$C$-parallel and $C$-proper Slant Curves of $S$-manifolds

TL;DR

This paper explores special classes of slant curves in $S$-manifolds, characterizing their properties and providing examples, thereby advancing the understanding of geometric structures in these manifolds.

Contribution

It introduces and analyzes $C$-parallel and $C$-proper slant curves in $S$-manifolds, establishing conditions for their classification and providing explicit examples.

Findings

Characterization of $C$-parallel and $C$-proper slant curves in $S$-manifolds.

Equivalence conditions linking these curves to non-Legendre and Legendre helices.

Explicit examples of such curves in $ ext{R}^{2m+s}(-3s)$.

Abstract

In the present paper, we define and study $C$-parallel and $C$-proper slant curves of $S$-manifolds. We prove that a curve $\gamma $ in an $S$-manifold of order $r\geq 3,$ under certain conditions, is $C$-parallel or $C$-parallel in the normal bundle if and only if it is a non-Legendre slant helix or Legendre helix, respectively. Moreover, under certain conditions, we show that $\gamma $ is $C$-proper or $C$-proper in the normal bundle if and only if it is a non-Legendre slant curve or Legendre curve, respectively. We also give two examples of such curves in $\mathbb{R}^{2m+s}(-3s).$

Equations

$$({\nabla }^{\perp }B)(X,Y,Z)={\nabla }_X^{\perp }B(Y,Z)-B({\nabla }_{X}Y,Z)-B(Y,{\nabla }_{X}Z)$$

$$\begin{array}[]{cccc}\varphi^{2}=-I+\overset{s}{\underset{\alpha=1}{\sum}}\eta^{\alpha}\otimes\xi_{\alpha},&\eta^{\alpha}(\xi_{\beta})=\delta_{\beta}^{\alpha},&\varphi\left(\xi_{\alpha}\right)=0,&\eta^{\alpha}\circ\varphi=0\end{array}$$

$$g(\phi X,\phi Y)=g(X,Y)-\stackrel{s}{\sum _{\alpha =1}}{\eta }^{\alpha }(X){\eta }^{\alpha }(Y),$$

$$\begin{array}[]{cc}d\eta^{\alpha}(X,Y)=g(X,\varphi Y)=-d\eta^{\alpha}(Y,X),&\eta^{\alpha}(X)=g(X,\xi),\end{array}$$

$$({\nabla }_X\phi )Y=\underset{\alpha =1}{\sum ^s}\left\{g(\phi X,\phi Y){\xi }_{\alpha }+{\eta }^{\alpha }(Y){\phi }^{2}X\right\},$$

$${\nabla }_X{\xi }_{\alpha }=-\phi X,\alpha \in \left\{1,...,s\right\}.$$

$${\nabla }_{T}H=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$${\nabla }_T^{\perp }H=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$$\Delta H=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$${\Delta }^{\perp }H=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$${\nabla }_{T}H=-{\kappa }_1^2{E}_1+{\kappa }_1^{\prime }{E}_2+{\kappa }_1{\kappa }_2{E}_3,$$

$${\nabla }_T^{\perp }H={\kappa }_1^{\prime }{E}_2+{\kappa }_1{\kappa }_2{E}_3,$$

$$\Delta H$$

$${\Delta }^{\perp }H$$

$$-{\kappa }_1^2{E}_1+{\kappa }_1^{\prime }{E}_2+{\kappa }_1{\kappa }_2{E}_3=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$${\kappa }_1^{\prime }{E}_2+{\kappa }_1{\kappa }_2{E}_3=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$$3{\kappa }_1{\kappa }_1^{\prime }{E}_1+\left({\kappa }_1^3+{\kappa }_1{\kappa }_2^2-{\kappa }_1^{\prime \prime }\right)E_2-(2{\kappa }_1^{\prime }{\kappa }_2+{\kappa }_1{\kappa }_2^{\prime })E_3-{\kappa }_1{\kappa }_2{\kappa }_3{E}_4=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$$\left({\kappa }_1{\kappa }_2^2-{\kappa }_1^{\prime \prime }\right)E_2-\left(2{\kappa }_1^{\prime }{\kappa }_2+{\kappa }_1{\kappa }_2^{\prime }\right)E_3-{\kappa }_1{\kappa }_2{\kappa }_3{E}_4=\lambda \stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha }.$$

$${\eta }^{\alpha }(T)=\cos \theta ,$$

$${\eta }^{\alpha }(E_2)=0,$$

$$\frac{-1}{\sqrt{s}}\le \cos \theta \le \frac{1}{\sqrt{s}}.$$

$$T=\frac{\pm 1}{\sqrt{s}}\stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha },$$

$$\stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha }\in sp\left\{T,E_3\right\},$$

$$\phi T\in sp\left\{E_2,E_4\right\},$$

$${\kappa }_2=\frac{-{\kappa }_1\sqrt{1-s{\cos }^2\theta }}{\sqrt{s}\cos \theta },{\kappa }_2\ne 0,$$

$$\lambda =\frac{-{\kappa }_1^2}{s\cos \theta }=\text{constant,}$$

$${\kappa }_1=-s\cos \theta \sqrt{1-s{\cos }^2\theta },$$

$${\kappa }_2=\sqrt{s}\left(1-s{\cos }^2\theta \right).$$

$$\lambda s\cos \theta =-{\kappa }_1^2.$$

$$\lambda =\frac{-{\kappa }_1^2}{s\cos \theta }=\text{constant.}$$

Full text

1. Introduction

Let $M^{m}$ be an integral submanifold of a Sasakian manifold $(N^{2n+1},\varphi,\xi,\eta,g)$. Then $M$ is called integral $C$-parallel if $\nabla^{\perp}B$ is parallel to the characteristic vector field $\xi$, where $B$ is the second fundamental form of $M$ and $\nabla^{\perp}B$ is given by

[TABLE]

where $X,Y,Z$ are vector fields on $M$, $\nabla^{\perp}$ and $\nabla$ are the normal connection and the Levi-Civita connection on $M$, respectively [8]. Now, let $\gamma$ be a curve in an almost contact metric manifold $(M,\varphi,\xi,\eta,g)$. Lee, Suh and Lee introduced the notions of $C$-parallel and $C$-proper curves along slant curves of Sasakian $3$-manifolds in the tangent and normal bundles [12]. A curve $\gamma$ in an almost contact metric manifold $(M,\varphi,\xi,\eta,g)$ is said to be $C$-parallel if $\nabla_{T}H=\lambda\xi$, $C$-proper if $\Delta H=\lambda\xi$, $C$-parallel in the normal bundle if $\nabla_{T}^{\perp}H=\lambda\xi$, $C$-proper in the normal bundle if $\Delta^{\perp}H=\lambda\xi$, where $T$ is the unit tangent vector field of $\gamma$, $H$ is the mean curvature vector field, $\Delta$ is the Laplacian, $\lambda$ is a non-zero differentiable function along the curve $\gamma,$ $\nabla^{\perp}$ and $\Delta^{\perp}$ denote the normal connection and Laplacian in the normal bundle, respectively [12]. For a submanifold $M$ of an arbitrary Riemannian manifold $\widetilde{M}$, if $\Delta H=\lambda H$, then $M$ is called submanifold with a proper mean curvature vector field $H$ [6]. If $\Delta^{\perp}H=\lambda H$, then $M$ is said to be submanifold with a proper mean curvature vector field $H$ in the normal bundle [1].

Let $\gamma(s)$ be a Frenet curve parametrized by the arc-length parameter $s$ in an almost contact metric manifold $M$. The function $\theta(s)$ defined by $cos[\theta(s)]=g(T(s),\xi)$ is called the contact angle function. A curve $\gamma$ is called a slant curve if its contact angle is a constant [7]. If a slant curve is with contact angle $\frac{\pi}{2}$, then it is called a Legendre curve [4].

Lee, Suh and Lee studied $C$-parallel and $C$-proper slant curves of Sasakian $3$-manifolds in [12]. As a generalization of this paper, in [9], the present authors studied $C$-parallel and $C$-proper curves in trans-Sasakian manifolds. In the present paper, our aim is to consider $C$-parallel and $C$-proper curves of $S$-manifolds.

The paper is organized as follows: In Section 2, we give a brief introduction about $S$-manifolds. Futhermore, we define the notions of $C$-parallel and $C$-proper curves in $S$-manifolds both in tangent and normal bundles. In Section 3, we consider $C$-parallel slant curves in $S$-manifolds in tangent and normal bundles, respectively. In Section 4, we study $C$-proper slant curves in $S$-manifolds in tangent and normal bundles, respectively. In the final section, we present two examples of these kinds of curves in $\mathbb{R}^{2m+s}(-3s)$.

2. Preliminaries

Let $(M,g)$ be a $(2m+s)$-dimensional Riemann manifold. $M$ is called framed metric manifold [16] with a framed metric structure $(\varphi,\xi_{\alpha},\eta^{\alpha},g)$, $\alpha\in\left\{1,...,s\right\},$ if this structure satisfies the following equations:

[TABLE]

[TABLE]

[TABLE]

where, $\varphi$ is a ($1,1$) tensor field* of rank $2m$; $\xi_{1},...,\xi_{s}$ are vector fields; $\eta^{1},...,\eta^{s}$ are $1$-forms and $g$ is a Riemannian metric on $M$; $X,Y\in TM$ and $\alpha,\beta\in\left\{1,...,s\right\}$. $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ is also called framed $\varphi$-manifold [13] or almost $r$-contact metric manifold [15]. $(\varphi,\xi_{\alpha},\eta^{\alpha},g)$ is said to be an $S$-structure*,* *if the Nijenhuis tensor of $\varphi$ is equal to $-2d\eta^{\alpha}\otimes\xi_{\alpha}$, where $\alpha\in\left\{1,...,s\right\}$ [3, 5].

When $s=1$, a framed metric structure turns into an almost contact metric structure and an $S$-structure turns into a Sasakian structure. For an $S$-structure, the following equations are satisfied [3, 5]:

[TABLE]

[TABLE]

If $M$ is Sasakian ($s=1$), (2.5) can be directly calculated from (2.4).

Firstly, we give the following definition:

Definition 1**.**

Let $\gamma:I\rightarrow(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be a unit speed curve in an $S$-manifold. Then $\gamma$ is called

i) $C$-parallel (in the tangent bundle) if

[TABLE]

ii) $C$-parallel in the normal bundle if

[TABLE]

iii) $C$-proper (in the tangent bundle) if

[TABLE]

iv) $C$-proper in the normal bundle if

[TABLE]

where $H$ is the mean curvature field of $\gamma$, $\lambda$ is a real-valued non-zero differentiable function and $\Delta$ is the Laplacian.

Let $\gamma:I\rightarrow M$  be a curve parametrized by arc length in an $n$-dimensional Riemannian manifold $(M,g)$. Denote by the Frenet frame and curvatures of $\gamma$ by $\left\{E_{1},E_{2},...,E_{r}\right\}$ and $\kappa_{1},...,\kappa_{r-1},$ respectively. We know that (see [1])

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

So we can directly state the following Proposition:

Proposition 1**.**

Let $\gamma:I\rightarrow(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be a unit speed curve in an $S$-manifold. Then

i) $\gamma$ is $C$-parallel (in the tangent bundle) if and only if

[TABLE]

ii) $\gamma$ is $C$-parallel in the normal bundle if and only if

[TABLE]

iii) $\gamma$ is $C$-proper (in the tangent bundle) if and only if

[TABLE]

iv) $\gamma$ is $C$-proper in the normal bundle if and only if

[TABLE]

Now, our aim is to apply Proposition 1 to slant curves in $S$-manifolds.

Let $\gamma:I\rightarrow(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be a slant curve. Then, if we differentiate

[TABLE]

we get

[TABLE]

where, $\theta$ denotes the constant contact angle satisfying

[TABLE]

The equality case is only valid for geodesics corresponding to the integral curves of

[TABLE]

(see [10]).

3. $C$-parallel Slant Curves of $S$-manifolds

Our first Theorem below is a result of Proposition 1 i).

Theorem 1**.**

Let $\gamma:I\rightarrow M^{2m+s}$ be a unit-speed slant curve. Then $\gamma$ is $C$-parallel (in the tangent bundle) if and only if it is a non-Legendre slant helix of order $r\geq 3$ satisfying

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and moreover if $\kappa_{3}=0$, then

[TABLE]

[TABLE]

Proof.

Let us assume that $\gamma$ is $C$-parallel (in the tangent bundle). Then, if we apply $E_{2}$ to equation (2.6), we find $\kappa_{1}^{\prime}=0$, that is, $\kappa_{1}=$constant. Now, applying $T$ to (2.6), we have

[TABLE]

Here, $\theta\neq\frac{\pi}{2}$ since $\kappa_{1}\neq 0$. Hence, $\gamma$ is non-Legendre slant. So, we get

[TABLE]

Equation (2.6) can be rewritten as

[TABLE]

which is equivalent to

[TABLE]

If we calculate the norm of both sides, we obtain

[TABLE]

If we assume $\kappa_{2}=0$, we have $\overset{s}{\underset{\alpha=1}{\sum}}\xi_{\alpha}$ is parallel to $T$. Then $\kappa_{1}=0$ or $\theta=\frac{\pi}{2},$ both of which is a contradiction. So, we have $\kappa_{2}\neq 0$ and $r\geq 3$. If we write equation (3.4) in (3.3), we get

[TABLE]

If we differentiate this last equation along the curve $\gamma,$ we find

[TABLE]

If we calculate $g(\varphi T,\varphi T)$, we have

[TABLE]

which gives us $\kappa_{3}=$constant. In particular, if $\kappa_{3}=0$, then we find equations (3.1) and (3.2). If $\kappa_{3}\neq 0,$ we differentiate equation (3.5) along the curve $\gamma$ and find that $\kappa_{4}=$constant. If we continue differentiating and calculating the norm of both sides, we easily obtain $\kappa_{i}=$constant for all $i=\overline{1,r}$, that is, $\gamma$ is a slant helix of order $r$. Thus, we have just proved the necessity.

To prove sufficiency, if $\gamma$ satisfies the equations given in the Theorem, then it is easy to show that equation (2.6) is satisfied. So, $\gamma$ is $C$-parallel (in the tangent bundle). ∎

For $C$-parallel slant curves in the normal bundle, we have the following Theorem:

Theorem 2**.**

Let $\gamma:I\rightarrow M^{2m+s}$ be a unit-speed slant curve. Then $\gamma$ is $C$-parallel in the normal bundle if and only if it is a Legendre helix of order $r\geq 3$ satisfying

[TABLE]

[TABLE]

[TABLE]

and moreover if $\kappa_{3}=0$, then

[TABLE]

Proof.

Let us assume that $\gamma$ is $C$-parallel in the normal bundle. Then, if we apply $T$ to equation (2.7), we have $\eta^{\alpha}(T)=0$, so $\gamma$ is Legendre. Next, we apply $E_{2}$ and find $\kappa_{1}=$constant. Thus, equation (2.7) becomes

[TABLE]

which gives us

[TABLE]

[TABLE]

If we differentiate equation (3.6), we get

[TABLE]

If we differentiate this last equation, we obtain

[TABLE]

If we apply $E_{2}$ to both sides, we find $\kappa_{2}=$constant. Then, the norm of equation (3.7) gives us $\kappa_{3}=$constant. In particular, if $\kappa_{3}=0$, from equation (3.7), we have

[TABLE]

Otherwise, from the norm of both sides in (3), we also have $\kappa_{4}=$constant. If we continue differentiating equation (3), we find that $\gamma$ is a helix of order $r$.

Conversely, let $\gamma$ be a Legendre helix of order $r\geq 3$ satisfying the stated equations. Then, it is easy to show that equation (2.7) is verified. Thus, $\gamma$ is $C$-parallel in the normal bundle. ∎

4. $C$-proper Slant Curves of $S$-manifolds

For $C$-proper slant curves in the tangent bundle, we can state the following Theorem:

Theorem 3**.**

Let $\gamma:I\rightarrow M^{2m+s}$ be a unit-speed slant curve. Then $\gamma$ is $C$-proper (in the tangent bundle) if and only if it is a non-Legendre slant curve satisfying

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and moreover if $\kappa_{3}=0$, then

[TABLE]

[TABLE]

[TABLE]

Proof.

Let $\gamma$ be $C$-proper (in the tangent bundle). If we apply $T$ to equation (2.8), we find

[TABLE]

Let us assume that $\gamma$ is Legendre. Then we have $\kappa_{1}^{\prime}=0$, that is, $\kappa_{1}=$constant. if we apply $E_{2}$ to equation (2.8), we get

[TABLE]

which gives us $\kappa_{1}=0$. Then equation (2.8) becomes

[TABLE]

which is a contradiction. Thus, $\gamma$ is non-Legendre slant and $\kappa_{1}\neq$constant. We find equations (4.1), (4.2), (4.3) and (4.4) applying $T$, $E_{2}$, $E_{3}$ and $E_{4}$, respectively. Then, we write these equations in (2.8) and calculate the norm of boths sides to obtain equation (4.5). Now, let us assume $\kappa_{2}=0$. Then, from equation (2.8), we have

[TABLE]

which is only possible when

[TABLE]

If we calculate $\nabla_{T}T$, we find $\kappa_{1}=0$, which is a contradiction. Hence, $\kappa_{2}\neq 0$. Differentiating equation (2.8), we can easily see that

[TABLE]

In particular, if $\kappa_{3}=0$, we obtain equations (4.6), (4.7) and (4.8). Please see our paper [10], Case III, equation (4.9), which is also valid when $\kappa_{1}$ and $\kappa_{2}$ are not constants.

Conversely, if $\gamma$ is a non-Legendre slant curve satisfying the stated equations, then Proposition 1 iii) is valid. So, $\gamma$ is C-proper (in the tangent bundle). ∎

Finally, we give the following Theorem for $C$-proper slant curves in the normal bundle:

Theorem 4**.**

Let $\gamma:I\rightarrow M^{2m+s}$ be a unit-speed slant curve. Then $\gamma$ is $C$-proper in the normal bundle if and only if it is a Legendre curve satisfying

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and moreover if $\kappa_{3}=0$, then

[TABLE]

[TABLE]

Proof.

The proof is similar to the proof of Theorem 3. For the case $\kappa_{3}=0$, please refer to [14]. ∎

5. Examples

In this section, we give the following two examples in the well-known $S$-manifold $\mathbb{R}^{2m+s}(-3s)$ [11]:

Example 1**.**

Let us consider $\mathbb{R}^{2m+s}(-3s)$ with $m=2$ and $s=2$. The curve $\gamma:I\rightarrow\mathbb{R}^{6}(-6)$ given by

[TABLE]

is a unit-speed non-Legendre slant helix with

[TABLE]

It has the Frenet frame field

[TABLE]

and it is C-parallel (in the tangent bundle) with $\lambda=\frac{1}{2}$.

Example 2**.**

Let us consider $\mathbb{R}^{2m+s}(-3s)$ with $m=1$ and $s=4$. We define real valued functions on an open interval $I$ as

[TABLE]

[TABLE]

The curve $\gamma:I\rightarrow\mathbb{R}^{6}(-12)$, $\gamma(t)=\left(\gamma_{1}(t),...,\gamma_{6}(t)\right)$ is a unit-speed Legendre curve with

[TABLE]

[TABLE]

and it is $C$-proper in the normal bundle with $\lambda=-8e^{2t}$.

Acknowledgements. This work is financially supported by Balikesir University Research Project Grant no. BAP 2018/016. The authors would like to thank the Balikesir University.