Biharmonic Legendre Curves on Totally η-Umbilical, Ruled, and Hopf Hypersurfaces in Four-Dimensional Complex Space Forms

Abstract

The main purpose of the present paper is to investigate biharmonic Legendre curves on hypersurfaces in four-dimensional complex space forms. We examine the necessary conditions for the existence of such curves on totally $\eta$-umbilical, ruled, and Hopf hypersurfaces for which the shape operator $A_N$ satisfies a symmetry property. The obtained results are discussed in the complex projective space $C{P}^2$. For any real hypersurfaces $(M,g)$ of $\tilde{M}\left(c\right)$, we have the relations ${\nabla }_X\xi =\varphi A_{N}X$ and $\left({\nabla }_X\varphi \right)Y=\eta \left(Y\right)A_{N}X-g(A_{N}X,Y)\xi$ for any $X,Y\in \mathrm{\Gamma }\left(TM\right)$. Using these equations, we provide the main results and theorems for biharmonic curves on the mentioned hypersurfaces.

1. Introduction

The real hypersurfaces of complex manifolds have very interesting and rich geometric properties with their contact structures induced from the complex structure. There are considerable classifications of such hypersurfaces according to the forms taken by the shape operators. For example, if the characteristic vector field is in the principal direction of this shape operator, such hypersurfaces are called Hopf hypersurfaces. Similarly, ruled hypersurfaces and totally $\eta$-umbilical hypersurfaces are characterized according to the suitable properties of the shape operator. The geometric and physical properties of such hypersurfaces have been widely studied by many authors, for example [1,2,3,4,5,6,7,8,9,10,11].

In addition to these facts, the subject of harmonic maps is used to determine the energy minimization point of a mapping between different metric manifolds. A harmonic map $\mathrm{\Psi }:(M_1,g_1)\to (M_2,g_2)$ between Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ is the critical points of the energy

$$E\left(\mathrm{\Psi }\right)={\displaystyle \frac{1}{2}}{\int }_M\mid d\mathrm{\Psi }{\mid }^2{\vartheta }_g,$$

where $d{v}_g$ denotes the measure on M induced by its metric. Therefore, these maps are the solutions of the celebrated Euler–Lagrange equation. The harmonic equation is provided by the vanishing of the tension field, given by

$$\tau \left(\mathrm{\Psi }\right)=trace\nabla d\mathrm{\Psi }=0.$$

As suggested by J. Eells and J.H. Sampson [12], one can define the bi-energy of a map $\mathrm{\Psi }:(M,g)\to (N,h)$ by

$$E_2\left(\mathrm{\Psi }\right)={\displaystyle \frac{1}{2}}{\int }_M\mid \tau \left(\mathrm{\Psi }\right){\mid }^2{\vartheta }_g$$

and say that $\mathrm{\Psi }$ is biharmonic if it is the critical point of the bi-energy.

In [13], G.Y. Jiang derived the first and second variation formula for the bi-energy, showing that the Euler–Lagrange equation associated with $E_2$ is

$$\begin{array}{ccc}\hfill {\tau }_2\left(\mathrm{\Psi }\right)& =& -▵\tau \left(\mathrm{\Psi }\right)-trace{R}^N(d\mathrm{\Psi },\tau \left(\mathrm{\Psi }\right))d\mathrm{\Psi }=0.\hfill \end{array}$$

If ${\tau }_2\left(\mathrm{\Psi }\right)=0$, then this equation is called the biharmonic equation. We examine proper biharmonic maps, as any hormonic map is biharmonic. These maps are non-harmonic biharmonic maps, and have been examined by several authors; see [14,15,16,17,18,19,20,21,22,23].

B.Y. Chen [24] defined biharmonic submanifolds of the Euclidean space and constructed a well-known hypothesis. If a biharmonic submanifold of the Euclidean space is harmonic, then this submanifold is minimal. For Riemannian immersions into Euclidean space, Chen’s definition of biharmonic submanifolds is equivalent to the definition obtained from the bi-energy functional.

In the theory of almost-contact manifolds, Legendre curves are of fundamental importance; for instance, a diffeomorphism of a contact manifold is a contact transformation precisely when it maps Legendre curves to Legendre curves [25]. C. Özgür and M. M. Tripathi showed that a Legendre curve in an $\alpha$-Sasakian manifold is biharmonic precisely when its curvature vanishes [26].

The aim of the present paper is to investigate biharmonic Legendre curves on hypersurfaces in four-dimensional complex space forms. With the aid of this investigation, some characterizations are obtained for certain hypersurfaces. We study totally $\eta$-umbilical, ruled, and Hopf hypersurfaces because they have constant sectional curvature.

2. Preliminaries

Let $(\tilde{M},\tilde{g})$ be an n-complex dimensional Hermitian manifold. Suppose that J is the complex structure on $\tilde{M}$. Then, we have

$$\begin{array}{c}\hfill J^2=-I\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \tilde{g}(JX,Y)=-\tilde{g}(X,JY)\end{array}$$

(2)

for any $X,Y\in \mathrm{\Gamma }\left(T\tilde{M}\right)$, where I denotes the identity map and $\tilde{g}$ is a compatible metric with the property of symmetry.

Let $\tilde{\nabla }$ denote the Levi-Civita connection of $(\tilde{M},\tilde{g})$. A Hermitian manifold is called Kaehlerian if $\tilde{\nabla }J=0$.

We denote the Riemannian curvature tensor field by $\tilde{R}$. Then, it is well known that

$$\begin{array}{c}\hfill \tilde{R}(JX,JY)Z=\tilde{R}(X,Y)Z\mathrm{and}\tilde{R}(X,Y)JZ=J\tilde{R}(X,Y)Z\end{array}$$

for any $X,Y\in \mathrm{\Gamma }\left(T\tilde{M}\right)$. A Kaehlerian manifold with constant holomorpic sectional curvature $4c$ is called a complex space form, denoted by $\tilde{M}\left(4c\right)$, if the Riemannian curvature tensor field satisfies the following relation:

$$\begin{array}{c}\hfill \tilde{R}(X,Y)Z=c\{\tilde{g}(X,Z)Y-\tilde{g}(Y,Z)X+\tilde{g}(JX,Z)JY-\tilde{g}(JY,Z)JX+2\tilde{g}(JX,Y)JZ\}.\end{array}$$

Let $(M,g)$ be a $(2n-1)$-dimensional real hypersurface of $\left(\tilde{M}\left(4c\right)\right)$ and let N denotes the unit normal vector field of M. For any $X\in \mathrm{\Gamma }\left(TM\right)$, we write

$$\begin{array}{c}\hfill JX=\varphi X+\eta \left(X\right)N,JN=\xi ,\end{array}$$

(3)

where $\varphi X$ is the tangential part of $JX$ to M, $\varphi$ is a $(1,1)$-type tensor field, $\eta$ is a 1-form, and $\xi$ is the structure vector field on M such that $(\varphi ,\xi ,\eta ,g)$ defines an almost-contact metric structure. Therefore, we have the following equalities:

$$\begin{array}{c}\hfill {\varphi }^{2}X=-X+\eta \left(X\right)\xi ,\\ \hfill \varphi \left(\xi \right)=0,\eta \left(\varphi X\right)=0,\end{array}$$

(4)

and

$$g(\varphi X,\varphi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right),$$

(5)

where $\eta \left(X\right)=g(X,\xi )$.

Let ∇ be the induced connection of $(M,g)$. The Gauss and Weingarten formulas are given by

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{X}Y& =& {\nabla }_{X}Y+g(A_{N}X,Y)N,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{X}N& =& -A_{N}X,\hfill \end{array}$$

(7)

for any $X,Y\in \mathrm{\Gamma }\left(TM\right)$. Here, $A_N$ is the shape operator of M. A real hypersurface is called totally geodesic if $A_N=0$, and is called totally $\eta$-umbilical [11] if there exists some smooth functions f and g such that the shape operator $A_N$ forms as

$$\begin{array}{c}\hfill A_{N}X=fX+g\eta \left(X\right)\xi \end{array}$$

(8)

for any $X\in \mathrm{\Gamma }\left(TM\right)$. We note that if M is a totally $\eta$-umbilical hypersurface of $\tilde{M}\left(c\right)$, then M has two constant principal curvatures (see [10]). Furthermore, if $\xi$ is principal direction on $(M,g)$, that is, there exists a smooth function $\alpha$ on M such that the shape operator $A_N$ forms as

$$\begin{array}{c}\hfill A_N\xi =\alpha \xi ,\end{array}$$

(9)

then $(M,g)$ is called a Hopf hypersurface. It is clear that totally-$\eta$ hypersurfaces are a special class of Hopf hypersurfaces.

Let D be a distribution on $(M,g)$ defined by

$$\begin{array}{c}\hfill D_p=\{X_p\in T_{p}M:g_p(X,\xi )=0\}\end{array}$$

for any $p\in M$. If D is integrable and if its integral manifold is a totally geodesic submanifold of M, then M is said to be a ruled hypersurface [7]. We note that ruled hypersurfaces are a special class of non-Hopf hypersurfaces.

For any real hypersurfaces $(M,g)$ of $\tilde{M}\left(c\right)$, we have the following relations for any $X,Y\in \mathrm{\Gamma }\left(TM\right)$:

$$\begin{array}{c}\hfill {\nabla }_X\xi =\varphi A_{N}X\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \left({\nabla }_X\varphi \right)Y=\eta \left(Y\right)A_{N}X-g(A_{N}X,Y)\xi .\end{array}$$

(11)

We denote by R the Riemannian tensor field of $(M,g)$. Then, the Gauss equation is provided by

$$\begin{array}{ccc}R(X,Y)Z& =& c\{\tilde{g}(X,Z)Y-\tilde{g}(Y,Z)X+\tilde{g}(JX,Z)JY-\tilde{g}(JY,Z)JX+2\tilde{g}(JX,Y)JZ\}\\ & & +g(A_{N}Y,Z)A_{N}X-g(A_{N}X,Z)A_{N}Y\hfill \end{array}$$

(12)

for any $X,Y,Z\in \mathrm{\Gamma }\left(TM\right)$.

3. Biharmonic Frenet Curves on Totally $\mathit{\eta }$-Umbilical and Ruled Hypersurfaces

Let $(\tilde{M},\tilde{g})$ be an n-dimensional Riemannian manifold, where $\tilde{g}$ is a compatible metric with the property of symmetry. A unit speed curve $\gamma :I\to \tilde{M}$ is said to be an s-Frenet curve if there exist orthonormal s-frame fields $(E_1={\gamma }^{\prime },E_2,\dots ,E_s)$ along $\gamma$ such that there exist positive functions ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{s-1}$ satisfying

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{E}_1& =& {\kappa }_1{E}_2,\hfill \\ \hfill {\nabla }_{{\gamma }^{\prime }}{E}_2& =& -{\kappa }_1{E}_1+{\kappa }_2{E}_3,\hfill \\ \hfill \dots \dots \\ \hfill {\nabla }_{{\gamma }^{\prime }}{E}_n& =& -{\kappa }_{n-1}{E}_{n-1},\hfill \end{array}$$

where ${\gamma }^{\prime }$ is the derivative of $\gamma$ with respect to the arc length parameter. The function ${\kappa }_i$ is called the ith curvature of $\gamma$ for $i\in \{1,\dots ,n\}$. A Frenet curve is said to be

i.

  A geodesic if s = 1, i.e., ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=0$.

ii.

 A Riemannain circle if $s=2$ and ${\kappa }_1$ is non-zero constant.

iii.

A helix of order s if ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{s-1}$ are constant.

In the particular case of $n=3$, we have

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{E}_1& =& \kappa E_2,\hfill \\ \hfill {\nabla }_{{\gamma }^{\prime }}{E}_2& =& -\kappa E_1+\tau E_3,\hfill \\ \hfill {\nabla }_{{\gamma }^{\prime }}{E}_3& =& -\tau E_2.\hfill \end{array}$$

(13)

The first curvature $\kappa ={\kappa }_1$ and second curvature $\tau ={\kappa }_2$ are called the geodesic curvature and geodesic torsion of $\gamma$, respectively.

Definition 1

([25]). Let $\gamma =\gamma \left(s\right)$ be a curve parameterized by the arc length parameter s on a contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$. The curve γ is called a Legendre curve if it is an integral curve, i.e.,

$$\eta \left({\gamma }^{\prime }\right)=0,$$

(14)

where ${\gamma }^{\prime }$ is the differentiation of γ.

Remark 1.

Let γ be a Legendre curve on a three-dimensional almost-contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$. Differentiating (14) with respect to ${\gamma }^{\prime }$ and using (13), it follows that

$$\begin{array}{ccc}\hfill 0& =& g({\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },\xi )+g({\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}\xi )\hfill \\ & =& g({\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },\xi )+g({\gamma }^{\prime },\varphi {\gamma }^{\prime }),\hfill \end{array}$$

which implies that

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=\kappa \varphi {\gamma }^{\prime }.$$

(15)

It is clear from (15) that if there exists a Legendre Frenet curve on a three-dimensional almost-contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$, then the set $\{{\gamma }^{\prime },\varphi {\gamma }^{\prime },\xi \}$ is an orthonormal basis of $T_{p}M$ at $\gamma \left(t\right)=p$.

Now, we shall compute the biharmonic equation for any real hypersurface of four-dimensional complex space forms:

Proposition 1.

Let $(M,g)$ be a hypersurface of a four-dimensional complex space form $\tilde{M}\left(c\right)$ and let $\gamma :I\to M$ be a Legendre curve on $(M,g)$, where I is an open internal on M. Then, the bi-tension field ${\tau }_2$ of M is given by the following equation:

$$\begin{array}{ccc}\hfill {\tau }_2& =& (-3\kappa {\kappa }^{\prime }){\gamma }^{\prime }+({\kappa }^{″}-{\kappa }^3+4c\kappa )\varphi {\gamma }^{\prime }\hfill \\ \hfill & & +\left(-2{\kappa }^{\prime }g(A_N{\gamma }^{\prime },{\gamma }^{\prime })-\kappa g({\nabla }_{{\gamma }^{\prime }}{A}_N{\gamma }^{\prime },{\gamma }^{\prime })-{\kappa }^{2}g(A_N{\gamma }^{\prime },\varphi {\gamma }^{\prime })\right)\xi \hfill \\ \hfill & & -\kappa g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\varphi A_N{\gamma }^{\prime }-\kappa g(A_N\varphi {\gamma }^{\prime },{\gamma }^{\prime })A_N{\gamma }^{\prime }+\kappa g(A_N{\gamma }^{\prime },{\gamma }^{\prime })A_N\varphi {\gamma }^{\prime }.\hfill \end{array}$$

(16)

Proof. 

Because $\gamma$ is a Legendre curve on M, we can consider the set $\{{\gamma }^{\prime },\varphi {\gamma }^{\prime },\xi \}$ as an orthonormal frame field on M such that (15) is satisfied. In view of (15), we have

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }& =& {\kappa }^{\prime }\varphi {\gamma }^{\prime }+\kappa {\nabla }_{{\gamma }^{\prime }}\left(\varphi {\gamma }^{\prime }\right)\hfill \\ \hfill & =& {\kappa }^{\prime }\varphi {\gamma }^{\prime }+\kappa \left({\gamma }^{\prime }{\nabla }_{{\gamma }^{\prime }}\varphi +\varphi {\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }.\right)\hfill \end{array}$$

(17)

If we consider (11) in (17), we obtain

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }& =& {\kappa }^{\prime }\varphi {\gamma }^{\prime }-\kappa g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\xi -{\kappa }^2{\gamma }^{\prime }.\hfill \end{array}$$

(18)

Taking the covariant derivative in (18), we derive

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }& =& {\kappa }^{″}\varphi {\gamma }^{\prime }+{\kappa }^{\prime }{\nabla }_{{\gamma }^{\prime }}\left(\varphi {\gamma }^{\prime }\right)-{\kappa }^{\prime }g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\xi \hfill \\ \hfill & & -\kappa g({\nabla }_{{\gamma }^{\prime }}{A}_N{\gamma }^{\prime },{\gamma }^{\prime })\xi -\kappa g(A_N{\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime })\xi -\kappa g(A_N{\gamma }^{\prime },{\gamma }^{\prime }){\nabla }_{{\gamma }^{\prime }}\xi \hfill \\ \hfill & & -2\kappa {\kappa }^{\prime }{\gamma }^{\prime }-{\kappa }^2{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },\hfill \end{array}$$

(19)

which is equivalent to

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }& =& {\kappa }^{″}\varphi {\gamma }^{\prime }-{\kappa }^{\prime }g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\xi -\kappa {\kappa }^{\prime }{\gamma }^{\prime }-\kappa g({\nabla }_{{\gamma }^{\prime }}{A}_N{\gamma }^{\prime },{\gamma }^{\prime })\xi -{\kappa }^{\prime }g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\xi \hfill \\ \hfill & & -{\kappa }^{2}g(A_N{\gamma }^{\prime },\varphi {\gamma }^{\prime })\xi -\kappa g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\varphi A_N{\gamma }^{\prime }-2\kappa {\kappa }^{\prime }{\gamma }^{\prime }-{\kappa }^3\varphi {\gamma }^{\prime }\hfill \end{array}$$

(20)

  □

Now, we shall compute the term of $R({\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }){\gamma }^{\prime }$.

Using (15) and (12), we have

$$\begin{array}{ccc}\hfill R({\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }){\gamma }^{\prime }& =& \kappa R({\gamma }^{\prime },\varphi {\gamma }^{\prime }){\gamma }^{\prime }\hfill \\ \hfill & =& \kappa \left[c\left(g(\varphi {\gamma }^{\prime },{\gamma }^{\prime }){\gamma }^{\prime }-g({\gamma }^{\prime },{\gamma }^{\prime })\varphi {\gamma }^{\prime }+g({\varphi }^2{\gamma }^{\prime },{\gamma }^{\prime })\varphi {\gamma }^{\prime }-g(\varphi {\gamma }^{\prime },{\gamma }^{\prime })\varphi {\gamma }^{\prime }\right.\right.\hfill \\ \hfill & & \left.\left.-2g(\varphi {\gamma }^{\prime },\varphi {\gamma }^{\prime })\varphi {\gamma }^{\prime }\right)+g(A_N\varphi {\gamma }^{\prime },{\gamma }^{\prime })A_N{\gamma }^{\prime }-g(A_N{\gamma }^{\prime },{\gamma }^{\prime })A_N\varphi {\gamma }^{\prime }\right].\hfill \end{array}$$

(21)

If we write (20) and (21) in the biharmonic equation given by

$$\begin{array}{c}\hfill {\tau }_2={\nabla }_{{\gamma }^{\prime }}^3{\gamma }^{\prime }-R({\gamma }^{\prime },{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }){\gamma }^{\prime },\end{array}$$

(22)

then the proof is straightforward.

Theorem 1.

Let $(M,g)$ be a three-dimensional totally η-umbilical real hypersurface of $\tilde{M}\left(c\right)$ and let $\gamma =\gamma \left(s\right)$ be a Legendre curve parameterized by arc-length parameter s. Then, γ is a biharmonic curve if and only if the curvature of γ is constant and the following relations are satisfied:

$$\begin{array}{c}\hfill {\kappa }^2=4c\mathrm{and}{\gamma }^{\prime }\left(f\right)=0.\end{array}$$

(23)

Proof. 

Using the fact that $(M,g)$ is a totally $\eta$-umbilical real hypersurface, from $\left(8\right)$ we have

$$\begin{array}{c}\hfill A_N{\gamma }^{\prime }=f{\gamma }^{\prime }\mathrm{and}{A}_N\varphi {\gamma }^{\prime }=f\varphi {\gamma }^{\prime },\end{array}$$

(24)

where f is a smooth function on M. If $\gamma$ is a biharmonic Legendre curve, then ${\tau }_2=0$. If we put ${\tau }_2=0$, then using (24) in (16), we obtain

$$\begin{array}{c}\hfill (-3\kappa {\kappa }^{\prime }){\gamma }^{\prime }+({\kappa }^{″}-{\kappa }^3+4c\kappa )\varphi {\gamma }^{\prime }+(-2{\kappa }^{\prime }f+\kappa {\gamma }^{\prime }\left(f\right))\xi =0.\end{array}$$

(25)

In view of (25), we get

$$\begin{array}{c}\hfill \kappa {\kappa }^{\prime }=0,{\kappa }^{″}-{\kappa }^3+4c\kappa =0\mathrm{and}-2{\kappa }^{\prime }f+\kappa {\gamma }^{\prime }\left(f\right)=0,\end{array}$$

showing that ${\kappa }^2=4c=\mathrm{constant}$. This implies that M has constant sectional curvature and that ${\gamma }^{\prime }\left(f\right)=0$. The converse part of this theorem is straightforward.   □

As a result of Theorem 1, we obtain the following corollaries.

Corollary 1.

Let $(M,g)$ be a three-dimensional totally η-umbilical real hypersurface of $\tilde{M}\left(c\right)$. If there exists a biharmonic Legendre curve lying on $(M,g)$, then the holomorphic sectional curvature of $\tilde{M}\left(c\right)$ is positive semi-definite.

Corollary 2.

A biharmonic Legendre curve lying on a three-dimensional totally η-umbilical real hypersurface of complex Euclidean space $C^2$ is a geodesic.

Now, we shall investigate biharmonic curves on ruled hypersurfaces. Before doing so, we recall the following proposition.

Proposition 2

([4,9]). Let $(M,g)$ be a ruled hypersurface of $\tilde{M}\left(c\right)$. Then, the shape operator satisfies the following formulas:

$$\begin{array}{c}\hfill A_N\xi =\alpha \xi +\beta U,A_{N}U=\beta \xi ,A_{N}X=0,\end{array}$$

(26)

where U and X are unit vectors, the set $\{\xi ,U,X\}$ is orthonormal, and α and β are smooth functions such that $\beta \ne 0$.

Theorem 2.

Let $(M,g)$ be a three-dimensional ruled real hypersurface of $\tilde{M}\left(c\right)$ and let $\gamma =\gamma \left(s\right)$ be a Legendre curve parameterized by arc-length parameter s. Then, γ is a biharmonic curve if and only if the curvature κ of γ is constant and the following relation is satisfied:

$$\begin{array}{c}\hfill {\kappa }^2=4c.\end{array}$$

(27)

Proof. 

From Proposition 2, there exist some smooth functions $\alpha$ and $\beta$ such that

$$\begin{array}{c}\hfill A_{N}U=\beta \xi ,A_{N}X=0,\end{array}$$

(28)

where U and X are unit vectors $U\perp \xi$ and $X\perp \xi$. Considering the orthonormal frame $\{{\gamma }^{\prime },\varphi {\gamma }^{\prime },\xi \}$ of $\mathrm{\Gamma }\left(TM\right)$, using (28) we have

$$\begin{array}{c}\hfill a{A}_N{\gamma }^{\prime }+b{A}_N\varphi {\gamma }^{\prime }=\beta \xi \end{array}$$

(29)

and

$$\begin{array}{c}\hfill c{A}_N{\gamma }^{\prime }+d{A}_N\varphi {\gamma }^{\prime }=0,\end{array}$$

(30)

where $a,b,c,d$ are some functions on M. In view of (29) and (30), we can write

$$\begin{array}{c}\hfill A_N{\gamma }^{\prime }={\mu }_1\xi \mathrm{and}{A}_N\varphi {\gamma }^{\prime }={\mu }_2\xi ,\end{array}$$

(31)

where ${\mu }_1={\displaystyle \frac{d}{ad-bc}}$ and ${\mu }_2={\displaystyle \frac{c}{ad-bc}}$. If $\gamma$ is a biharmonic curve, then we can use (31) in (16) to obtain

$$\begin{array}{c}\hfill (-3\kappa {\kappa }^{\prime }){\gamma }^{\prime }+({\kappa }^{″}-{\kappa }^3+4c\kappa )\varphi {\gamma }^{\prime }=0,\end{array}$$

(32)

which shows that ${\kappa }^2=4c=\mathrm{constant}$. This in turn implies that M has constant sectional curvature. The converse part of this theorem is straightforward.   □

As a result of the above theorem, we obtain the following corollaries.

Corollary 3.

Let $(M,g)$ be a three-dimensional ruled real hypersurface of $\tilde{M}\left(c\right)$. If there exists a biharmonic Legendre curve on lying $(M,g)$, then the holomorphic sectional curvature of $\tilde{M}\left(c\right)$ is positive semi-definite.

Corollary 4.

A biharmonic Legendre curve lying on a three-dimensional ruled real hypersurface of complex Euclidean space $C^2$ is a geodesics.

Now, we shall investigate biharmonic Legendre curves on real hypersurfaces for which the Ricci tensor is pseudoparallel.

Definition 2

([3,5]). Let $(M,g)$ be real hypersurface and let S denote the Ricci operator of $(M,g)$. The Ricci operator S is called pseudoparallel if there exists a function F on M such that the following condition is satisfied:

$$\begin{array}{c}\hfill R(X,Y)·S=F(X\wedge Y)·S\end{array}$$

for any $X,Y\in \mathrm{\Gamma }\left(TM\right)$.

Lemma 1.

Let $(M,g)$ be a three-dimensional real hypersurface of a non-flat complex space with form $\tilde{M}\left(c\right)$. If the Ricci operator S is pseudoparallel, then M is a Hopf hypersurface or the shape operator A is represented by the matrix

$$\begin{array}{c}\hfill A_N=\left(\begin{array}{ccc}\alpha & h& 0\\ h& f& 0\\ 0& 0& g\end{array}\right).\end{array}$$

(33)

Theorem 3.

Let $(M,g)$ be a three-dimensional pseudoparallel hypersurface of $\tilde{M}\left(c\right)$ and let $\gamma =\gamma \left(s\right)$ be a Legendre curve parameterized by arc-length s. Then, γ is a biharmonic curve if and only if κ is constant and

$$\begin{array}{c}\hfill \kappa =\mp \sqrt{4c-f^2+fg}\mathrm{and}{\gamma }^{\prime }\left(f\right)=0.\end{array}$$

(34)

Proof. 

Suppose that $(M,g)$ is a three-dimensional pseudoparallel hypersurface. Considering (33), we can write $A_N\xi =\alpha \xi +h{\gamma }^{\prime }$, $A_N{\gamma }^{\prime }=h\xi +f{\gamma }^{\prime }$ and $A_N\varphi {\gamma }^{\prime }=g\varphi {\gamma }^{\prime }$, where $\alpha ,h,f,g$ are smooth functions. Using these facts in (16) and with $\gamma$ as a biharmonic curve, we obtain

$$\begin{array}{c}\left(-3\kappa {\kappa }^{\prime }+\right){\gamma }^{\prime }+\left({\kappa }^{″}-\kappa f^2+\kappa fg\right)\varphi {\gamma }^{\prime }\hfill \\ +\left(-2{\kappa }^{\prime }{f}_1-\kappa \left[g({\nabla }_{{\gamma }^{\prime }}(h\xi +f{\gamma }^{\prime }),{\gamma }^{\prime })\right]\right)\xi =0.\hfill \end{array}$$

(35)

  □

Here, we have from (15) that

$$\begin{array}{ccc}\hfill g({\nabla }_{{\gamma }^{\prime }}h\xi ,{\gamma }^{\prime })+g({\nabla }_{{\gamma }^{\prime }}f{\gamma }^{\prime },{\gamma }^{\prime })& =& hg({\nabla }_{{\gamma }^{\prime }}\xi ,{\gamma }^{\prime })+{\gamma }^{\prime }\left[f\right]\hfill \\ \hfill & =& -hg(\xi ,{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime })+{\gamma }^{\prime }\left[f\right]\hfill \\ \hfill & =& {\gamma }^{\prime }\left[f\right].\hfill \end{array}$$

(36)

Using (36), we obtain

$$\begin{array}{ccc}& & \left(-3\kappa {\kappa }^{\prime }+\right){\gamma }^{\prime }+\left({\kappa }^{″}-\kappa f^2+\kappa fg\right)\varphi {\gamma }^{\prime }+\left(-2{\kappa }^{\prime }{f}_1-\kappa {\gamma }^{\prime }\left[f\right]\right)\xi =0.\hfill \end{array}$$

(37)

In view of (37), $\gamma$ is a biharmonic curve if and only if $\kappa$ is constant and (34) is satisfied.

4. Biharmonic Curves on Hopf Hypersurfaces

In this section, we shall investigate biharmonic Frenet curves on Hopf hypersurfaces on complex space forms.

Suppose that $\gamma$ is a Frenet curve on a Hopf hypersurface $(M,g)$. Then, the shape operator $A_N$ can be represented by the matrix

$$\begin{array}{c}\hfill A_N=\left(\begin{array}{ccc}\alpha & 0& 0\\ 0& f_1& f_2\\ 0& g_1& g_2\end{array}\right),\end{array}$$

(38)

with respect to the orthonormal frame $\{{\gamma }^{\prime },\varphi {\gamma }^{\prime },\xi \}$.

In this case, we have the following theorem.

Theorem 4.

Let $(M,g)$ be a three-dimensional Hopf hypersurface of $\tilde{M}\left(c\right)$ and let $\gamma =\gamma \left(s\right)$ be a Legendre curve parameterized by arc-length s. Then, γ is a biharmonic curve if and only if the following relations are satisfied:

$$\begin{array}{ccc}\hfill & & \kappa ={\displaystyle \frac{f_1{f}_2}{3}}s+a,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & {\kappa }^{″}-{\kappa }^3+4c\kappa -\kappa f_1^2-\kappa f_2{g}_1+\kappa f_1{g}_2=0,\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& & 2{\kappa }^{\prime }{f}_1+\kappa {\gamma }^{\prime }\left(f_1\right)-{\kappa }^2{f}_1+{\kappa }^2{f}_2=0,\hfill \end{array}$$

(41)

where a is a real constant.

Proof. 

From (38), we can write $A_N{\gamma }^{\prime }=f_1{\gamma }^{\prime }+f_2\varphi {\gamma }^{\prime }$ and $A_N\varphi {\gamma }^{\prime }=g_1{\gamma }^{\prime }+g_2\varphi {\gamma }^{\prime }$, where $f_i,g_i$ and $i\in \{1,2\}$ are smooth functions on M. Using these facts in (16) along with the fact that $\gamma$ is biharmonic, we obtain

$$\begin{array}{c}\left(-3\kappa {\kappa }^{\prime }+\kappa f_1{f}_2\right){\gamma }^{\prime }+\left({\kappa }^{″}-{\kappa }^3+4c\kappa -\kappa f_1^2-\kappa f_2{g}_1+\kappa f_1{g}_2\right)\varphi {\gamma }^{\prime }\hfill \\ +\left(-2{\kappa }^{\prime }{f}_1-\kappa \left[g({\nabla }_{{\gamma }^{\prime }}{f}_1{\gamma }^{\prime },{\gamma }^{\prime })-g({\nabla }_{{\gamma }^{\prime }}{f}_2\varphi {\gamma }^{\prime },{\gamma }^{\prime })\right]-{\kappa }^2{f}_2\right)\xi =0.\hfill \end{array}$$

(42)

Using (11) and (15), we have

$$\begin{array}{c}\hfill {\nabla }_{{\gamma }^{\prime }}{f}_1{\gamma }^{\prime }={\gamma }^{\prime }\left(f_1\right){\gamma }^{\prime }+f_1{\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }={\gamma }^{\prime }\left(f_1\right){\gamma }^{\prime }+\kappa \varphi {\gamma }^{\prime }\end{array}$$

(43)

and

$$\begin{array}{ccc}\hfill {\nabla }_{{\gamma }^{\prime }}{f}_2\varphi {\gamma }^{\prime }& =& {\gamma }^{\prime }\left(f_2\right)\varphi {\gamma }^{\prime }+f_2{\nabla }_{{\gamma }^{\prime }}\varphi {\gamma }^{\prime }\hfill \\ \hfill & =& {\gamma }^{\prime }\left(f_2\right)\varphi {\gamma }^{\prime }+f_2\left(\left({\nabla }_{{\gamma }^{\prime }}\varphi \right){\gamma }^{\prime }+\varphi {\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }\right)\hfill \\ \hfill & =& {\gamma }^{\prime }\left(f_2\right)\varphi {\gamma }^{\prime }+f_{2}g(A_N{\gamma }^{\prime },{\gamma }^{\prime })\xi -\kappa f_1{\gamma }^{\prime }\hfill \\ \hfill & =& {\gamma }^{\prime }\left(f_2\right)\varphi {\gamma }^{\prime }+f_1{f}_2\xi -\kappa f_1{\gamma }^{\prime }.\hfill \end{array}$$

(44)

If we consider (43) and (44), we get

$$\begin{array}{c}\left(-3\kappa {\kappa }^{\prime }+\kappa f_1{f}_2\right){\gamma }^{\prime }+\left({\kappa }^{″}-{\kappa }^3+4\kappa g_1-\kappa f_1^2-\kappa f_2{g}_1+\kappa f_1{g}_2\right)\varphi {\gamma }^{\prime }\hfill \\ \left(-2{\kappa }^{\prime }{f}_1-\kappa {\gamma }^{\prime }\left(f_1\right)+{\kappa }^2{f}_1-{\kappa }^2{f}_2\right)\xi =0.\hfill \end{array}$$

(45)

The proof is completed from (45).   □

Example 1.

Let M be a real hypersurface of ${\mathtt{C}}^2$ provided by

$$\begin{array}{c}\hfill M=\{(z_1,z_2):|z_1|=1,z_1,z_2\in \mathtt{C}\}.\end{array}$$

In this case, we can consider a hypersurface M with the following form:

$$\begin{array}{c}\hfill M=\{(cos{x}_1,sin{x}_1,x_2,x_3):x_1\in (=,2\pi ],x_2,x_3\in \mathbb{R}\}\end{array}$$

and the natural complex structure J is defined by

$$\begin{array}{c}\hfill JX=(-x^2,x^1,-x^4,x^3)\end{array}$$

for any $X=(x^1,x^2,x^3,x^4)$. In this case, the natural vector fields on M are provided by

$$\begin{array}{c}\hfill e_1=(-sin{x}_1,cos{x}_1,0,0),e_2=(0,0,1,0),e_3=(0,0,0,1)\end{array}$$

and the unit normal vector field by

$$\begin{array}{c}\hfill N=(cos{x}_1,sin{x}_1,0,0).\end{array}$$

Therefore, it is clear that $e_1=JN=\xi$. By straightforward computation, we obtain

$$\begin{array}{c}\hfill A_N=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\end{array}$$

which shows that M is a Hopf hypersurface.

Let us now consider the following circle with radius r lying on M:

$$\begin{array}{c}\hfill \gamma \left(t\right)=(0,0,rcost,rsint).\end{array}$$

It is clear that this curve is a Legendre Frenet curve that also satisfies (39)–(41).

Now, we shall investigate biharmonic curves on some special hypersurfaces of complex projective spaces.

Let $C^{n+1}$ be an $(n+1)$-dimensional complex Euclidean space and let $C^{*}$ be the multiplicative group defined on $C^{n+1}-\left\{0\right\}$. It is well known that there exists a natural projection $C^{n+1}-\left\{0\right\}\to (C^{n+1}-\left\{0\right\})/C^{*}$ defined by

$$\begin{array}{c}\hfill (z_0,z_1,\dots ,z_j,z_{j+1},\dots ,z_n)=({\displaystyle \frac{z_0}{z_j}},\dots ,{\displaystyle \frac{z_{j-1}}{z_j}},{\displaystyle \frac{z_{j+1}}{z_j}},\dots ,{\displaystyle \frac{z_n}{z_j}}),\end{array}$$

(46)

where $C^{n+1}-\left\{0\right\})/C^{*}$ is called the n-dimemsional complex projective, usually denoted by $C{P}^n$.

Consider a real $(2n+1)$-dimensional hypersphere $S^{2n+1}$ of radius r in $C^{n+1}$ defined by

$$\begin{array}{c}\hfill S^{2n+1}=\{z\in C^{n+1}:\tilde{g}(z,z)=r^2\},\end{array}$$

where

$$\begin{array}{c}\hfill \tilde{g}(z,w)=ReF(z,w),\forall z,w\in C^{n+1}.\end{array}$$

Here, F is the Hermitian metric and $ReF$ denotes the real part of F. It is known that there exists a natural projection $\pi :S^{2n+1}\to C{P}^n$. If M is a hypersurface of $C{P}^n$, then ${\pi }^{-1}\left(M\right)=M^{\prime }$ is also a hypersurface of $S^{2n+1}$.

Let $p,q\ge 0$, $p+q=n-1$ and $C^{n+1}=C^{p+1}\times C^{q+1}$. Consider a hypersurface $M^{\prime }$ defined by

$$\begin{array}{c}\hfill M^{\prime }=S^{2p+1}\left(\sqrt{r^2-b^2}\right)\times S^{2q+1}\left(b\right),\end{array}$$

where $0<b<r$. We note that $M^{\prime }$ is the product of two spheres. A hypersurface defined by $\pi \left(M^{\prime }\right)=M$ is called a tube in the complex projective space $C{P}^n$. If $p=0$, then M is called a geodesic hypersphere.

Theorem 5

(Theorem 3.13 in [9]). Geodesic hyperspheres in complex projective space have two distinct principal curvatures: ${\lambda }_1=-{\displaystyle \frac{1}{r}}tanu$ of multiplicity $2n-2$, and ${\lambda }_2=-{\displaystyle \frac{1}{r}}cotu$ of multiplicity 1 for $0<u<{\displaystyle \frac{\pi }{2}}$ and $b=rsinu$.

Theorem 6.

There does not exist any biharmonic Legendre curve on geodesic hyperspheres of ${\mathrm{C}\mathrm{P}}^2$.

Proof. 

Let $\gamma$ be a biharmonic Frenet curve. From Theorems 4 and 5, either $f_1=-{\displaystyle \frac{1}{r}}tanu$ or $f_2=-{\displaystyle \frac{1}{r}}cotu$ and $f_2=g_1=0$. Using these facts in (39)–(41), we find that $\kappa$ is constant and

$$\begin{array}{ccc}& & -{\kappa }^2+4c-{\displaystyle \frac{1}{r^2}}{tan}^{2}u=0,\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& & {\gamma }^{\prime }(tanu)-\kappa tanu=0,\hfill \end{array}$$

(48)

for $0<u<{\displaystyle \frac{\pi }{2}}$ and $b=rsinu$. Because $C{P}^2$ is of constant holomorpic curvature with $c={\displaystyle \frac{1}{r}}$, in view of (47) and (48) we get

$$\begin{array}{c}\hfill \kappa ={\displaystyle \frac{\sqrt{4r-{tan}^{2}u}}{r}}=0.\end{array}$$

(49)

This shows that there does not exist any biharmonic Frenet curve on geodesic hyperspheres of $C{P}^2$.   □

Let $M^{\prime }$ be a tube in $C{P}^n$. For $t<r^4$, if $M^{\prime }$ is defined with

$$\begin{array}{c}\hfill M^{\prime }=\{z\in C^{n+1}:\tilde{g}(z,z)=r^2,F(z,\overline{z})=t\},\end{array}$$

then $M^{\prime }$ is called a complex quadric hypersurface. Here, $\overline{z}$ denotes the conjugate of z.

Theorem 7

(Theorem 3.15 in [9]). The complex quadric hypersurfaces in complex projective space have three distinct principal curvatures: ${\lambda }_1=-{\displaystyle \frac{1}{r}}cotu$ of multiplicity $(n-1)$, ${\lambda }_2={\displaystyle \frac{1}{r}}tanu$ of multiplicity $(n-1)$, and ${\lambda }_3={\displaystyle \frac{2}{r}}tan2u$ of multiplicity 1 for $0<u<{\displaystyle \frac{\pi }{2}}$.

From Theorems 4 and 7, we obtain the following corollary.

Corollary 5.

The curvature of a biharmonic Legendre curve on complex quadric hypersurfaces of $C{P}^2$ is non-constant.

5. Discussion

We investigate biharmonic Legendre curves on hypersurfaces with four-dimensional complex space forms. Through these investigations, some characterizations are obtained for totally $\eta$-umbilical, ruled, and Hopf hypersurfaces.

The common speciality of these hypersurfaces is that they have constant sectional curvature. We examine the conditions of the Legendre curve $\gamma$ to be a biharmonic vurve in these hypersurfaces.

Legendre curves describe physically admissible paths under geometric constraints. These curves can be used in geometric optics, since light propagation is described in a contact manifold (position plus direction). Moreover, a Legendre curve represents a light ray that satisfies the optical constraint; thus, wave fronts are envelopes of families of Legendre curves.

In further studies, the conditions for Legendre curves to be biharmonic could be investigated in different hypersurfaces, for example non-Hopf hypersurfaces and hypersurfaces on product-like manifolds.

6. Conclusions

The main results of Theorems 1–4 are provided in the Table 1.

Table 1. Table of main results.

M is a totally $\mathit{\eta }$-umbilical real hypersurface	${\mathit{\kappa }}^2=4\mathit{c}$ and ${\mathit{\gamma }}^{\prime }\left(\mathit{f}\right)=0$
M is a ruled real hypersurface	${\kappa }^2=4c$
M is a pseudoparallel hypersurface	$\kappa =\mp \sqrt{4c-f^2+fg}$ and ${\gamma }^{\prime }\left(f\right)=0$
M is a Hopf hypersurface	$\kappa ={\displaystyle \frac{f_1{f}_2}{3}}s+a$,
	${\kappa }^{″}-{\kappa }^3+4c\kappa -\kappa f_1^2-\kappa f_2{g}_1+\kappa f_1{g}_2=0$,
	$2{\kappa }^{\prime }{f}_1+\kappa {\gamma }^{\prime }\left(f_1\right)-{\kappa }^2{f}_1+{\kappa }^2{f}_2=0$