Hindawi Publishing Corporation ISRN Geometry Volume 2013, Article ID 821429, 5 pages http://dx.doi.org/10.1155/2013/821429

Research Article Slant Curves in the Unit Tangent Bundles of Surfaces

Zhong Hua Hou and Lei Sun

Institute of Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China

Correspondence should be addressed to Lei Sun; [email protected]

Received 26 September 2013; Accepted 25 October 2013

Academic Editors: T. Friedrich and M. Pontecorvo

Copyright © 2013 Z. H. Hou and L. Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let (𝑀, 𝑔) be a surface and let (𝑈(𝑇𝑀), 𝐺) be the unit of 𝑀 endowed with the Sasaki metric. We know that any curve Γ(𝑠) in 𝑈(𝑇𝑀) consistofacurve𝛾(𝑠) in 𝑀 and as unit vector field 𝑋(𝑠) along 𝛾(𝑠). In this paper we study the geometric properties 𝛾(𝑠) and 𝑋(𝑠) satisfying when Γ(𝑠) is a slant .

1. Introduction Theorem 1. Let Γ(𝑠) = (𝛾(𝑠), 𝑋(𝑠)) be a Legendrian geodesic parameterized by arc length in 𝑈(𝑇𝑀) with domain 𝑠∈[𝑎,𝑏]. (𝑀,𝑔,𝜙,𝜉,𝜂) Let be a 3-dimensional contact metric mani- If the set consisting of points 𝑠∈[𝑎,𝑏]such that 𝐾𝑀(𝑠) = 1 𝑀 fold.Theslantcurvesin are generalization of Legendrian is discrete, then 𝛾(𝑠) is a geodesic of velocity 2 and 𝑋(𝑠) is the curves which form a constant angle with the Reeb vector field normal direction of 𝛾 in 𝑀. 𝜉.Choetal.[1] studied Lancret type problem for curves in Sasakian 3-manifold. They showed that a curve 𝛾(𝑠) ⊂𝑀 Theorem 2. Let Γ(𝑠) = (𝛾(𝑠), 𝑋(𝑠)) be a slant geodesic param- is slant if and only if (𝜏 ± 1)/𝑘𝛾 is constant where 𝜏 and 𝑘𝛾 eterized by arc length in 𝑈(𝑇𝑀) which is not Legendrian. 𝛾(𝑠) are torsion and curvature of ,respectively,andtheyalso Under the assumptions of 𝐾𝑀(𝑠) as in Theorem 1,wehavethe gave some examples of slant curves. One can find some other following. papers about slant curves in almost contact metric manifolds. 𝑋(𝑠) =±𝐸(𝑠)̸ 𝛾(𝑠) For examples, Calin˘ et al. [2] studied the slant curves in 𝑓- (1) If ,then is a geodesic of velocity 2 and 𝑋 𝛾(𝑠) Kenmotsu manifolds. In [3], Calin˘ and Crasmareanu studied is a parallel vector field along . slant curves in normal almost contact manifolds. (2) If 𝑋(𝑠) = ±𝐸(𝑠),then𝛾(𝑠) is a curve of velocity 2| cos 𝜃| (𝑀, 𝑔) Let be a . Sasaki [4, 5]stud- with constant curvature 𝑘𝛾 =±tan 𝜃. ied the geometries of 𝑇𝑀 endowed with the Sasaki metric 𝐽 𝑇𝑀 and introduced the almost complex structure in which 2. Preliminaries is compatible with 𝐺𝑠.Tashiro[6] constructed an almost (𝐺󸀠,𝜙󸀠,𝜉󸀠,𝜂󸀠) contact metric structure in the unit tangent Firstly, we introduce the (almost) contact metric structure on 𝑈(𝑇𝑀) 𝑀 bundle of which is induced from the almost com- a Riemannian manifold of odd dimension. With the same (𝑇𝑀, 𝐺 ,𝐽) 𝑇𝑀 ∞ plex structure 𝑠 in .KlingenbergandSasaki[7] notations as in [9]; let 𝑀 be a real (2𝑛 + 1)-dimensional C 2 ∞ studied in the unit tangent bundle of -sphere manifold and X(𝑀) the Lie algebra of C vector fields on 𝑆2 𝑈(𝑇𝑆2) ∞ endowed with Sasaki metric and showed that is 𝑀.Analmost cocomplex structure on 𝑀 is defined by a C 𝑅𝑃3(1/4) ∞ ∞ isometric to .Sasaki[8] studied the geodesics on the (1,1)-tensor 𝜙,aC vector field 𝜉 and a C 1-form 𝜂 on 𝑀 unit tangent bundles over space forms. such that for any point 𝑥∈𝑀we have In this paper, we study the slant geodesics in the unit 𝑈(𝑇𝑀) 𝑀 𝛾(𝑠) 2 tangent bundle of some surface .Foranycurve 𝜙𝑥 =−𝐼+𝜂𝑥 ⨂ 𝜉𝑥,𝜂𝑥 (𝜉𝑥)=1, (1) in 𝑀,let𝑇(𝑠) = |𝑇(𝑠)|𝐸(𝑠) be the tangent vector field of 𝛾(𝑠) and let 𝐾𝑀(𝑠) be the sectional curvature of 𝑀 at 𝛾(𝑠),wehave where 𝐼 denotes the identity transformation of the tan- the following theorems. gent space 𝑇𝑥𝑀 at 𝑥. Manifolds equipped with an almost 2 ISRN Geometry

󸀠 ∗ cocomplex structure are called almost contact manifolds.A from 𝑈(𝑇𝑀) to 𝑇𝑀 and this map induces a metric 𝐺 =𝑖 𝐺𝑠 Riemannian manifold with a metric 𝑔 andanalmostcontact in 𝑈(𝑇𝑀) as follows: (𝜙, 𝜉, 𝜂) structure is called almost contact metric or almost co- 󸀠 ℎ ℎ Hermitian manifold if 𝑔 and (𝜙, 𝜉, 𝜂) satisfies 𝐺 (𝑋 ,𝑌 )=𝑔(𝑋, 𝑌) , 𝐺󸀠 (𝑋ℎ,𝑌𝑡)=0, 𝑔 (𝜙𝑋, 𝜙𝑌) =𝑔(𝑋, 𝑌) −𝜂(𝑋) 𝜂 (𝑌) , (2) (7) 𝐺󸀠 (𝑋𝑡,𝑌𝑡)=𝑔(𝑋, 𝑌) −𝑔(𝑋, 𝑢) 𝑔 (𝑌, 𝑢) , for any vector fields 𝑋, 𝑌 ∈ X(𝑀). As in Kahler geometry, we can define the fundamental 2- 𝑋, 𝑌 ∈ X(𝑀) Φ (𝑀,𝑔,𝜙,𝜉,𝜂) for any vector fields . form on the almost contact metric manifold From Tashiro [6], we know that there is an almost contact 󸀠 󸀠 󸀠 󸀠 as metric structure (𝐺 ,𝜙 ,𝜉 ,𝜂 ) in 𝑈(𝑇𝑀) which is induced from the almost complex structure 𝐽 in 𝑇𝑀 such that Φ (𝑋, 𝑌) =𝑔(𝑋, 𝜙𝑌) , (3) 󸀠 󸀠 V 󸀠 V 𝐽∘𝑖∗ =𝑖∗ ∘𝜙 +𝜂 ⨂ 𝑢 ,𝐽∘𝑖∗ (𝜉 )=𝑢. (8) for any vector fields 𝑋, 𝑌 ∈ X(𝑀).Obviously,thisform 𝑛 satisfies 𝜂∧Φ =0̸ which means every almost contact metric This implies that manifold is orientable and that (𝜂, Φ) defines an almost 𝜂󸀠 (𝜉󸀠)=1, 𝜙󸀠 (𝜉󸀠)=0, 𝜂󸀠 ∘𝜙󸀠 =0. cosymplectic structure on 𝑀.Ifthefundamental2-formΦ= (9) 𝑑𝜂,thenwecall(𝑀,𝑔,𝜙,𝜉,𝜂)acontactmetricmanifold. 𝑖 𝑖 From Blair et al. [10], at any point (𝑥, 𝑢) ∈ 𝑈(𝑇𝑀) we have Let (𝑥 ,𝑢) be the locally coordinate systems on the 𝑇𝑀 𝑀 𝐺 tangent bundle of .Sasaki[4, 5] defined a metric 𝑠 on 𝜉󸀠 =𝑢ℎ, 𝑇𝑀 which is the natural lifts of the metric 𝑔 on 𝑀 as follows: 𝜂󸀠 (𝑋) =󸀠 𝐺 (𝑋, 𝑢ℎ), ℎ ℎ 𝐺𝑠 (𝑋 ,𝑌 )=𝑔(𝑋, 𝑌) , (10) Φ󸀠 (𝑋, 𝑌) =󸀠 𝐺 (𝑋,󸀠 𝜙 𝑌) = 2𝑑𝜂󸀠 (𝑋, 𝑌) , ℎ V 𝐺𝑠 (𝑋 ,𝑌 )=0, (4) 𝜙󸀠𝑋ℎ =𝑋𝑡,𝜙󸀠𝑋𝑡 =−𝑋ℎ +𝜂󸀠 (𝑋ℎ)𝜉󸀠, V V 𝐺𝑠 (𝑋 ,𝑌 )=𝑔(𝑋, 𝑌) , where 𝑋∈X(𝑀) and 𝑋, 𝑌∈X(𝑈(𝑇𝑀)).Hence,weknow ℎ 𝑖 𝑖 𝑗 𝑘 V (𝐺,𝜙,𝜉,𝜂) 𝑈(𝑇𝑀) where 𝑋, 𝑌 ∈ X(𝑀) and 𝑋 =𝑋𝜕𝑥𝑖 −𝑋𝑢 Γ𝑖𝑗 𝜕𝑢𝑘 and 𝑋 = that there is a contact metric structure in 𝑋𝑖𝜕 𝑋 (𝑥, 𝑢) such that 𝑢𝑖 are the horizontal and vertical lifts of at with respect to Levi-Civita connection ∇ of 𝑔,respectively, 1 󸀠 󸀠 󸀠 1 󸀠 𝑘 𝜂= 𝜂 ,𝜉=2𝜉,𝜙=𝜙,𝐺=𝐺 . (11) and {Γ𝑖𝑗 } are the Christoffel symbols of ∇. The Levi-Civita 2 4 ∇ 𝐺 connection of 𝑠 is defined as By (5), we know that the Levi-Civita connection 𝐷 of (𝑈(𝑇𝑀), 𝐺) is determined by the following: ℎ ℎ 1 𝑀 V ℎ 1 𝑀 ℎ ∇𝑋ℎ 𝑌 =(∇𝑋𝑌) − (𝑅 𝑢) , ∇𝑋V 𝑌 = (𝑅 𝑌) , ⊤ 1 2 𝑋𝑌 2 𝑢𝑋 𝐷 𝑌ℎ =(∇ 𝑌ℎ) =(∇ 𝑌)ℎ − (𝑅 𝑢)𝑡, 𝑋ℎ 𝑋ℎ 𝑋 2 𝑋𝑌 V V 1 𝑀 ℎ V ∇ ℎ 𝑌 =(∇ 𝑌) + (𝑅 𝑋) , ∇ V 𝑌 =0, 𝑋 𝑋 2 𝑢𝑌 𝑋 ℎ ℎ ⊤ 1 ℎ 𝐷𝑋𝑡 𝑌 =(∇𝑋𝑡 𝑌 ) = (𝑅𝑢𝑋𝑌) , (5) 2 (12) 𝑡 𝑡 ⊤ 𝑡 1 ℎ 𝑀 𝐷 ℎ 𝑌 =(∇ ℎ 𝑌 ) =(∇ 𝑌) + (𝑅 𝑋) , where 𝑋, 𝑌 ∈ X(𝑀) and 𝑅 is the curvature tensor on 𝑋 𝑋 𝑋 2 𝑢𝑌 (𝑀, 𝑔).Thealmostcomplexstructure𝐽 on 𝑇𝑀 which is 𝑡 𝑡 ⊤ 𝑡 compatible with 𝐺𝑠 is given by 𝐷𝑋𝑡 𝑌 =(∇𝑋𝑡 𝑌 ) =−𝑔(𝑌, 𝑢) 𝑋 ,

ℎ V V ℎ for any (𝑥, 𝑢) ∈ 𝑈(𝑇𝑀) and 𝑋, 𝑌 ∈ X(𝑈(𝑇𝑀)). 𝐽𝑋 =𝑋,𝐽𝑋=−𝑋 , (6) 𝑈(𝑇𝑀) for any vector field 𝑋∈X(𝑀). 3. Geodesic Slant Curves in We know that the normal vector field of the unit tangent Let (𝑀, 𝑔) be a surface and let 𝛾:[𝑎,𝑏]→be 𝑀 a curve in bundle 𝑈(𝑇𝑀) = {(𝑥, 𝑢) ∈ 𝑇𝑀; 𝑔(𝑢,𝑢)=1} of 𝑀 at (𝑥, 𝑢) ∈ V ℎ 𝑡 𝑀.LetΓ(𝑠) = (𝛾(𝑠), 𝑋(𝑠)) be a curve in (𝑈(𝑇𝑀), 𝐺, 𝜙,, 𝜉,𝜂) 𝑈(𝑇𝑀) in (𝑇𝑀,𝑠 𝐺 ) is 𝑢 .LetX(𝑈(𝑇𝑀)) ={𝑋 +𝑌;𝑋,𝑌∈ ∞ where the contact metric structure is given by (11). X(𝑀)} be the Lie algebra of C vector fields on 𝑈(𝑇𝑀) ℎ 𝑖 𝑖 𝑗 𝑘 𝑡 V where 𝑋 =𝑋𝜕𝑥𝑖 −𝑋𝑢 Γ𝑖𝑗 and 𝑋 = (𝑋 − 𝑔(𝑋, 𝑢)𝑢) are Definition 3. We say that Γ(𝑠) is a slant curve in 𝑈(𝑇𝑀) if the 󸀠 the horizontal and tangential lifts of 𝑋 at (𝑥, 𝑢) ∈ 𝑈(𝑇𝑀), angle between the tangent vector field Γ (𝑠) of Γ(𝑠) and 𝜉 is respectively. Let 𝑖 : 𝑈(𝑇𝑀) →𝑇𝑀 be the including map constant. ISRN Geometry 3

Assume that Γ(𝑠) is parameterized by the arc length. We Substituting (19)into(14), we have know that 2 2 cos 𝜃 󸀠 󸀠 2 𝑖 𝑖 𝛼 +4( 𝛼 +𝛼𝛽𝑘𝛾) +4(𝑘𝛾 cos 𝜃+𝛽) =4. (20) 󸀠 𝑑𝑥 𝑑𝑋 𝛼2 Γ (𝑠) = 𝜕 𝑖 + 𝜕 𝑖 𝑑𝑠 𝑥 𝑑𝑠 𝑢 Since 𝑔(𝑋,𝑇 ∇ 𝑋) =,from( 0 15)and(19)wehave 𝑖 𝑖 𝑗 𝑑𝑥 ℎ 𝑑𝑋 𝑑𝑥 𝑘 𝑖 = (𝜕𝑥𝑖 ) (Γ (𝑠)) +( + 𝑋 Γ𝑗𝑘)𝜕𝑢𝑖 (Γ (𝑠)) 2 cos 𝜃 󸀠 2 𝑑𝑠 𝑑𝑠 𝑑𝑠 𝑇=2cos 𝜃𝑋 − ( 𝛼 +𝛼 𝛽𝑘𝛾)∇𝑇𝑋. 𝑔(∇𝑇𝑋,𝑇 ∇ 𝑋) 𝛼 ℎ V =(𝑇 +(∇𝑇𝑋) ) (Γ (𝑠)) , (21) (13) It follows that 󸀠 󸀠 󸀠 󸀠 𝑅 (𝑇, 𝑋, 𝑋,∇ 𝑋) where 𝑇(𝑠) =𝛾 (𝑠).Since𝐺(Γ ,Γ )=1and 𝐺=𝐺/4 we have 𝑇 󸀠 2 𝑔 (𝑇, 𝑇) +𝑔(∇ 𝑋, ∇ 𝑋) =4. 2((cos 𝜃/𝛼) 𝛼 +𝛼 𝛽𝑘𝛾)𝑅(∇𝑇𝑋, 𝑋,𝑇 𝑋,∇ 𝑋) 𝑇 𝑇 (14) =− 𝑔(∇𝑇𝑋,𝑇 ∇ 𝑋) (22) 󸀠 Let 𝜃 be the angle between Γ (𝑠) and 𝜉.Wehave cos 𝜃 󸀠 2 =−2( 𝛼 +𝛼 𝛽𝑘𝛾)𝐾𝑀 (𝑠) . 󸀠 𝑔 (𝑇 (𝑠) ,𝑋(𝑠)) 𝛼 𝜃 (𝑠) =𝐺(Γ (𝑠) ,𝜉)= . (15) cos 2 It follows from (19)and(22)that(17)turnsinto Taking the derivative on both sides of (15)withrespectto𝑠, 𝜃 𝑘 𝜂(𝑁 )+2(cos 𝛼󸀠 +𝛼2𝛽𝑘 )(𝐾 −1)=0. we derive that Γ Γ 𝛼 𝛾 𝑀 (23)

󸀠 𝑑 󸀠 − sin 𝜃⋅𝜃 = 𝐺(Γ (𝑠) ,𝜉) For the Legendrian curves, we have the following theo- 𝑑𝑠 rem. 󸀠 =𝐺(𝐷 󸀠 Γ (𝑠) ,𝜉) Γ (𝑠) Theorem 1. Let Γ(𝑠) = (𝛾(𝑠) and let 𝑋(𝑠)) be a Legendrian 󸀠 geodesic parameterized by arc length in (𝑈(𝑇𝑀),𝐺,𝜙,𝜉,𝜂) +𝐺(Γ (𝑠) ,𝐷 󸀠 𝜉) Γ (𝑠) with domain 𝑠∈[𝑎,𝑏]. Suppose that the set consisting of points 𝑠∈[𝑎,𝑏]such that the sectional curvature of 𝑀 at 𝛾(𝑠) =𝑘𝜂(𝑁 )+𝐺(Γ󸀠 (𝑠) ,2(∇ 𝑋)ℎ Γ Γ 𝑇 satisfying 𝐾𝑀(𝑠) = 1 is discrete. Then 𝛾(𝑠) is a geodesic of velocity 2 and 𝑋(𝑠) is the normal direction of 𝛾(𝑠) in 𝑀. ℎ 𝑡 +(𝑅𝑢∇ 𝑋𝑢) −(𝑅𝑇𝑢𝑢) ) 𝑇 Proof. Since Γ(𝑠) is a Legendrian geodesic, we have

=𝑘Γ𝜂(𝑁Γ)+8𝑔(∇𝑇𝑋, 𝑇) − 8𝑅 (𝑇, 𝑋,𝑇 𝑋,∇ 𝑋) , 1 cos 𝜃=0, 𝛽=± , 𝑋 = ±𝑁,Γ 𝑘 =0. (24) (16) 2

󸀠 ℎ Substituting these into (20)and(23), we have where 𝜃 =𝑑𝜃/𝑑𝑠, 𝜉(𝑠) = 2𝑋 (𝑠),and𝑘Γ and 𝑁Γ are the Γ(𝑠) 2 2 curvature and the direction of the acceleration of ,respec- 𝛼 (1 + 𝑘𝛾)=4, tively. (25) Γ(𝑠) 𝜃≡ 2 If is a slant curve, that is, Const, from (16)we 𝛼 𝑘𝛾 (𝐾𝑀 −1)=0. have Bythesecondequationof(25) and the assumption of 𝐾(𝑠) we 𝑘Γ𝜂(𝑁Γ)+8𝑔(∇𝑇𝑋, 𝑇) − 8𝑅 (𝑇, 𝑋,𝑇 𝑋,∇ 𝑋) = 0. (17) have 𝑘𝛾 =0which means 𝛾(𝑠) is a geodesic in 𝑀.Substituting 𝑘𝛾 =0into the first equation of (25)wehave𝛼=2.This Let {𝐸, 𝑁} be the Frenet frame on 𝛾(𝑠).Itfollowsfrom(15) completes the proof of Theorem 1. that we have For the non-Legendrian slant geodesics, we have the 2 cos 𝜃 following theorem. 𝑋= 𝐸+2𝛽𝑁, (18) 𝛼 Theorem 2. Let Γ(𝑠) = (𝛾(𝑠), 𝑋(𝑠)) be a slant geodesic in ∞ where 𝛼 = |𝑇| and 𝛽 is a C function. It follows that 𝑈(𝑇𝑀) which is not Legendrian. Under the assumptions of 𝐾𝑀 as in Theorem 1,wehavethefollowing. 2 𝜃 󸀠 ∇ 𝑋=( cos ) 𝐸+2 𝜃⋅𝑘 𝑁+2𝛽󸀠𝑁−2𝛼𝛽𝑘 𝐸 (1) If 𝑋(𝑠) =±𝐸(𝑠)̸ ,then𝛾(𝑠) is a geodesic of velocity 2 and 𝑇 𝛼 cos 𝛾 𝛾 𝑋 is a parallel vector field along 𝛾(𝑠). (19) cos 𝜃 󸀠 󸀠 (2) If 𝑋(𝑠) = ±𝐸(𝑠),then𝛾(𝑠) is a curve of velocity 2| cos 𝜃| =−2( 𝛼 +𝛼𝛽𝑘𝛾)𝐸+2(𝑘𝛾 cos 𝜃+𝛽)𝑁. 𝛼2 with constant curvature 𝑘𝛾 =±tan 𝜃. 4 ISRN Geometry

Proof. By the assumption we know that 𝜃 =𝜋/2̸ ,andform Let 𝐼 be a subset of [𝑎, 𝑏] such that 𝛽|𝐼 =0.Supposethat𝐼 is which and (18)wehave𝛽 =±1/2̸ .Since𝑔(𝑋, 𝑋),from =1 notempty.From(34)wehave (18)wehave ±√1−cos2𝜃 2 𝛽 (𝑠) = , (35) cos 𝜃 2 2 4( ) +4𝛽 =1, (26) 𝛼 for any 𝑠 ∈ 𝐼 − [𝑎, 𝑏]. It is a contradiction to the continuity of 𝛽 unless cos 𝜃=±1;thatis,𝜃=0or 𝜋.Substituting𝜃=0 from which we derive that or 𝜋 into (34)wehave𝛽=0which means 𝐼=[𝑎,𝑏].This completes the proof of the Lemma. {√1−4𝛽2 𝐼=0 𝛽 =0̸ { 𝜋 Firstly, we consider which implies .Itfollows { ,𝜃∈[0,), 𝑋 =±𝐸̸ cos 𝜃 { 2 2 from which and (18)wehave . = { (27) From (34)wehave 𝛼 { { √1−4𝛽2 𝜋 { ± sin 𝜃 − ,𝜃∈(,𝜋]. 𝛽≡ . (36) { 2 2 2

Suppose that 𝜃∈[0,𝜋/2),thenwehave It follows from which and (27)wehave𝛼=2.Substituting𝛽 into (33)wehave𝑘𝛾 =0. 2 cos 𝜃 Substituting 𝛼, 𝛽,and𝑘𝛾 into (19), we have ∇𝑇𝑋=0.This 𝛼= . (1) √1−4𝛽2 (28) proves Case in Theorem 2. Secondly, we consider 𝐼=[𝑎,𝑏]which implies 𝛽≡0. It follows from which and (18)wehave𝑋=±𝐸and 𝛼= Taking derivative on both side of this equation with respect 2| cos 𝜃|. to 𝑠,wehave Substituting 𝛼 and 𝛽 into (33)wehave 𝜃 2𝛽𝛽󸀠 1 cos 𝛼󸀠 = . 𝑘 =±√ −1=± 𝜃, (37) 2 (29) 𝛾 2 tan 𝛼 √1−4𝛽2 cos 𝜃 which proves Case (2) in Theorem 2.Thiscompletestheproof It follows that of Theorem 2. 𝜃 4𝛽 𝜃 cos 𝛼󸀠 +𝛼2𝛽𝑘 = cos (𝛽󸀠 +𝑘 𝜃) . 𝛼 𝛾 1−4𝛽2 𝛾 cos (30) Acknowledgment This work was supported in part by the fundamental research Substituting (30)into(23), we have funds for the Central University. 8𝛽 𝜃 𝑘 𝜂(𝑁 )+ cos (𝛽󸀠 +𝑘 𝜃) (𝐾 −1)=0. References Γ Γ 1−4𝛽2 𝛾 cos 𝑀 (31) [1] J. T. Cho, J.-I. Inoguchi, and J.-E. Lee, “On slant curves in Remark 4. (31)alsoholdsfor𝜃 ∈ (𝜋/2, 𝜋]. Sasakian 3-manifolds,” Bulletin of the Australian Mathematical Society, vol. 74, no. 3, pp. 359–367, 2006. For proving Theorem 2, we need the following lemma. [2] C. Calin,˘ M. Crasmareanu, and M. I. Munteanu, “Slant curves in three-dimensional 𝑓-Kenmotsu manifolds,” Journal of Math- Lemma 5. If there is some 𝑠0 ∈ [𝑎, 𝑏] such that 𝛽(𝑠0)=0, ematical Analysis and Applications,vol.394,no.1,pp.400–407, then 𝛽 is vanishing everywhere. 2012. [3] C. Calin˘ and M. Crasmareanu, “Slant curves in 3-dimensional Proof. Substituting (30)into(20), we have normal almost ,” Mediterranean Journal of Mathematics, vol. 10, no. 2, pp. 1067–1077, 2013. 1 2 ( 2𝜃+(𝑘 𝜃+𝛽󸀠) )=1, [4] S. Sasaki, “On the differential geometry of tangent bundles of 1−4𝛽2 cos 𝛾 cos (32) Riemannian manifolds,” The Tohoku Mathematical Journal,vol. 10, pp. 338–354, 1958. and from which we derive that [5] S. Sasaki, “On the differential geometry of tangent bundles of Riemannian manifolds, II,” The Tohoku Mathematical Journal, 󸀠 √ 2 2 vol.14,no.2,pp.146–155,1962. 𝑘𝛾 cos 𝜃+𝛽 =± 1−4𝛽 − cos 𝜃. (33) [6] Y. Tashiro, “On contact structure of hypersurfaces in complex manifolds. I,” The Tohoku Mathematical Journal,vol.15,pp.62– Since 𝑘Γ =0and under the assumption of 𝐾(𝑠),from(31)and 78, 1963. (33)wehave [7] W.Klingenberg and S. Sasaki, “On the tangent sphere bundle of 𝑓 2 2 a -sphere,” The Tohoku Mathematical Journal,vol.27,pp.49– 𝛽(1−4𝛽 − cos 𝜃) = 0. (34) 56, 1975. ISRN Geometry 5

[8] S. Sasaki, “Geodesics on the tangent sphere bundles over space forms,” Journal fur¨ die Reine und Angewandte Mathematik,vol. 288, pp. 106–120, 1976. [9] D. Janssens and L. Vanhecke, “Almost contact structures and curvature tensors,” Kodai Mathematical Journal,vol.4,no.1,pp. 1–27, 1981. [10] D. E. Blair, of Contact and Symplectic Manifolds,vol.203ofProgress in Mathematics,Birkhauser,¨ Boston, Mass, USA, 2002. Copyright of ISRN Geometry is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.