C -parallel and C -proper Slant Curves of S -manifolds
aa r X i v : . [ m a t h . G M ] M a r C -PARALLEL AND C -PROPER SLANT CURVES OF S -MANIFOLDS S¸ABAN G ¨UVENC¸ AND CIHAN ¨OZG ¨UR
Abstract.
In the present paper, we define and study C -parallel and C -properslant curves of S -manifolds. We prove that a curve γ in an S -manifold of order r ≥ , under certain conditions, is C -parallel or C -parallel in the normal bundleif and only if it is a non-Legendre slant helix or Legendre helix, respectively.Moreover, under certain conditions, we show that γ is C -proper or C -proper inthe normal bundle if and only if it is a non-Legendre slant curve or Legendrecurve, respectively. We also give two examples of such curves in R m + s ( − s ) . Introduction
Let M m be an integral submanifold of a Sasakian manifold ( N n +1 , ϕ, ξ, η, g ).Then M is called integral C -parallel if ∇ ⊥ B is parallel to the characteristic vectorfield ξ , where B is the second fundamental form of M and ∇ ⊥ B is given by( ∇ ⊥ B )( X, Y, Z ) = ∇ ⊥ X B ( Y, Z ) − B ( ∇ X Y, Z ) − B ( Y, ∇ X Z )where X, Y, Z are vector fields on M , ∇ ⊥ and ∇ are the normal connection and theLevi-Civita connection on M , respectively [8]. Now, let γ be a curve in an almostcontact metric manifold ( M, ϕ, ξ, η, g ). Lee, Suh and Lee introduced the notions of C -parallel and C -proper curves along slant curves of Sasakian 3-manifolds in thetangent and normal bundles [12]. A curve γ in an almost contact metric manifold( M, ϕ, ξ, η, g ) is said to be C -parallel if ∇ T H = λξ , C -proper if ∆ H = λξ , C -parallelin the normal bundle if ∇ ⊥ T H = λξ , C -proper in the normal bundle if ∆ ⊥ H = λξ ,where T is the unit tangent vector field of γ , H is the mean curvature vector field,∆ is the Laplacian, λ is a non-zero differentiable function along the curve γ, ∇ ⊥ and∆ ⊥ denote the normal connection and Laplacian in the normal bundle, respectively[12]. For a submanifold M of an arbitrary Riemannian manifold f M , if ∆ H = λH ,then M is called submanifold with a proper mean curvature vector field H [6]. If∆ ⊥ H = λH , then M is said to be submanifold with a proper mean curvature vectorfield H in the normal bundle [1].Let γ ( s ) be a Frenet curve parametrized by the arc-length parameter s in analmost contact metric manifold M . The function θ ( s ) defined by cos [ θ ( s )] = g ( T ( s ) , ξ ) is called the contact angle function . A curve γ is called a slant curve if its contact angle is a constant [7]. If a slant curve is with contact angle π , thenit 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 Mathematics Subject Classification.
Key words and phrases. C -parallel curve, C -proper curve, slant curve, S -manifold. studied C -parallel and C -proper curves in trans-Sasakian manifolds. In the presentpaper, 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 curvesin 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 normalbundles, respectively. In the final section, we present two examples of these kindsof curves in R m + s ( − s ). 2. Preliminaries
Let (
M, g ) be a (2 m + s )-dimensional Riemann manifold. M is called framedmetric manifold [16] with a framed metric structure ( ϕ, ξ α , η α , g ), α ∈ { , ..., s } , ifthis structure satisfies the following equations: ϕ = − I + s P α =1 η α ⊗ ξ α , η α ( ξ β ) = δ αβ , ϕ ( ξ α ) = 0 , η α ◦ ϕ = 0 (2.1) g ( ϕX, ϕY ) = g ( X, Y ) − s X α =1 η α ( X ) η α ( Y ) , (2.2) dη α ( X, Y ) = g ( X, ϕY ) = − dη α ( Y, X ) , η α ( X ) = g ( X, ξ ) , (2.3)where, ϕ is a (1 ,
1) tensor field of rank 2 m ; ξ , ..., ξ s are vector fields; η , ..., η s are 1-forms and g is a Riemannian metric on M ; X, Y ∈ T M and α, β ∈ { , ..., s } .( M m + s , ϕ, ξ α , η α , g ) is also called framed ϕ -manifold [13] or almost r -contact met-ric manifold [15]. ( ϕ, ξ α , η α , g ) is said to be an S -structure , if the Nijenhuis tensorof ϕ is equal to − dη α ⊗ ξ α , where α ∈ { , ..., s } [3, 5].When s = 1, a framed metric structure turns into an almost contact metricstructure and an S -structure turns into a Sasakian structure. For an S -structure,the following equations are satisfied [3, 5]:( ∇ X ϕ ) Y = s X α =1 (cid:8) g ( ϕX, ϕY ) ξ α + η α ( Y ) ϕ X (cid:9) , (2.4) ∇ X ξ α = − ϕX, α ∈ { , ..., s } . (2.5)If M is Sasakian ( s = 1), (2.5) can be directly calculated from (2.4).Firstly, we give the following definition: Definition 1.
Let γ : I → ( M m + s , ϕ, ξ α , η α , g ) be a unit speed curve in an S -manifold. Then γ is calledi) C -parallel (in the tangent bundle) if ∇ T H = λ s X α =1 ξ α , ii) C -parallel in the normal bundle if ∇ ⊥ T H = λ s X α =1 ξ α , -PARALLEL AND C -PROPER CURVES 3 iii) C -proper (in the tangent bundle) if ∆ H = λ s X α =1 ξ α , iv) C -proper in the normal bundle if ∆ ⊥ H = λ s X α =1 ξ α , where H is the mean curvature field of γ , λ is a real-valued non-zero differentiablefunction and ∆ is the Laplacian. Let γ : I → M be a curve parametrized by arc length in an n -dimensionalRiemannian manifold ( M, g ). Denote by the Frenet frame and curvatures of γ by { E , E , ..., E r } and κ , ..., κ r − , respectively. We know that (see [1]) ∇ T H = − κ E + κ ′ E + κ κ E , ∇ ⊥ T H = κ ′ E + κ κ E , ∆ H = −∇ T ∇ T ∇ T T = 3 κ κ ′ E + (cid:0) κ + κ κ − κ ′′ (cid:1) E − (2 κ ′ κ + κ κ ′ ) E − κ κ κ E and ∆ ⊥ H = −∇ ⊥ T ∇ ⊥ T ∇ ⊥ T T = (cid:0) κ κ − κ ′′ (cid:1) E − (2 κ ′ κ + κ κ ′ ) E − κ κ κ E . So we can directly state the following Proposition:
Proposition 1.
Let γ : I → ( M m + s , ϕ, ξ α , η α , g ) be a unit speed curve in an S -manifold. Theni) γ is C -parallel (in the tangent bundle) if and only if − κ E + κ ′ E + κ κ E = λ s X α =1 ξ α , (2.6) ii) γ is C -parallel in the normal bundle if and only if κ ′ E + κ κ E = λ s X α =1 ξ α , (2.7) iii) γ is C -proper (in the tangent bundle) if and only if κ κ ′ E + (cid:0) κ + κ κ − κ ′′ (cid:1) E − (2 κ ′ κ + κ κ ′ ) E − κ κ κ E = λ s X α =1 ξ α , (2.8) iv) γ is C -proper in the normal bundle if and only if (cid:0) κ κ − κ ′′ (cid:1) E − (2 κ ′ κ + κ κ ′ ) E − κ κ κ E = λ s X α =1 ξ α . (2.9) S¸ABAN G¨UVENC¸ AND CIHAN ¨OZG¨UR
Now, our aim is to apply Proposition 1 to slant curves in S -manifolds.Let γ : I → ( M m + s , ϕ, ξ α , η α , g ) be a slant curve. Then, if we differentiate η α ( T ) = cos θ, we get η α ( E ) = 0 , where, θ denotes the constant contact angle satisfying − √ s ≤ cos θ ≤ √ s . The equality case is only valid for geodesics corresponding to the integral curves of T = ± √ s s X α =1 ξ α , (see [10]). 3. C -parallel Slant Curves of S -manifolds Our first Theorem below is a result of Proposition 1 i).
Theorem 1.
Let γ : I → M m + s be a unit-speed slant curve. Then γ is C -parallel(in the tangent bundle) if and only if it is a non-Legendre slant helix of order r ≥ satisfying s X α =1 ξ α ∈ sp { T, E } ,ϕT ∈ sp { E , E } ,κ = − κ √ − s cos θ √ s cos θ , κ = 0 ,λ = − κ s cos θ = constant,and moreover if κ = 0 , then κ = − s cos θ p − s cos θ, (3.1) κ = √ s (cid:0) − s cos θ (cid:1) . (3.2) Proof.
Let us assume that γ is C -parallel (in the tangent bundle). Then, if weapply E to equation (2.6), we find κ ′ = 0, that is, κ =constant. Now, applying T to (2.6), we have λs cos θ = − κ . Here, θ = π since κ = 0. Hence, γ is non-Legendre slant. So, we get λ = − κ s cos θ = constant.Equation (2.6) can be rewritten as s X α =1 ξ α = − κ λ T + κ κ λ E , -PARALLEL AND C -PROPER CURVES 5 which is equivalent to s X α =1 ξ α = s cos θT − κ s cos θκ E . (3.3)If we calculate the norm of both sides, we obtain κ = − κ √ − s cos θ √ s cos θ . (3.4)If we assume κ = 0, we have s P α =1 ξ α is parallel to T . Then κ = 0 or θ = π , bothof which is a contradiction. So, we have κ = 0 and r ≥
3. If we write equation(3.4) in (3.3), we get s X α =1 ξ α = s cos θT + √ s p − s cos θE . If we differentiate this last equation along the curve γ, we find ϕT = − κ s cos θ E − κ √ − s cos θ √ s E . (3.5)If we calculate g ( ϕT, ϕT ), we have s cos θ (cid:0) − s cos θ (cid:1) (cid:0) s cos θ − κ (cid:1) = κ , which gives us κ =constant. In particular, if κ = 0, then we find equations (3.1)and (3.2). If κ = 0 , we differentiate equation (3.5) along the curve γ and find that κ =constant. If we continue differentiating and calculating the norm of both sides,we easily obtain κ i =constant for all i = 1 , r , that is, γ is a slant helix of order r .Thus, we have just proved the necessity.To prove sufficiency, if γ satisfies the equations given in the Theorem, then itis easy to show that equation (2.6) is satisfied. So, γ is C -parallel (in the tangentbundle). (cid:3) For C -parallel slant curves in the normal bundle, we have the following Theorem: Theorem 2.
Let γ : I → M m + s be a unit-speed slant curve. Then γ is C -parallelin the normal bundle if and only if it is a Legendre helix of order r ≥ satisfying s X α =1 ξ α = √ sE ,ϕT = κ √ s E − κ √ s E ,κ = 0 , λ = κ κ √ s and moreover if κ = 0 , then κ = √ s, ϕT = E . Proof.
Let us assume that γ is C -parallel in the normal bundle. Then, if we apply T to equation (2.7), we have η α ( T ) = 0, so γ is Legendre. Next, we apply E andfind κ =constant. Thus, equation (2.7) becomes κ κ E = λ s X α =1 ξ α , S¸ABAN G¨UVENC¸ AND CIHAN ¨OZG¨UR which gives us E = 1 √ s s X α =1 ξ α , (3.6) κ = 0 , λ = κ κ √ s . If we differentiate equation (3.6), we get ϕT = κ √ s E − κ √ s E . (3.7)If we differentiate this last equation, we obtain ∇ T ϕT = s X α =1 ξ α + κ ϕE (3.8)= κ ′ √ s E + κ √ s ( − κ T + κ E ) − κ ′ √ s E − κ √ s ( − κ E + κ E ) . If we apply E to both sides, we find κ =constant. Then, the norm of equation(3.7) gives us κ =constant. In particular, if κ = 0, from equation (3.7), we have κ = √ s, ϕT = E . Otherwise, from the norm of both sides in (3.8), we also have κ =constant. If wecontinue differentiating equation (3.8), we find that γ is a helix of order r .Conversely, let γ be a Legendre helix of order r ≥ γ is C -parallelin the normal bundle. (cid:3) C -proper Slant Curves of S -manifolds For C -proper slant curves in the tangent bundle, we can state the followingTheorem: Theorem 3.
Let γ : I → M m + s be a unit-speed slant curve. Then γ is C -proper(in the tangent bundle) if and only if it is a non-Legendre slant curve satisfying s X α =1 ξ α ∈ sp { T, E , E } ,ϕT ∈ sp { E , E , E , E } ,κ = constant, κ = 0 ,λ = 3 κ κ ′ s cos θ , (4.1) κ + κ = κ ′′ κ , (4.2) λsη α ( E ) = − (2 κ ′ κ + κ κ ′ ) , (4.3) λsη α ( E ) = − κ κ κ , (4.4) η α ( E ) + η α ( E ) = 1 − s cos θs (4.5) and moreover if κ = 0 , then ϕT = p − s cos θE , (4.6) -PARALLEL AND C -PROPER CURVES 7 E = 1 √ s √ − s cos θ − s cos θT + s X α =1 ξ α ! , (4.7) κ = √ s (cid:18) κ cos θ √ − s cos θ (cid:19) . (4.8) Proof.
Let γ be C -proper (in the tangent bundle). If we apply T to equation (2.8),we find λs cos θ = 3 κ κ ′ . Let us assume that γ is Legendre. Then we have κ ′ = 0, that is, κ =constant. ifwe apply E to equation (2.8), we get0 = κ + κ κ − κ ′′ = κ (cid:0) κ + κ (cid:1) which gives us κ = 0. Then equation (2.8) becomes λ s X α =1 ξ α = 0 , which is a contradiction. Thus, γ is non-Legendre slant and κ =constant. Wefind equations (4.1), (4.2), (4.3) and (4.4) applying T , E , E and E , respectively.Then, we write these equations in (2.8) and calculate the norm of boths sides toobtain equation (4.5). Now, let us assume κ = 0. Then, from equation (2.8), wehave λ s X α =1 ξ α = 3 κ κ ′ T, which is only possible when T = 1 √ s s X α =1 ξ α . If we calculate ∇ T T , we find κ = 0, which is a contradiction. Hence, κ = 0.Differentiating equation (2.8), we can easily see that ϕT ∈ sp { E , E , E , E } . In particular, if κ = 0, we obtain equations (4.6), (4.7) and (4.8). Please see ourpaper [10], Case III, equation (4.9), which is also valid when κ and κ are notconstants.Conversely, if γ is a non-Legendre slant curve satisfying the stated equations,then Proposition 1 iii) is valid. So, γ is C-proper (in the tangent bundle). (cid:3) Finally, we give the following Theorem for C -proper slant curves in the normalbundle: Theorem 4.
Let γ : I → M m + s be a unit-speed slant curve. Then γ is C -properin the normal bundle if and only if it is a Legendre curve satisfying s X α =1 ξ α ∈ sp { E , E } ,ϕT ∈ sp { E , E , E , E } κ = constant, κ = 0 ,κ κ − κ ′′ = 0 ,λsη α ( E ) = − (2 κ ′ κ + κ κ ′ ) , S¸ABAN G¨UVENC¸ AND CIHAN ¨OZG¨UR λsη α ( E ) = − κ κ κ ,η α ( E ) + η α ( E ) = 1 s and moreover if κ = 0 , then s X α =1 ξ α = √ sE ,κ = √ s, ϕT = E . Proof.
The proof is similar to the proof of Theorem 3. For the case κ = 0, pleaserefer to [14]. (cid:3) Examples
In this section, we give the following two examples in the well-known S -manifold R m + s ( − s ) [11]: Example 1.
Let us consider R m + s ( − s ) with m = 2 and s = 2 . The curve γ : I → R ( − given by γ ( t ) = (sin t, t, − cos t, − cos t, − t − sin t cos t, − t − sin t cos t ) is a unit-speed non-Legendre slant helix with κ = κ = 1 √ , θ = 2 π . It has the Frenet frame field ( T, √ ϕT, T + X α =1 ξ α !) and it is C-parallel (in the tangent bundle) with λ = . Example 2.
Let us consider R m + s ( − s ) with m = 1 and s = 4 . We define realvalued functions on an open interval I as γ ( t ) = 2 t Z cos( e u ) du, γ ( t ) = − t Z sin( e u ) du,γ ( t ) = ... = γ ( t ) = − t Z cos( e u ) u Z sin( e v ) dv du. The curve γ : I → R ( − , γ ( t ) = ( γ ( t ) , ..., γ ( t )) is a unit-speed Legendre curvewith κ = 2 e t , κ = 2 , r = 3 ,ϕT = E , E = 12 X α =1 ξ α and it is C -proper in the normal bundle with λ = − e t . Acknowledgements.
This work is financially supported by Balikesir UniversityResearch Project Grant no. BAP 2018/016. The authors would like to thank theBalikesir University. -PARALLEL AND C -PROPER CURVES 9 References [1] Arroyo J., Barros M. and Garay O. J.,
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E-mail address , S¸. G¨uven¸c: [email protected]
E-mail address , C. ¨Ozg¨ur:, C. ¨Ozg¨ur: