On Interpolating Sesqui-Harmonic Legendre Curves in Sasakian Space Forms

TL;DR

This paper investigates the conditions under which Legendre curves in Sasakian space forms are interpolating sesqui-harmonic, providing theoretical criteria and an explicit example.

Contribution

It establishes necessary and sufficient conditions for interpolating sesqui-harmonic Legendre curves in Sasakian space forms and presents a concrete example.

Findings

Derived explicit conditions for interpolating sesqui-harmonic Legendre curves.

Provided an example of such a curve in a Sasakian space form.

Enhanced understanding of harmonic properties in Sasakian geometry.

Abstract

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

Equations

$$E(\phi )={\int }_{\Omega }{\Vert d\phi \Vert }^{2}d{\nu }_g,$$

$$E_2(\phi )={\int }_{\Omega }{\Vert \tau (\phi )\Vert }^{2}d{\nu }_g,$$

$$\tau (\phi )=tr(\nabla d\phi )=0,$$

$${\tau }_2(\phi )=tr({\nabla }^{\phi }{\nabla }^{\phi }-{\nabla }_{\nabla }^{\phi })\tau (\phi )-tr(R^N(d\phi ,\tau (\phi ))d\phi )=0,$$

$$E_{{\delta }_1,{\delta }_2}(\phi )={\delta }_1{\int }_{\Omega }{\Vert d\phi \Vert }^{2}d{\nu }_g+{\delta }_2{\int }_{\Omega }{\Vert \tau (\phi )\Vert }^{2}d{\nu }_g,$$

$${\tau }_{{\delta }_1,{\delta }_2}(\phi )={\delta }_2{\tau }_2(\phi )-{\delta }_1\tau (\phi )=0$$

$$N_{\varphi }=-2d\eta \otimes \xi$$

$$\left({\nabla }_X\varphi \right)Y=g(X,Y)\xi -\eta \left(Y\right)X$$

$${\nabla }_X\xi =-\varphi X$$

$$R\left(X,Y\right)Z=\frac{c+3}{4}\left\{g(Y,Z)X-g(X,Z)Y\right\}$$

$$+\frac{c-1}{4}\{g(X,\varphi Z)\varphi Y-g(Y,\varphi Z)\varphi X+2g(X,\varphi Y)\varphi Z$$

$$+\eta \left(X\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)X+g(X,Z)\eta \left(Y\right)\xi -g(Y,Z)\eta \left(X\right)\xi \}$$

$$E_1=T={\gamma }^{\prime },$$

$${\nabla }_T{E}_1=k_1{E}_2,$$

$${\nabla }_T{E}_i=-k_{i-1}{E}_{i-1}+k_i{E}_{i+1},2\le i\le n-1,$$

$${\nabla }_T{E}_n=-k_{n-1}{E}_{n-1},$$

$$-3{\delta }_2{k}_1{k}_1^{\prime }=0,$$

$${\delta }_2[k_1^{\prime \prime }-k_1^3-k_1{k}_2^2-\left(\frac{c+3}{4}\right)k_1$$

$$+3\left(\frac{c-1}{4}\right)k_1{\left[g(\varphi T,E_2)\right]}^2-\left(\frac{c-1}{4}\right)k_1{\left[\eta \left(E_2\right)\right]}^2]-{\delta }_1{k}_1=0,$$

$${\delta }_2[2k_1^{\prime }{k}_2+k_1{k}_2^{\prime }+3\left(\frac{c-1}{4}\right)k_{1}g(\varphi T,E_2)g(\varphi T,E_3)$$

$$-\left(\frac{c-1}{4}\right)k_1\eta \left(E_2\right)\eta \left(E_3\right)]=0,$$

$${\delta }_2[k_1{k}_2{k}_3+3\left(\frac{c-1}{4}\right)k_{1}g(\varphi T,E_2)g(\varphi T,E_4)$$

$$-\left(\frac{c-1}{4}\right)k_1\eta \left(E_2\right)\eta \left(E_4\right)]=0.$$

$$\tau (\gamma )={\nabla }_{T}T=k_1{E}_2.$$

$${\nabla }_T{\nabla }_{T}T=-k_1^2{E}_1+k_1^{\prime }{E}_2+k_1{k}_2{E}_3,$$

$${\nabla }_T{\nabla }_T{\nabla }_{T}T=-3k_1{k}_1^{{}^{\prime }}{E}_1+\left(k_1^{\prime \prime }-k_1^3-k_1{k}_2^2\right)E_2$$

$$+\left(2k_1^{\prime }{k}_2+k_1{k}_2^{\prime }\right)E_3+\left(k_1{k}_2{k}_3\right)E_4,$$

$$R(T,{\nabla }_{T}T)T=-\left(\frac{c+3}{4}\right)k_1{E}_2$$

$$-3\left(\frac{c-1}{4}\right)k_{1}g(\varphi T,E_2)\varphi T+\left(\frac{c-1}{4}\right)k_1\eta \left(E_2\right)\xi .$$

$${\tau }_{{\delta }_1,{\delta }_2}(\gamma )=\left(-3{\delta }_2{k}_1{k}_1^{\prime }\right)E_1+\left[{\delta }_2\left(k_1^{\prime \prime }-k_1^3-k_1{k}_2^2+\left(\frac{c+3}{4}\right)k_1\right)-{\delta }_1{k}_1\right]E_2$$

Full text

On Interpolating Sesqui-Harmonic Legendre Curves in Sasakian Space Forms

Fatma KARACA, Cihan ÖZGÜR and Uday Chand DE

Abstract. We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

**Mathematics Subject Classification. **53C25, 53C40, 53A05.

**Keywords and phrases. **Interpolating sesqui-harmonic curve, Legendre curve, Sasakian space form.

1. **Introduction **

A map $\varphi$ $:\left(M,g\right)\rightarrow\left(N,h\right)$ between Riemannian manifolds is called a *harmonic map *and  a *biharmonic map, respectively if it is a critical point of the $E(\varphi)$ *and $E_{2}(\varphi)$

[TABLE]

[TABLE]

where $\Omega$ is a compact domain of $M$. The harmonic map equation is

[TABLE]

and it is called the *tension field *of $\varphi$ [5]. The Euler-Lagrange equation of $E_{2}(\varphi)$ is

[TABLE]

and it is called the bitension field of $\varphi$ [11].

In [3], Branding defined and considered interpolating sesqui-harmonic maps between Riemannian manifolds. The author introduced an action functional for maps between Riemannian manifolds that interpolated between the actions for harmonic and biharmonic maps. The map $\varphi$ is said to be *interpolating sesqui-harmonic *if it is a critical point of $E_{\delta_{1},\delta_{2}}(\varphi)$ * *

[TABLE]

where $\Omega$ is a compact domain of $M$ and $\delta_{1},\delta_{2}\in\mathbb{R}$ [3]. The interpolating sesqui-harmonic map equation is

[TABLE]

for $\delta_{1},\delta_{2}\in\mathbb{R}$ [3]. An interpolating sesqui-harmonic map is biminimal if variations of (1.3) that are normal to the image $\varphi(M)\subset N$ and $\delta_{2}=1$, $\delta_{1}>0$ [13]. For some recent study of biminimal immersions see [8], [13], [14] and [15].

Interpolating sesqui-harmonic curves in a $3$-dimensional sphere were studied in [3]. In [6] and [7], Fetcu and Oniciuc considered biharmonic Legendre curves in Sasakian space forms. In [4], Cho, Inoguchi and Lee studied affine biharmonic curves in $3$-dimensional pseudo-Hermitian geometry. In [10], Inoguchi and Lee studied affine biharmonic curves in $3$-dimensional homogeneous geometries.  In [16], the second author and Güvenç studied biharmonic Legendre curves in generalized Sasakian space forms. In [9], Güvenç and the second author studied $f$-biharmonic Legendre curves in Sasakian space forms. Motivated by the above studies, in the present paper, we consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We obtain the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. We also give an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

2. Preliminaries

Let $M=(M^{2n+1},\phi,\xi,\eta,g)$ be an almost contact metric manifold with an almost contact metric structure $(\phi,\xi,\eta,g)$. A contact metric manifold $(M^{2n+1},\phi,\xi,\eta,g)$ is called a Sasakian manifold if it is normal, that is,

[TABLE]

where $N_{\phi}$ is the Nijenhuis tensor field of $\phi$ [1]. It is well-known that an almost contact metric manifold is Sasakian if and only if

[TABLE]

and

[TABLE]

[2]. The sectional curvature of a $\phi$-section is called a $\phi$-sectional curvature. When the $\phi$-sectional curvature is a constant, then the Sasakian manifold is called a* Sasakian space form and it is denoted by $M(c)$ [2]*. The curvature tensor $R$ of a Sasakian space form $M(c)$ is given by

[TABLE]

[TABLE]

[TABLE]

for all $X,Y,Z$ $\in TM$ [2].

A submanifold of a Sasakian manifold $M$ is called an integral submanifold if $\eta(X)=0$, for every tangent vector $X$. An integral curve of a Sasakian manifold $M$ is called a Legendre curve [2].

3. **Interpolating sesqui-harmonic Legendre curves in Sasakian

space forms **

Let $\gamma:I\subset\mathbb{R}\longrightarrow(M^{n},g)$ be a curve parametrized by arc length in a Riemannian manifold $(M^{n},g)$ . Then $\gamma$ is called a Frenet curve of osculating order $r$, $1\leq r\leq n$, if there exists orthonormal vector fields $\left\{E_{i}\right\}_{i=1,2,...n}$ along $\gamma$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the function $\left\{k_{1}=k,k_{2}=\tau,k_{3},...,k_{n-1}\right\}$ are called the curvatures of $\gamma$ [12].

Firstly, we have the following theorem for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form:

Theorem 3.1**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with constant $\phi$-sectional curvature $c$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ be a Legendre curve of osculating order $r$ and $m=min\{r,4\}$. Then $\gamma$ is interpolating sesqui-harmonic if and only if there exists real numbers $\delta_{1},\delta_{2}$ such that

$\left(1\right)$* $c=1$ or $\phi T\perp E_{2}$ or $\phi T\in\left\{E_{2},...,E_{m}\right\};$ and*

$\left(2\right)$* the first $m$ of the following equations are satisfied:*

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

Let $\gamma:I\longrightarrow M$ be a Legendre curve of osculating order $r$ in $M(c).$   By the use of (1.1) and (3.1), we have

[TABLE]

From (3.1), we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using the equations (3.6), (3.7) and (3.8) into the equation (4.1) in [3], we find

[TABLE]

[TABLE]

[TABLE]

Taking the scalar product of equation (3.9) with $E_{2},E_{3}$ and $E_{4}$ respectively, then we obtain the desired result. ∎

Now we shall discuss some special cases of Theorem 3.1:

Case I. $c=1.$

From Theorem 3.1,  we have:

Proposition 3.1**.**

Let $M(1)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c=1$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(1)$ be a Legendre curve of osculating order $r$ such that $\frac{\delta_{1}}{\delta_{2}}\neq 0$. Then $\gamma$ is interpolating sesqui-harmonic if and only if

[TABLE]

[TABLE]

[TABLE]

where $1-\frac{\delta_{1}}{\delta_{2}}>0,$ $\delta_{1},$ $\delta_{2}$ is a constant.

Proof.

Assume that $\gamma$ is an interpolating sesqui-harmonic Legendre curve of osculating order $r$ in $M(1)$ such that $\frac{\delta_{1}}{\delta_{2}}\neq 0$ and $c=1.$ From Theorem 3.1, we obtain the result. ∎

Using Proposition 3.1, we have:

Theorem 3.2**.**

Let $M(1)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c=1$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(1)$ be a non geodesic Legendre curve of osculating order $r$. Then

$(1)$* It is a Legendre geodesic or*

$(2)$* $\gamma$ is interpolating sesqui-harmonic with $\frac{\delta_{1}}{\delta_{2}}\neq 0$ if and only if it is a Legendre circle with $k_{1}=\sqrt{1-\frac{\delta_{1}}{\delta_{2}}}$ where $1-\frac{\delta_{1}}{\delta_{2}}>0$ is a constant or*

$(3)$* $\gamma$ is interpolating sesqui-harmonic with $\frac{\delta_{1}}{\delta_{2}}\neq 0$ if and only if it is a Legendre helix with $k_{1}^{2}+k_{2}^{2}=1-\frac{\delta_{1}}{\delta_{2}}$ where $1-\frac{\delta_{1}}{\delta_{2}}>0,$ $\delta_{1},$ $\delta_{2}$ is a constant.*

In both cases, if $1-\frac{\delta_{1}}{\delta_{2}}<0$, then such an interpolating sesqui-harmonic Legendre curve does not exist.

Proof.

Let $\gamma:I\longrightarrow M(1)$ be an interpolating sesqui-harmonic curve with* $\frac{\delta_{1}}{\delta_{2}}\neq 0$. From Theorem 3.1, if we consider the osculating order $r=2$, then $\gamma$ is a Legendre circle with $k_{1}=\sqrt{1-\frac{\delta_{1}}{\delta_{2}}}$ where $1-\frac{\delta_{1}}{\delta_{2}}>0$ is a constant. *Similarly, if we consider the osculating order $r=3$, then we obtain that $k_{2}$ is a non-zero constant. Thus, $\gamma$ is a Legendre helix with $k_{1}^{2}+k_{2}^{2}=1-\frac{\delta_{1}}{\delta_{2}}$ where $1-\frac{\delta_{1}}{\delta_{2}}>0$ is a constant. On the other hand, assume that $\gamma$ is a Legendre circle with $k_{1}=\sqrt{1-\frac{\delta_{1}}{\delta_{2}}}$ or a Legendre helix with $k_{1}^{2}+k_{2}^{2}=1-\frac{\delta_{1}}{\delta_{2}}$ where $1-\frac{\delta_{1}}{\delta_{2}}>0$ is a constant. Obviously, $\gamma$ satisfies Theorem 3.1, respectively. It is trivial that $1-\frac{\delta_{1}}{\delta_{2}}<0$ cannot be possible. If $1-\frac{\delta_{1}}{\delta_{2}}=0,$ we obtain a geodesic. This proves the theorem. ∎

Case II. $c\neq 1$ and $\phi T\perp E_{2}.$

From Theorem 3.1,  we can state:

Proposition 3.2**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1,$ $\phi T\perp E_{2}$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ be a Legendre curve of osculating order $r$ such that $\frac{\delta_{1}}{\delta_{2}}\neq 0$. Then $\gamma$ is interpolating sesqui-harmonic if and only if

[TABLE]

[TABLE]

[TABLE]

where $\delta_{1},$ $\delta_{2}$ is a constant.

Proof.

Let $\gamma$ be an interpolating sesqui-harmonic Legendre curve of osculating order $r$ in $M(c)$ such that $c\neq 1,$ $\phi T\perp E_{2}$ and $\frac{\delta_{1}}{\delta_{2}}\neq 0.$ From Theorem 3.1, we get the result. ∎

From [7], we have the following lemma:

Lemma 3.1**.**

[7]** Let $\gamma$ be a Legendre Frenet curve of osculating order $3$ in a Sasakian space form $M(c)$ and $\phi T\perp E_{2}$. Then $\left\{T=E_{1},E_{2},E_{3},\phi T,\nabla_{T}\phi T,\xi\right\}$ is linearly independent at any point of $\gamma$ and therefore $n\geq 3$.

Hence we can state:

Theorem 3.3**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1,$ $\phi T\perp E_{2}$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ a Legendre curve of osculating order $r$.

$(1)$* If $c\leq-3$ and $\frac{\delta_{1}}{\delta_{2}}$ $\geq 0$, then $\gamma$ is interpolating sesqui-harmonic if and only if it is a geodesic.*

$(2)$* If $c>-3$ and $\frac{\delta_{1}}{\delta_{2}}$ $<0$, then $\gamma$ is interpolating sesqui-harmonic if and only if either*

$(a)$* $\gamma$ is of osculating order $r=2$, $n\geq 2$ and $\gamma$ is a circle with $k_{1}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}},$ in which case $\left\{T,E_{2},\phi T,\nabla_{T}\phi T,\xi\right\}$ are linearly independent, or*

$(b)$* $\gamma$ is of osculating order $r=3$, $n\geq 3$ and $\gamma$ is a helix with $k_{1}^{2}+k_{2}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}},$ in which case $\left\{T,E_{2},E_{3},\phi T,\nabla_{T}\phi T,\xi\right\}$ are linearly independent, where $\delta_{1},$ $\delta_{2}$ $\in\mathbb{R}$.*

Proof.

$(1)$ From Proposition 3.2, if we take $c\leq-3$ and $\frac{\delta_{1}}{\delta_{2}}$ $\geq 0$, it is easy to see that $\gamma$ is interpolating sesqui-harmonic if and only if it is a geodesic.

$(2)$ Assume that $c>-3$, $\frac{\delta_{1}}{\delta_{2}}$ $<0$ and $\gamma:I\longrightarrow M(c)$ be an interpolating sesqui-harmonic curve. From Proposition 3.2, if we take $n\geq 2$ and $\gamma$ is of osculating order $r=2$, then $\gamma$ is a circle with $k_{1}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}}.$ Using Lemma 3.1, we have that $\left\{T,E_{2},\phi T,\nabla_{T}\phi T,\xi\right\}$ are linearly independent. Similarly, if we take $n\geq 3$ and $\gamma$ is of osculating order $r=3$, then we obtain that $k_{2}$ is a non-zero constant. Thus, $\gamma$ is a helix with $k_{1}^{2}+k_{2}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}}.$ Using Lemma 3.1, we have that $\left\{T,E_{2},E_{3},\phi T,\nabla_{T}\phi T,\xi\right\}$ are linearly independent. Conversely, assume that $\gamma$ is a Legendre circle with $k_{1}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}}$ or a Legendre helix with $k_{1}^{2}+k_{2}^{2}=\frac{c+3}{4}-\frac{\delta_{1}}{\delta_{2}}$. Obviously, $\gamma$ satisfies Theorem 3.1, respectively. Hence, we obtain the desired result. ∎

Case III. $c\neq 1$ and $\phi T\parallel E_{2}.$

From Theorem 3.1,  we have:

Proposition 3.3**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ be a Legendre curve of osculating order $r$ with $\phi T\parallel E_{2}$ and $\frac{\delta_{1}}{\delta_{2}}\neq 0$. Then $\gamma$ is interpolating sesqui-harmonic if and only if

[TABLE]

[TABLE]

[TABLE]

where $\delta_{1},$ $\delta_{2}$ is a constant.

Proof.

Assume $\gamma$ is an interpolating sesqui-harmonic Legendre curve in $M(c)$ such that $c\neq 1$, $\phi T\parallel E_{2}$ and $\frac{\delta_{1}}{\delta_{2}}\neq 0.$ From Theorem 3.1, we get the result. ∎

Hence we can state:

Theorem 3.4**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ a Legendre curve of osculating order $r$ such that $\phi T\parallel E_{2}$. Then $\left\{T,\phi T,\xi\right\}$ is the Frenet frame field of $\gamma.$

$(1)$* If $c<1$ and $\frac{\delta_{1}}{\delta_{2}}$ $\geq 0$, then $\gamma$ is interpolating sesqui-harmonic if and only if it is a geodesic.*

$(2)$* If $c>1$ and $\frac{\delta_{1}}{\delta_{2}}$ $<0$, then $\gamma$ is interpolating sesqui-harmonic if and only if it is a helix with $k_{1}^{2}=c-1-\frac{\delta_{1}}{\delta_{2}},$ $\left(k_{2}=1\right)$ where $\delta_{1},$ $\delta_{2}$ $\in\mathbb{R}$.*

Proof.

If we take $\phi T\parallel E_{2},$ we get $g\left(\phi T,E_{2}\right)=\pm 1,$ $g\left(\phi T,E_{3}\right)=g\left(\phi T,E_{4}\right)=0.$

$(1)$ From Proposition 3.3 and the above equations and if we take $c\leq 1$ and $\frac{\delta_{1}}{\delta_{2}}$ $\geq 0$, it is easy to see that $\gamma$ is interpolating sesqui-harmonic if and only if it is a geodesic.

$(2)$ If $c>1$, $\frac{\delta_{1}}{\delta_{2}}$ $<0$ from Proposition 3.3 and the above equations, we have $k_{1}=$constant and $k_{1}^{2}=c-1-\frac{\delta_{1}}{\delta_{2}},$ and $k_{2}=1$. Conversely, assume that $\gamma$ is a Legendre helix with $k_{1}^{2}=c-1-\frac{\delta_{1}}{\delta_{2}}$ and $k_{2}=1$. Then $\gamma$ satisfies Theorem 3.1 obviously. This completes the proof of the theorem. ∎

Case IV. $c\neq 1$ and $g(\phi T,E_{2})\neq 0,1,-1$.

Proposition 3.4**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1,$ $g(\phi T,E_{2})\neq 0,1,-1$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ a Legendre curve of osculating order $r$ such that $4\leq r\leq 2n+1$, $n\geq 2$. Then $\gamma$ is interpolating sesqui-harmonic with $\frac{\delta_{1}}{\delta_{2}}\neq 0$ if and only if

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

Assume that $\gamma$ is an interpolating sesqui-harmonic Legendre Frenet curve such that $g(\phi T,E_{2})$ is not a constant equal to $0,1$ or $-1$. In this case, we get $4\leq r\leq 2n+1$, $n\geq 2$ and $\phi T\in span\left\{E_{2},E_{3},E_{4}\right\}$.

Hence, we can take $f(t)=$ $g(\phi T,E_{2}).$ So by a differentiation, we obtain

[TABLE]

[TABLE]

Since $\gamma$ is a Legendre curve and $\phi$ is anti-symmetric, we have $\eta(E_{2})=0,$ $g\left(T,\phi T\right)=0$ and $g\left(E_{2},\phi E_{2}\right)=0.$ Thus we obtain

[TABLE]

Additionally, we can write

[TABLE]

From Theorem 3.1, the equations (3.10) and (3.11), the curve $\gamma$ is interpolating sesqui-harmonic if and only if

[TABLE]

[TABLE]

[TABLE]

[TABLE]

If $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ satisfies the converse statement, it is obvious that the first four of the equations in Theorem 3.1 are satisfied. Thus $\gamma$ is interpolating sesqui-harmonic. This proves the theorem. ∎

Using the equation (3.10) and the third equation of Proposition 3.4, we obtain

[TABLE]

[TABLE]

[TABLE]

where $w_{0}=$constant. Substituting the equation (3.12) in the second equation of Proposition 3.4, we get

[TABLE]

Then we have $f=$constant. Thus $k_{2}=$constant $>0$, $g\left(E_{3},\phi T\right)=0$ and then $\phi T=fE_{2}+g\left(\phi T,E_{4}\right)E_{4}$. We obtain that there exists a unique constant $\alpha_{0}\in\left(0,2\pi\right)\backslash\left\{\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\}$ such that $f=\cos\alpha_{0}$ and $g\left(E_{4},\phi T\right)=\sin\alpha_{0}.$

So we can state:

Theorem 3.5**.**

Let $M(c)=(M^{2n+1},\phi,\xi,\eta,g)$ be a Sasakian space form with $c\neq 1,$ $n\geq 2$ and $\gamma:I\subset\mathbb{R}\longrightarrow M(c)$ a Legendre curve of osculating order $r$ such that $g(\phi T,E_{2})\neq 0,1,-1$.

$(1)$* If $c\leq-3$ and $\frac{\delta_{1}}{\delta_{2}}$ $\geq 0$, then $\gamma$ is interpolating sesqui-harmonic if and only if it is a geodesic.*

$(2)$* If $c>-3$ and $\frac{\delta_{1}}{\delta_{2}}$ $<0$, then $\gamma$ is interpolating sesqui-harmonic if and only if $\phi T=\cos\alpha_{0}E_{2}+\sin\alpha_{0}E_{4},$*

[TABLE]

[TABLE]

[TABLE]

where $\alpha_{0}\in\left(0,2\pi\right)\backslash\left\{\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\}$ is constant such that $\left(c+3+3\left(c-1\right)\cos^{2}\alpha_{0}\right)\delta_{2}-4\delta_{1}>0$ and $3\left(c-1\right)\sin 2\alpha_{0}<0.$

Remark 3.1**.**

For $c\neq 1$ and $g(\phi T,E_{2})\neq 0,1,-1$, there are also interpolating sesqui-harmonic curves which are not helices.

Now, we give brief information about the Sasakian space form $\mathbb{R}^{2n+1}(-3)$ [2]:

Let us take $M=\mathbb{R}^{2n+1}$ with the standard coordinate functions $\left(x_{1},...,x_{n},y_{1},...,y_{n},z\right),$ the contact structure $\eta=\frac{1}{2}(dz-\sum\limits_{i=1}^{n}y_{i}dx_{i}),$ the characteristic vector field $\xi=2\frac{\partial}{\partial z}$ and the tensor field $\phi$ given by

[TABLE]

The Riemannian metric is $g=\eta\otimes\eta+\frac{1}{4}\sum\limits_{i=1}^{n}\left((dx_{i})^{2}+(dy_{i})^{2}\right).$ Thus, $\mathbb{R}^{2n+1}(-3)$ is a Sasakian space form with constant $\phi-$sectional curvature $c=-3$. The vector fields

[TABLE]

form a $g$-orthonormal basis and Levi-Civita connection is obtained as

[TABLE]

[TABLE]

(see [1]).

Now, we give an example for interpolating sesqui-harmonic Legendre curves in $\mathbb{R}^{5}(-3):$

**Example. **Let $\gamma=(\gamma_{1},...,\gamma_{5})$ be a unit speed Legendre curve in $\mathbb{R}^{5}(-3).$ We can write the tangent vector field $T$ of

[TABLE]

Using the above equation, $\eta(T)=0$ and $g(T,T)=1,$ we have

[TABLE]

and

[TABLE]

So for a Legendre curve (3.14), (3.15) and (3.13) gives us

[TABLE]

and

[TABLE]

From (3.16) and (3.17), $\phi T\perp E_{2}$ if and only if

[TABLE]

So we can state the following example:

Let us take $\gamma(t)=\left(\sin 2t,-\cos 2t,0,0,1\right)$ in $\mathbb{R}^{5}(-3).$ By the use of Theorem 3.1 and the above equations, $\gamma$ is an interpolating sesqui-harmonic Legendre curve with osculating order $r=2$, $k_{1}=2$, $\delta_{1}=-8$, $\delta_{2}=2$ and $\phi T\perp E_{2}$. We can see that Theorem 3.1 are verified. From the equations (3-1) in [7], the curve $\gamma$ is not biharmonic. Hence the biharmonicity and interpolating sesqui-harmonic of $\gamma$ are different.