On some curves in three-dimensional $\beta$-Kenmotsu manifolds

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.

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]:

$$\begin{aligned}{} & {} \varphi ^{2}=-I+\eta \otimes \xi , \nonumber \\{} & {} \eta (\xi )=1, \quad \varphi \xi =0, \quad \eta \circ \varphi =0, \quad \eta (X)=g(X,\xi ),\nonumber \\{} & {} g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(2.1)

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

$$\begin{aligned} \left( \nabla _{X}\,\varphi \right) Y= & {} f \left( g(\varphi X,Y\right) \xi -\eta (Y)\varphi X),\end{aligned}$$

(2.2)

$$\begin{aligned} \nabla _{X}\,\xi= & {} f(X-\eta (X)\xi ), \end{aligned}$$

(2.3)

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

$$\begin{aligned} R(X,Y)Z= & {} \left( \frac{r}{2}+2( f^{2}+f^{'})\right) ( g(Y,Z)X-g(X,Z)Y ) \nonumber \\{} & {} -\! \left( \frac{r}{2}\!+\!3( f^{2}\!+\!f^{'})\right) (g(Y,Z)\eta (X)\xi \!-\!g(X,Z)\eta (Y)\xi \!-\! \eta (X)\eta (Z)Y\!+\!\eta (Y)\eta (Z)X ), \end{aligned}$$

(2.4)

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]

$$\begin{aligned} \eta (N)=-\dfrac{\beta }{k_{1}}(\sin \theta )^{2}, \end{aligned}$$

(2.5)

where $\left| \sin \theta \right| \le \min {\dfrac{k_{1}}{\beta }}$ and

$$\begin{aligned} \eta (B)=\dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}. \end{aligned}$$

(2.6)

Remark 2.3

For a Legendre curve in a $\beta$-Kenmotsu manifold, we have

$$\begin{aligned} N=-\xi , \quad k_{1}=\beta , \quad k_{2}=0. \end{aligned}$$

(2.7)

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

$$\begin{aligned} E(\psi )=\frac{1}{2}\int _{M}|\textrm{d}\psi |^{2}v_{g}, \end{aligned}$$

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

$$\begin{aligned} \tau (\psi ):=\textrm{trace}\nabla \textrm{d}\psi =0. \end{aligned}$$

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

$$\begin{aligned} E_{2}(\psi )=\frac{1}{2}\int _{M}|\tau (\psi ) |^{2}v_{g}. \end{aligned}$$

Namely, $\psi$ is a biharmonic map if $\tau _{2} (\psi )$ which is the bitension field of $\psi$ equals to

$$\begin{aligned} \tau _{2} (\psi )=\textrm{trace}(\nabla ^{\psi }\nabla ^{\psi }-\nabla ^{\psi }_{\nabla })\tau (\psi )-\textrm{trace}(R^{{\bar{M}}}(\textrm{d}\psi ,\tau (\psi ))\textrm{d}\psi ) =0. \end{aligned}$$

Here, $R^{{\bar{M}}}$, the curvature tensor field of ${\bar{M}}$, is defined as

$$\begin{aligned} R^{{\bar{M}}}(X,Y)Z=\nabla ^{{\bar{M}}}_{X}\nabla ^{{\bar{M}}}_{Y}Z-\nabla ^{{\bar{M}}}_{Y}\nabla ^{{\bar{M}}}_{X}Z-\nabla ^{{\bar{M}}}_{[X,Y]}Z, \end{aligned}$$

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

$$\begin{aligned} E_{f}(\psi )=\frac{1}{2}\int _{M}f|\textrm{d}\psi |^{2}v_{g}, \end{aligned}$$

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:

$$\begin{aligned} \tau _{f}(\psi )=f\tau (\psi )+\textrm{d}\psi (\textrm{grad}f)=0, \end{aligned}$$

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

$$\begin{aligned} E_{2,f}(\psi )=\frac{1}{2}\int _{M}f|\tau (\psi ) |^{2}v_{g}. \end{aligned}$$

The Euler–Lagrange equation for the f-biharmonic map is given by

$$\begin{aligned} \tau _{2,f}(\psi )=f\tau _{2}(\psi )+\Delta f\tau (\psi )+2\nabla ^{\psi }_{\textrm{grad}f}\tau (\psi )=0, \end{aligned}$$

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

$$\begin{aligned} E_{f,2}(\psi )=\frac{1}{2}\int _{M}|\tau _{f}(\psi ) |^{2}v_{g}. \end{aligned}$$

The Euler–Lagrange equation for the bi-f-harmonic map is given by

$$\begin{aligned} \tau _{f,2}(\psi )=\textrm{trace}\big ((\nabla ^{\psi }f(\nabla ^{\psi }\tau _{f}(\psi )) -f \nabla _{\nabla ^{M}}^{\psi }\tau _{f}(\psi )+fR^{{\bar{M}}}(\tau _{f}(\psi ),\textrm{d}\psi )\textrm{d}\psi \big )=0, \end{aligned}$$

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

$$\begin{aligned} E_{2,\lambda }(\psi )=E_{2}(\psi )+\lambda E(\psi ) \end{aligned}$$

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

$$\begin{aligned}{}[\tau _{2,\lambda }(\psi )]^{\perp }=[\tau _{2}(\psi )]^{\perp }-\lambda [\tau (\psi )]^{\perp }=0, \end{aligned}$$

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

$$\begin{aligned} E_{2,\lambda ,f}(\psi )=E_{2,f}(\psi )+\lambda E_{f}(\psi ), \end{aligned}$$

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

$$\begin{aligned}{}[\tau _{2,\lambda ,f}(\psi )]^{\perp }=[\tau _{2,f}(\psi )]^{\perp }-\lambda [\tau _{f}(\psi )]^{\perp }=0, \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} \nabla _{T}T=k_{1}N \\ \nabla _{T}N=-k_{1}T+k_{2}B\\ \nabla _{T}B=-k_{2}N, \end{array}\right. } \end{aligned}$$

(3.1)

where $k_{1}$ and $k_{2}$ are the curvature and the torsion of the curve, respectively.

Using these Serret–Frenet formulas, we get

$$\begin{aligned} \nabla _{T}T= & {} k_{1}N, \end{aligned}$$

(3.2)

$$\begin{aligned} \nabla _{T}^{2}T= & {} -k_{1}^{2}T+k_{1}^{'}N+k_{1}k_{2}B, \end{aligned}$$

(3.3)

$$\begin{aligned} \nabla _{T}^{3}T= & {} -3k_{1}k_{1}^{'}T+(-k_{1}^{3}-k_{1}k_{2}^{2}+k_{1}^{''})N+(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})B, \end{aligned}$$

(3.4)

and by substituting (3.2) to the curvature tensor formula (2.4), we have

$$\begin{aligned} R(T,\nabla _{T}T)T= & {} -k_{1}\left( \frac{r}{2}+2\beta ^{2}\right) N \nonumber \\{} & {} -k_{1}\left( \frac{r}{2}+3\beta ^{2}\right) [\eta (N)\eta (T)T-\eta (T)^{2}N-\eta (N)\xi ]. \end{aligned}$$

(3.5)

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

$$\begin{aligned} \tau _{f}(\gamma )=f^{'}T+f\nabla _{T}T=f^{'}T+f(k_{1}N)=0. \end{aligned}$$

(4.1)

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

$$\begin{aligned} \tau _{2,f}(\gamma )= & {} f \tau _{2}(\gamma )+(\Delta f)\tau (\gamma )+2\nabla _{\text {grad}f}^{\gamma }\tau (\gamma ) \nonumber \\ {}= & {} f(\nabla _{T}^{3}T-R(T,\nabla _{T}T)T)+f^{''}\nabla _{T}T+2f^{'}\nabla _{T}^{2}T\nonumber \\ {}= & {} \left[ -3k_{1}k_{1}^{'}f+k_{1}f\left( \frac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (T)-2k_{1}^{2}f^{'}\right] T \nonumber \\ {}{} & {} +\Big [\left( -k_{1}^{3}-k_{1}k_{2}^{2}+k_{1}^{''}+k_{1}\left( \frac{r}{2}+2\beta ^{2}\right) -k_{1}\left( \frac{r}{2}+3\beta ^{2}\right) \eta (T)^{2}\right) f +2k_{1}^{'}f^{'}+k_{1}f^{''}\Big ]N \nonumber \\ {}{} & {} +\Big [(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})f+2k_{1}k_{2}f^{'}\Big ]B\nonumber \\ {}{} & {} - k_{1}f\left( \frac{r}{2}+3\beta ^{2}\right) \eta (N)\xi \nonumber \\ {}= & {} 0. \end{aligned}$$

(5.1)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} 3k_{1}^{'}f+2k_{1}f^{'}=0,\\ k_{1}^{2}+k_{2}^{2}-\dfrac{k_{1}^{''}}{k_{1}}-\dfrac{r}{2}-2\beta ^{2}+\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})-2\dfrac{k_{1}^{'}}{k_{1}}\dfrac{f^{'}}{f}-\dfrac{f^{''}}{f}=0, \\ \dfrac{2k_{1}^{'}k_{2}}{k_{1}}+k_{2}^{'}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)+2k_{2}\dfrac{f^{'}}{f}=0. \end{array}\right. } \end{aligned}$$

(5.2)

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:

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2} -\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ k_{2}^{'}-k_{2}\dfrac{k_{1}^{'}}{k_{1}}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B) =0, \end{array}\right. } \end{aligned}$$

(5.3)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ \left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B) =0. \end{array}\right. } \end{aligned}$$

(5.4)

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

$$\begin{aligned} k_{1}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}. \end{aligned}$$

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

$$\begin{aligned} k_{1}=\sqrt{\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}-\beta ^{2}}, \end{aligned}$$

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

$$\begin{aligned} k_{1}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}). \end{aligned}$$

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:

$$\begin{aligned} k_{1}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}-\beta ^{2}, \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ k_{2}\dfrac{k_{1}^{'}}{k_{1}}+\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B) =0. \end{array}\right. } \end{aligned}$$

(5.5)

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

$$\begin{aligned} f=c(\textrm{e}^{\int \frac{1}{k_{2}}\left( \frac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)\textrm{d}s})^{\frac{3}{2}} \end{aligned}$$

and $k_{1}$, $k_{2},r$ satisfy the following differential equation:

$$\begin{aligned} k_{1}^{2}+k_{2}^{2}=\dfrac{3}{4}(\dfrac{k_{1}^{'}}{k_{1}})^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}+\dfrac{k_{2}k_{1}^{'}(\eta (T)^{2}+\eta (N)^{2})}{k_{1}\eta (N)\eta (B)}. \end{aligned}$$

Proof

From second equation of (5.5), we obtain that

$$\begin{aligned} k_{1}=\textrm{e}^{-\int \frac{1}{k_{2}}\left( \frac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)\textrm{d}s}. \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} 3k_{1}^{'}f+2k_{1}f^{'}=0, \\ k_{1}^{2}+k_{2}^{2}=\dfrac{k_{1}^{''}}{k_{1}}+2\dfrac{k_{1}^{'}}{k_{1}}\dfrac{f^{'}}{f}+\dfrac{f^{''}}{f}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ \dfrac{2k_{1}^{'}k_{2}}{k_{1}}+k_{2}^{'}+2k_{2}\dfrac{f^{'}}{f} -\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0, \end{array}\right. } \end{aligned}$$

(5.6)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}- \left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ k_{2}^{'}-\dfrac{k_{1}^{'}k_{2}}{k_{1}}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0, \end{array}\right. } \end{aligned}$$

(5.7)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} 3k_{1}^{'}f+2k_{1}f^{'}=0, \\ k_{1}^{2}=\dfrac{k_{1}^{''}}{k_{1}}+2\dfrac{k_{1}^{'}}{k_{1}}\dfrac{f^{'}}{f}+\dfrac{f^{''}}{f}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ \left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| sin\theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(sin\theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

(5.8)

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

$$\begin{aligned} f=ck_{1}^{-\frac{3}{2}} ,\quad k_{1}=\sqrt{\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}-\beta ^{2}}. \end{aligned}$$

Case 5-IV: If $k_{1} \ne \textrm{constant}$ and $k_{2}=\textrm{constant}>0$, then (5.6) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} 3k_{1}^{'}f+2k_{1}f^{'}=0, \\ k_{1}^{2}+k_{2}^{2}=\dfrac{k_{1}^{''}}{k_{1}}+2\dfrac{k_{1}^{'}}{k_{1}}\dfrac{f^{'}}{f}+\dfrac{f^{''}}{f}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ \dfrac{2k_{1}^{'}k_{2}}{k_{1}}+2k_{2}\dfrac{f^{'}}{f}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

(5.9)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{3}{4}\left( \dfrac{k_{1}^{'}}{k_{1}}\right) ^{2}-\dfrac{k_{1}^{''}}{2k_{1}}+\dfrac{r}{2}+2\beta ^{2}- \left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ \dfrac{k_{1}^{'}k_{2}}{k_{1}}+\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0, \end{array}\right. } \end{aligned}$$

(5.10)

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

$$\begin{aligned} \tau _{f,2}(\gamma )= & {} \text {trace}\big (\nabla ^{\gamma }f(\nabla ^{\gamma }\tau _{f}(\gamma )) -f \nabla _{\nabla }^{\gamma }\tau _{f}(\gamma )+fR(\tau _{f}(\gamma ),d\gamma )d\gamma \big ) \nonumber \\ {}= & {} (ff^{''})^{'}T+(3ff^{''}+2(f^{'})^{2})\nabla _{T}T +4ff^{'}\nabla ^{2}_{T}T +f^{2}\nabla ^{3}_{T}T+f^{2}R(\nabla _{T}T,T)T \nonumber \\ {}= & {} \big [(ff^{''})^{'}-4k_{1}^{2}ff^{'}-3k_{1}k_{1}^{'}f^{2}+f^{2}k_{1}(\dfrac{r}{2}+3\beta ^{2})\eta (N)\eta (T)\big ]T\nonumber \\ {}{} & {} + \big [(3ff^{''}+2(f^{'})^{2})k_{1}+4ff^{'}k_{1}^{'}-f^{2}(k_{1}^{3}+k_{1}k_{2}^{2}-k_{1}^{''}) +f^{2}k_{1}\big (\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (T)^{2}\big )\big ]N\nonumber \\ {}{} & {} + \big [4ff^{'}k_{1}k_{2}+f^{2}(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})\big ]B\nonumber \\ {}{} & {} -f^{2}k_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\xi \nonumber \\ {}= & {} 0. \end{aligned}$$

(6.1)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}-3k_{1}k_{1}^{'}f^{2}=0,\\ {\left\{ \begin{array}{ll} (3ff^{''}+2(f^{'})^{2})k_{1}+4ff^{'}k_{1}^{'}-f^{2}(k_{1}^{3}+k_{1}k_{2}^{2}-k_{1}^{''})\\ +f^{2}k_{1}\big [\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})\big ]=0,\end{array}\right. } \\ 4f^{'}k_{1}k_{2}+f(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})-fk_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. } \end{aligned}$$

(6.2)

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

$$\begin{aligned} \small { {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ (3ff^{''}+2(f^{'})^{2})k_{1}+f^{2}k_{1}\left( -k_{1}^{2}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})\right) =0, \\ \left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. }} \end{aligned}$$

(6.3)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ (3ff^{''}+2(f^{'})^{2})-f^{2}\big (k_{1}^{2}+\beta ^{2})=0. \end{array}\right. } \end{aligned}$$

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:

$$\begin{aligned} 2 f^{'}f^{''}+(5k_{1}^{2}-\beta ^{2})ff^{'}=0. \end{aligned}$$

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

$$\begin{aligned} f(s)=c_1\cos \left( \sqrt{\dfrac{\beta ^{2}-5k_{1}^{2}}{2}}s\right) +c_2\sin \left( \sqrt{\dfrac{\beta ^{2}-5k_{1}^{2}}{2}}s\right) , \end{aligned}$$

where $\beta ^{2}-5k_{1}^{2} <0,$ or

$$\begin{aligned} f(s)=c_3e^{-\sqrt{\dfrac{\beta ^{2}-5k_{1}^{2}}{2}}s}+c_4e^{\sqrt{\dfrac{\beta ^{2}-5k_{1}^{2}}{2}}s}, \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ (3ff^{''}+2(f^{'})^{2})k_{1}+f^{2}k_{1}\left( -k_{1}^{2}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})\right) =0. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0,\\ (3ff^{''}+2(f^{'})^{2})-f^{2}\big (k_{1}^{2}+\beta ^{2}\big )=0. \end{array}\right. } \end{aligned}$$

Case 6-II: If $k_{1}=\textrm{constant}>0$ and $k_{2}=\textrm{constant}>0$, then (6.2) reduces to

$$\begin{aligned} \small { {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ (3ff^{''}+2(f^{'})^{2})k_{1}-f^{2}k_{1}\left( k_{1}^{2}+k_{2}^{2}-\dfrac{r}{2}-2\beta ^{2}+\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})\right) =0, \\ 4f^{'}k_{1}k_{2}-fk_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. } } \end{aligned}$$

(6.4)

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

$$\begin{aligned} \small { {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ 3ff^{''}+2(f^{'})^{2}-f^{2}(k_{1}^{2}+k_{2}^{2})+f^{2}\left[ \dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2})\right] =0, \\ 4f^{'}k_{2}-f\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. }} \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}-3k_{1}k_{1}^{'}f^{2}=0, \\ {\left\{ \begin{array}{ll} (3ff^{''}+2(f^{'})^{2})k_{1}+4ff^{'}k_{1}^{'}-f^{2}(k_{1}^{3}+k_{1}k_{2}^{2}-k_{1}^{''})\\ +f^{2}k_{1}\left[ \dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) \right] =0, \end{array}\right. } \\ {\left\{ \begin{array}{ll} 4ff^{'}k_{1}k_{2}+f^{2}(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})\\ +f^{2}k_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0.\end{array}\right. } \end{array}\right. } \end{aligned}$$

(6.5)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ (3ff^{''}+2(f^{'})^{2})k_{1}-f^{2}k_{1}^{3}+f^{2}k_{1}\left[ \dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(sin\theta )^{4}\right) \right] =0, \\ f^{2}k_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} 2 f^{'}f^{''}+(5k_{1}^{2}-\beta ^{2})ff^{'}=0. \end{aligned}$$

Case 6-IV: If $k_{1}=\textrm{constant}>0$ and $k_{2}=\textrm{constant}>0$, then (6.5) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ {\left\{ \begin{array}{ll} (3ff^{''}+2(f^{'})^{2})k_{1}-f^{2}(k_{1}^{3}+k_{1}k_{2}^{2})\\ +f^{2}k_{1}\left[ \dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) \right] =0,\end{array}\right. } \\ 4ff^{'}k_{1}k_{2}-f^{2}k_{1}\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} (ff^{''})^{'}-4k_{1}^{2}ff^{'}=0, \\ 3ff^{''}+2(f^{'})^{2}+f^{2}\left[ -k_{1}^{2}-k_{2}^{2}+\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(sin\theta )^{4}\right) \right] =0, \\ 4ff^{'}k_{2}-f\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

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:

$$\begin{aligned} (ff^{''})^{'}+3f^{''}f+2(f^{'})^{2}-4\beta ^{2}f^{'}f-2\beta ^{2}f^{2}=0. \end{aligned}$$

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

$$\begin{aligned}{}[\tau _{2,\lambda ,f}(\gamma )]^{\perp }= & {} [\tau _{2,f}(\gamma )]^{\perp }-\lambda [\tau _{f}(\gamma )]^{\perp }\nonumber \\= & {} \Big [\left( -k_{1}^{3}-k_{1}k_{2}^{2}+k_{1}^{''}+k_{1}\left( \frac{r}{2}+2\beta ^{2}-\left( \frac{r}{2}+3\beta ^{2}\right) \eta (T)^{2}-\lambda \right) \right) f+2k_{1}^{'}f^{'}+k_{1}f^{''}\Big ]N \nonumber \\ {}{} & {} +[(2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})f+2k_{1}k_{2}f^{'}]B\nonumber \\ {}{} & {} - k_{1}f\left( \frac{r}{2}+3\beta ^{2}\right) \eta (N)\xi \nonumber \\ {}= & {} 0. \end{aligned}$$

(7.1)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{k_{1}^{''}}{k_{1}}+2\dfrac{k_{1}^{'}f^{'}}{k_{1}f} +\dfrac{f^{''}}{f}+2\beta ^{2}-\lambda +\dfrac{r}{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ (2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})f+2k_{1}k_{2}f^{'}-k_{1}f\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. } \end{aligned}$$

(7.2)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}=\dfrac{f^{''}}{f}-\lambda +\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ k_{1}f\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. } \end{aligned}$$

(7.3)

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

$$\begin{aligned} k_{1}^{2}=\dfrac{f^{''}}{f}+\dfrac{r}{2}+2\beta ^{2}-\lambda . \end{aligned}$$

(7.4)

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:

$$\begin{aligned} k_{1}^{2}=\dfrac{f^{''}}{f}-\lambda -\beta ^{2}. \end{aligned}$$

(7.5)

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

$$\begin{aligned} f(s)=c_1\cos ((\sqrt{k_{1}^2+\beta ^2+\lambda })s)+c_2\sin ((\sqrt{k_{1}^2+\beta ^2+\lambda })s), \end{aligned}$$

where $k_{1}^2+\beta ^2+\lambda <0,$ and

$$\begin{aligned} f(s)=c_3\textrm{e}^{-(\sqrt{k_{1}^2+\beta ^2+\lambda })s}+c_4\textrm{e}^{(\sqrt{k_{1}^2+\beta ^2+\lambda })s}, \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{f^{''}}{f}-\lambda +\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) (\eta (T)^{2}+\eta (N)^{2}), \\ 2k_{2}f^{'}-f\left( \dfrac{r}{2}+3\beta ^{2}\right) \eta (N)\eta (B)=0. \end{array}\right. } \end{aligned}$$

(7.6)

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

$$\begin{aligned} k_{1}^{2}+k_{2}^{2}=\dfrac{f^{''}}{f}-\dfrac{2k_{2}f^{'}(\eta (T)^{2}+\eta (N)^{2})}{f\eta (N)\eta (B)}+2\beta ^{2}+\dfrac{r}{2}-\lambda . \end{aligned}$$

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

$$\begin{aligned} \small { {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{k_{1}^{''}}{k_{1}}+2\dfrac{k_{1}^{'}f^{'}}{k_{1}f}+\dfrac{f^{''}}{f}-\lambda +\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ (2k_{1}^{'}k_{2}+k_{1}k_{2}^{'})f+2k_{1}k_{2}f^{'}+k_{1}f\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } } \end{aligned}$$

(7.7)

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}=\dfrac{f^{''}}{f}-\lambda +\dfrac{r}{2}+2\beta ^{2}-\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ k_{1}f\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} f(s)=c_1\cos ((\sqrt{k_{1}^2+\beta ^2+\lambda })s)+c_2\sin ((\sqrt{k_{1}^2+\beta ^2+\lambda }))s), \end{aligned}$$

where $k_{1}^2+\beta ^2+\lambda <0,$ or

$$\begin{aligned} f(s)=c_3\textrm{e}^{\big (\sqrt{k_{1}^2+\beta ^2+\lambda }\big )s}+c_4\textrm{e}^{-\big (\sqrt{k_{1}^2+\beta ^2+\lambda }\big )s}, \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} k_{1}^{2}+k_{2}^{2}=\dfrac{f^{''}}{f}-\lambda +\dfrac{r}{2}+2\beta ^{2} -\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( (\cos \theta )^{2}+\dfrac{f^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) , \\ 2k_{1}k_{2}f^{'}+k_{1}f\left( \dfrac{r}{2}+3\beta ^{2}\right) \left( \dfrac{f}{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-f^{2}(\sin \theta )^{2}}\right) =0. \end{array}\right. } \end{aligned}$$

(7.8)

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

$$\begin{aligned} k_{1}^{2}+k_{2}^{2}=- & {} \dfrac{2k_{2}f^{'}\left( (\cos \theta )^{2}+\dfrac{\beta ^{2}}{k_{1}^{2}}(\sin \theta )^{4}\right) -3\beta ^{2}\left( f\dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) }{f\left( \dfrac{\beta }{k_{1}}(\sin \theta )^{2}\right) \left( \dfrac{\left| \sin \theta \right| }{k_{1}}\sqrt{k_{1}^{2}-\beta ^{2}(\sin \theta )^{2}}\right) } \\+ & {} \dfrac{f^{''}}{f}+2\beta ^{2}-\lambda . \end{aligned}$$

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

$$\begin{aligned} f(s)=c_1\cos (\big (\sqrt{2\beta ^2+\lambda }\big )s)+c_2\sin (\big (\sqrt{2\beta ^2+\lambda }\big )s), \end{aligned}$$

where $2\beta ^2+\lambda <0,$ or

$$\begin{aligned} f(s)=c_3\textrm{e}^{-\big (\sqrt{2\beta ^2+\lambda }\big )s}+c_4\textrm{e}^{\big (\sqrt{2\beta ^2+\lambda }\big )s}, \end{aligned}$$

where $2\beta ^2+\lambda >0,$ $c_i$ $(1 \le i \le 4)$ are real constants.