Slant Curves in the Unit Tangent Bundles of Surfaces

Abstract

Let (𝑀,𝑔) be a surface and let (𝑈(𝑇𝑀),𝐺) be the unit tangent bundle of 𝑀 endowed with the Sasaki metric. We know that any curve Γ(𝑠) in 𝑈(𝑇𝑀) consist of a curve 𝛾(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 geodesic.

1. Introduction

Let (𝑀,𝑔,𝜙,𝜉,𝜂) be a 3-dimensional contact metric manifold. The slant curves in 𝑀 are generalization of Legendrian curves which form a constant angle with the Reeb vector field 𝜉. Cho et al. [1] studied Lancret type problem for curves in Sasakian 3-manifold. They showed that a curve 𝛾(𝑠)⊂𝑀 is slant if and only if (𝜏±1)/𝑘_𝛾 is constant where 𝜏 and 𝑘_𝛾 are torsion and curvature of 𝛾(𝑠), respectively, and they also gave some examples of slant curves. One can find some other papers about slant curves in almost contact metric manifolds. For examples, Călin et al. [2] studied the slant curves in 𝑓-Kenmotsu manifolds. In [3], Călin and Crasmareanu studied slant curves in normal almost contact manifolds.

Let (𝑀,𝑔) be a Riemannian manifold. Sasaki [4, 5] studied the geometries of 𝑇𝑀 endowed with the Sasaki metric and introduced the almost complex structure 𝐽 in 𝑇𝑀 which is compatible with 𝐺_𝑠. Tashiro [6] constructed an almost contact metric structure (𝐺^′,𝜙^′,𝜉^′,𝜂^′) in the unit tangent bundle 𝑈(𝑇𝑀) of 𝑀 which is induced from the almost complex structure (𝑇𝑀,𝐺_𝑠,𝐽) in 𝑇𝑀. Klingenberg and Sasaki [7] studied geodesics in the unit tangent bundle of 2-sphere 𝑆² endowed with Sasaki metric and showed that 𝑈(𝑇𝑆²) is isometric to 𝑅𝑃³(1/4). Sasaki [8] studied the geodesics on the unit tangent bundles over space forms.

In this paper, we study the slant geodesics in the unit tangent bundle 𝑈(𝑇𝑀) of some surface 𝑀. For any curve 𝛾(𝑠) in 𝑀, let 𝑇(𝑠)=|𝑇(𝑠)|𝐸(𝑠) be the tangent vector field of 𝛾(𝑠) and let 𝐾_𝑀(𝑠) be the sectional curvature of 𝑀 at 𝛾(𝑠), we have the following theorems.

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]. If the set consisting of points 𝑠∈[𝑎,𝑏] such that 𝐾_𝑀(𝑠)=1 is discrete, then 𝛾(𝑠) is a geodesic of velocity 2 and 𝑋(𝑠) is the normal direction of 𝛾 in 𝑀.

Theorem 2. Let Γ(𝑠)=(𝛾(𝑠),𝑋(𝑠)) be a slant geodesic parameterized by arc length in 𝑈(𝑇𝑀) which is not Legendrian. Under the assumptions of 𝐾_𝑀(𝑠) as in Theorem 1, we have the following.(1)If 𝑋(𝑠)≠±𝐸(𝑠), then 𝛾(𝑠) is a geodesic of velocity 2 and 𝑋 is a parallel vector field along 𝛾(𝑠).(2)If 𝑋(𝑠)=±𝐸(𝑠), then 𝛾(𝑠) is a curve of velocity 2|cos𝜃| with constant curvature 𝑘_𝛾=±tan𝜃.

2. Preliminaries

Firstly, we introduce the (almost) contact metric structure on a Riemannian manifold of odd dimension. With the same notations as in [9]; let 𝑀 be a real (2𝑛+1)-dimensional 𝒞^∞ manifold and 𝔛(𝑀) the Lie algebra of 𝒞^∞ vector fields on 𝑀. An almost cocomplex structure on 𝑀 is defined by a 𝒞^∞ (1,1)-tensor 𝜙, a 𝒞^∞ vector field 𝜉 and a 𝒞^∞ 1-form 𝜂 on 𝑀 such that for any point 𝑥∈𝑀 we have 𝜙²_𝑥=−𝐼+𝜂_𝑥𝜉_𝑥,𝜂_𝑥𝜉_𝑥=1, (1) where 𝐼 denotes the identity transformation of the tangent space 𝑇_𝑥𝑀 at 𝑥. Manifolds equipped with an almost cocomplex structure are called almost contact manifolds. A Riemannian manifold with a metric 𝑔 and an almost contact structure (𝜙,𝜉,𝜂) is called almost contact metric or almost co-Hermitian manifold if 𝑔 and (𝜙,𝜉,𝜂) satisfies 𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌), (2) for any vector fields 𝑋,𝑌∈𝔛(𝑀).

As in Kahler geometry, we can define the fundamental 2-form Φ on the almost contact metric manifold (𝑀,𝑔,𝜙,𝜉,𝜂) as Φ(𝑋,𝑌)=𝑔𝑋,𝜙𝑌, (3) for any vector fields 𝑋,𝑌∈𝔛(𝑀). Obviously, this form satisfies 𝜂∧Φ^𝑛≠0 which means every almost contact metric manifold is orientable and that (𝜂,Φ) defines an almost cosymplectic structure on 𝑀. If the fundamental 2-form Φ=𝑑𝜂, then we call (𝑀,𝑔,𝜙,𝜉,𝜂) a contact metric manifold.

Let (𝑥^𝑖,𝑢^𝑖) be the locally coordinate systems on the tangent bundle 𝑇𝑀 of 𝑀. Sasaki [4, 5] defined a metric 𝐺_𝑠 on 𝑇𝑀 which is the natural lifts of the metric 𝑔 on 𝑀 as follows: 𝐺_𝑠𝑋^ℎ,𝑌^ℎ(),𝐺=𝑔𝑋,𝑌_𝑠𝑋^ℎ,𝑌^𝑣𝐺=0,_𝑠𝑋^𝑣,𝑌^𝑣=𝑔(𝑋,𝑌), (4) where 𝑋,𝑌∈𝔛(𝑀) and 𝑋^ℎ=𝑋^𝑖𝜕_𝑥𝑖−𝑋^𝑖𝑢^𝑗Γ^𝑘_𝑖𝑗𝜕_𝑢𝑘 and 𝑋^𝑣=𝑋^𝑖𝜕_𝑢𝑖 are the horizontal and vertical lifts of 𝑋 at (𝑥,𝑢) with respect to Levi-Civita connection ∇ of 𝑔, respectively, and {Γ^𝑘_𝑖𝑗} are the Christoffel symbols of ∇. The Levi-Civita connection ∇ of 𝐺_𝑠 is defined as ∇_𝑋^ℎ𝑌^ℎ=∇_𝑋𝑌^ℎ−12𝑅^𝑀_𝑋𝑌𝑢^𝑣,∇_𝑋^𝑣𝑌^ℎ=12𝑅^𝑀_𝑢𝑋𝑌^ℎ,∇_𝑋^ℎ𝑌^𝑣=∇_𝑋𝑌^𝑣+12𝑅^𝑀_𝑢𝑌𝑋^ℎ,∇_𝑋^𝑣𝑌^𝑣=0, (5) where 𝑋,𝑌∈𝔛(𝑀) and 𝑅^𝑀 is the curvature tensor on (𝑀,𝑔). The almost complex structure 𝐽 on 𝑇𝑀 which is compatible with 𝐺_𝑠 is given by 𝐽𝑋^ℎ=𝑋^𝑣,𝐽𝑋^𝑣=−𝑋^ℎ, (6) for any vector field 𝑋∈𝔛(𝑀).

We know that the normal vector field of the unit tangent bundle 𝑈(𝑇𝑀)={(𝑥,𝑢)∈𝑇𝑀;𝑔(𝑢,𝑢)=1} of 𝑀 at (𝑥,𝑢)∈𝑈(𝑇𝑀) in (𝑇𝑀,𝐺_𝑠) is 𝑢^𝑣. Let 𝔛(𝑈(𝑇𝑀))={𝑋^ℎ+𝑌^𝑡;𝑋,𝑌∈𝔛(𝑀)} be the Lie algebra of 𝒞^∞ vector fields on 𝑈(𝑇𝑀) where 𝑋^ℎ=𝑋^𝑖𝜕_𝑥𝑖−𝑋^𝑖𝑢^𝑗Γ^𝑘_𝑖𝑗 and 𝑋^𝑡=(𝑋−𝑔(𝑋,𝑢)𝑢)^𝑣 are the horizontal and tangential lifts of 𝑋 at (𝑥,𝑢)∈𝑈(𝑇𝑀), respectively. Let 𝑖∶𝑈(𝑇𝑀)→𝑇𝑀 be the including map from 𝑈(𝑇𝑀) to 𝑇𝑀 and this map induces a metric 𝐺^′=𝑖^∗𝐺_𝑠 in 𝑈(𝑇𝑀) as follows: 𝐺^′𝑋^ℎ,𝑌^ℎ(),𝐺=𝑔𝑋,𝑌_′𝑋^ℎ,𝑌^𝑡𝐺=0,_′𝑋^𝑡,𝑌^𝑡=𝑔(𝑋,𝑌)−𝑔(𝑋,𝑢)𝑔(𝑌,𝑢), (7) for any vector fields 𝑋,𝑌∈𝔛(𝑀).

From Tashiro [6], we know that there is an almost contact metric structure (𝐺^′,𝜙^′,𝜉^′,𝜂^′) in 𝑈(𝑇𝑀) which is induced from the almost complex structure 𝐽 in 𝑇𝑀 such that 𝐽∘𝑖_∗=𝑖_∗∘𝜙^′+𝜂^′𝑢^𝑣,𝐽∘𝑖_∗𝜉^′=𝑢^𝑣. (8) This implies that 𝜂^′𝜉^′=1,𝜙^′𝜉^′=0,𝜂^′∘𝜙^′=0. (9) From Blair et al. [10], at any point (𝑥,𝑢)∈𝑈(𝑇𝑀) we have 𝜉^′=𝑢^ℎ,𝜂^′𝑋=𝐺^′𝑋,𝑢^ℎ,Φ^′𝑋,𝑌=𝐺^′𝑋,𝜙^′𝑌=2𝑑𝜂^′𝑋,𝑌,𝜙^′𝑋^ℎ=𝑋^𝑡,𝜙^′𝑋^𝑡=−𝑋^ℎ+𝜂^′𝑋^ℎ𝜉^′, (10) where 𝑋∈𝔛(𝑀) and 𝑋,𝑌∈𝔛(𝑈(𝑇𝑀)). Hence, we know that there is a contact metric structure (𝐺,𝜙,𝜉,𝜂) in 𝑈(𝑇𝑀) such that 1𝜂=2𝜂^′,𝜉=2𝜉^′,𝜙=𝜙^′1,𝐺=4𝐺^′. (11)

By (5), we know that the Levi-Civita connection 𝐷 of (𝑈(𝑇𝑀),𝐺) is determined by the following: 𝐷_𝑋^ℎ𝑌^ℎ=∇_𝑋^ℎ𝑌^ℎ⊤=∇_𝑋𝑌^ℎ−12𝑅_𝑋𝑌𝑢^𝑡,𝐷_𝑋^𝑡𝑌^ℎ=∇_𝑋^𝑡𝑌^ℎ⊤=12𝑅_𝑢𝑋𝑌^ℎ,𝐷_𝑋^ℎ𝑌^𝑡=∇_𝑋^ℎ𝑌^𝑡⊤=∇_𝑋𝑌^𝑡+12𝑅_𝑢𝑌𝑋^ℎ,𝐷_𝑋^𝑡𝑌^𝑡=∇_𝑋^𝑡𝑌^𝑡⊤=−𝑔(𝑌,𝑢)𝑋^𝑡, (12) for any (𝑥,𝑢)∈𝑈(𝑇𝑀) and 𝑋,𝑌∈𝔛(𝑈(𝑇𝑀)).

3. Geodesic Slant Curves in 𝑈(𝑇𝑀)

Let (𝑀,𝑔) be a surface and let 𝛾∶[𝑎,𝑏]→𝑀 be a curve in 𝑀. Let Γ(𝑠)=(𝛾(𝑠),𝑋(𝑠)) be a curve in (𝑈(𝑇𝑀),𝐺,𝜙,𝜉,𝜂), where the contact metric structure is given by (11).

Definition 3. We say that Γ(𝑠) is a slant curve in 𝑈(𝑇𝑀) if the angle between the tangent vector field Γ^′(𝑠) of Γ(𝑠) and 𝜉 is constant.

Assume that Γ(𝑠) is parameterized by the arc length. We know that Γ^′(𝑠)=𝑑𝑥^𝑖𝜕𝑑𝑠^𝑥𝑖+𝑑𝑋^𝑖𝜕𝑑𝑠^𝑢𝑖=𝑑𝑥^𝑖𝜕𝑑𝑠^𝑥𝑖ℎ(Γ(𝑠))+𝑑𝑋^𝑖+𝑑𝑠𝑑𝑥^𝑗𝑋𝑑𝑠^𝑘Γ^𝑖_𝑗𝑘𝜕_𝑢𝑖(Γ(𝑠=𝑇))_ℎ+∇_𝑇𝑋^𝑣(Γ(𝑠)), (13) where 𝑇(𝑠)=𝛾^′(𝑠). Since 𝐺(Γ^′,Γ^′)=1 and 𝐺=𝐺^′/4 we have 𝑔(𝑇,𝑇)+𝑔∇_𝑇𝑋,∇_𝑇𝑋=4. (14) Let 𝜃 be the angle between Γ^′(𝑠) and 𝜉. We have cos𝜃(𝑠)=𝐺Γ^′(𝑠)=𝑔(𝑇(𝑠)(𝑠,𝜉,𝑋))2. (15) Taking the derivative on both sides of (15) with respect to 𝑠, we derive that −sin𝜃⋅𝜃^′=𝑑𝐺Γ𝑑𝑠^′(𝑠)𝐷,𝜉=𝐺_Γ′(𝑠)Γ^′(𝑠)Γ,𝜉+𝐺^′(𝑠),𝐷_Γ′(𝑠)𝜉=𝑘_Γ𝜂𝑁_Γ+𝐺Γ^′(𝑠),2∇_𝑇𝑋^ℎ+𝑅_{𝑢∇𝑇𝑋}𝑢^ℎ−𝑅_𝑇𝑢𝑢^𝑡=𝑘_Γ𝜂𝑁_Γ+8𝑔∇_𝑇𝑋,𝑇−8𝑅𝑇,𝑋,𝑋,∇_𝑇𝑋, (16) where 𝜃^′=𝑑𝜃/𝑑𝑠, 𝜉(𝑠)=2𝑋^ℎ(𝑠), and 𝑘_Γ and 𝑁_Γ are the curvature and the direction of the acceleration of Γ(𝑠), respectively.

If Γ(𝑠) is a slant curve, that is, 𝜃≡Const, from (16) we have 𝑘_Γ𝜂𝑁_Γ+8𝑔∇_𝑇𝑋,𝑇−8𝑅𝑇,𝑋,𝑋,∇_𝑇𝑋=0. (17) Let {𝐸,𝑁} be the Frenet frame on 𝛾(𝑠). It follows from (15) that we have 𝑋=2cos𝜃𝛼𝐸+2𝛽𝑁, (18) where 𝛼=|𝑇| and 𝛽 is a 𝒞^∞ function. It follows that ∇_𝑇𝑋=2cos𝜃𝛼^′𝐸+2cos𝜃⋅𝑘_𝛾𝑁+2𝛽^′𝑁−2𝛼𝛽𝑘_𝛾𝐸=−2cos𝜃𝛼²𝛼^′+𝛼𝛽𝑘_𝛾𝐸+2𝑘_𝛾cos𝜃+𝛽^′𝑁. (19) Substituting (19) into (14), we have 𝛼²+4cos𝜃𝛼²𝛼^′+𝛼𝛽𝑘_𝛾²+4𝑘_𝛾cos𝜃+𝛽^′2=4. (20) Since 𝑔(𝑋,∇_𝑇𝑋)=0, from (15) and (19) we have 2𝑇=2cos𝜃𝑋−𝑔∇_𝑇𝑋,∇_𝑇𝑋cos𝜃𝛼𝛼^′+𝛼²𝛽𝑘_𝛾∇_𝑇𝑋. (21) It follows that 𝑅𝑇,𝑋,𝑋,∇_𝑇𝑋2()𝛼=−cos𝜃/𝛼^′+𝛼²𝛽𝑘_𝛾𝑅∇_𝑇𝑋,𝑋,𝑋,∇_𝑇𝑋𝑔∇_𝑇𝑋,∇_𝑇𝑋=−2cos𝜃𝛼𝛼^′+𝛼²𝛽𝑘_𝛾𝐾_𝑀(𝑠). (22) It follows from (19) and (22) that (17) turns into 𝑘_Γ𝜂𝑁_Γ+2cos𝜃𝛼𝛼^′+𝛼²𝛽𝑘_𝛾𝐾_𝑀−1=0. (23)

For the Legendrian curves, we have the following theorem.

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) 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) satisfying 𝐾_𝑀(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.

Proof. Since Γ(𝑠) is a Legendrian geodesic, we have 1cos𝜃=0,𝛽=±2,𝑋=±𝑁,𝑘_Γ=0. (24) Substituting these into (20) and (23), we have 𝛼²1+𝑘²_𝛾𝛼=4,₂𝑘_𝛾𝐾_𝑀−1=0. (25) By the second equation of (25) and the assumption of 𝐾(𝑠) we have 𝑘_𝛾=0 which means 𝛾(𝑠) is a geodesic in 𝑀. Substituting 𝑘_𝛾=0 into the first equation of (25) we have 𝛼=2. This completes the proof of Theorem 1.

For the non-Legendrian slant geodesics, we have the following theorem.

Theorem 2. Let Γ(𝑠)=(𝛾(𝑠),𝑋(𝑠)) be a slant geodesic in 𝑈(𝑇𝑀) which is not Legendrian. Under the assumptions of 𝐾_𝑀 as in Theorem 1, we have the following.(1)If 𝑋(𝑠)≠±𝐸(𝑠), then 𝛾(𝑠) is a geodesic of velocity 2 and 𝑋 is a parallel vector field along 𝛾(𝑠).(2)If 𝑋(𝑠)=±𝐸(𝑠), then 𝛾(𝑠) is a curve of velocity 2|cos𝜃| with constant curvature 𝑘_𝛾=±tan𝜃.

Proof. By the assumption we know that 𝜃≠𝜋/2, and form which and (18) we have 𝛽≠±1/2. Since 𝑔(𝑋,𝑋)=1, from (18) we have 4cos𝜃𝛼²+4𝛽²=1, (26) from which we derive that cos𝜃𝛼=1−4𝛽²2𝜋,𝜃∈0,2,−1−4𝛽²2𝜋,𝜃∈2,𝜋. (27) Suppose that 𝜃∈[0,𝜋/2), then we have 𝛼=2cos𝜃1−4𝛽². (28) Taking derivative on both side of this equation with respect to 𝑠, we have cos𝜃𝛼²𝛼^′=2𝛽𝛽^′1−4𝛽². (29) It follows that cos𝜃𝛼𝛼^′+𝛼²𝛽𝑘_𝛾=4𝛽cos𝜃1−4𝛽²𝛽^′+𝑘_𝛾cos𝜃. (30) Substituting (30) into (23), we have 𝑘_Γ𝜂𝑁_Γ+8𝛽cos𝜃1−4𝛽²𝛽^′+𝑘_𝛾cos𝜃𝐾_𝑀−1=0. (31)
Remark  4. (31) also holds for 𝜃∈(𝜋/2,𝜋].
For proving Theorem 2, we need the following lemma.
Lemma  5.  If there is some   𝑠₀∈[𝑎,𝑏]  such that   𝛽(𝑠₀)=0, then   𝛽  is vanishing everywhere.
Proof. Substituting (30) into (20), we have 11−4𝛽²cos²𝜃+𝑘_𝛾cos𝜃+𝛽^′2=1, (32) and from which we derive that 𝑘_𝛾cos𝜃+𝛽^′=±1−4𝛽²−cos²𝜃. (33) Since 𝑘_Γ=0 and under the assumption of 𝐾(𝑠), from (31) and (33) we have 𝛽1−4𝛽²−cos²𝜃=0. (34) Let 𝐼 be a subset of [𝑎,𝑏] such that 𝛽|_𝐼=0. Suppose that 𝐼 is not empty. From (34) we have 𝛽(𝑠)=±√1−cos²𝜃2, (35) for any 𝑠∈𝐼−[𝑎,𝑏]. It is a contradiction to the continuity of 𝛽 unless cos𝜃=±1; that is, 𝜃=0 or 𝜋. Substituting 𝜃=0 or 𝜋 into (34) we have 𝛽=0 which means 𝐼=[𝑎,𝑏]. This completes the proof of the Lemma.
Firstly, we consider 𝐼=∅ which implies 𝛽≠0. It follows from which and (18) we have 𝑋≠±𝐸.
From (34) we have 𝛽≡±sin𝜃2. (36) It follows from which and (27) we have 𝛼=2. Substituting 𝛽 into (33) we have 𝑘_𝛾=0.
Substituting 𝛼, 𝛽, and 𝑘_𝛾 into (19), we have ∇_𝑇𝑋=0. This proves Case (1) in Theorem 2.
Secondly, we consider 𝐼=[𝑎,𝑏] which implies 𝛽≡0. It follows from which and (18) we have 𝑋=±𝐸 and 𝛼=2|cos𝜃|.
Substituting 𝛼 and 𝛽 into (33) we have 𝑘_𝛾=±1cos²𝜃−1=±tan𝜃, (37) which proves Case (2) in Theorem 2. This completes the proof of Theorem 2.

Acknowledgment

This work was supported in part by the fundamental research funds for the Central University.