7.7) reduces to
Hence, we get
Theorem 7.6
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic slant curve with \(k_{1}=\textrm{constant}>0\) and \(k_{2}=\textrm{constant}>0\). Then, \(\gamma \) is an f-biminimal curve if and only if
Now, assume that \(\gamma :I\longrightarrow M\) is a Legendre curve. Via Definition 2.1, Remark 2.3, and Theorem 7.1, the following theorem is obtained.
Theorem 7.7
Let \((M,\varphi ,\xi ,\eta ,g)\) be a three-dimensional \(\beta \)-Kenmotsu manifold and \(\gamma :I\longrightarrow M\) be a non-geodesic Legendre curve. Then, \(\gamma \) is an f-biminimal curve if and only if either
where \( 2\beta ^2+\lambda <0,\) or
where \( 2\beta ^2+\lambda >0,\) \(c_i\) \( (1 \le i \le 4)\) are real constants.
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Bozdağ, Ş.N., Perktaş, S.Y. & Erdoğan, F.E. On some curves in three-dimensional \(\beta \)-Kenmotsu manifolds. Arab. J. Math. 12, 89–103 (2023). https://doi.org/10.1007/s40065-022-00405-w
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DOI: https://doi.org/10.1007/s40065-022-00405-w