# Slant Curves in the Unit Tangent Bundles of Surfaces

Authors: Zhong Hua Hou; Lei Sun
Journal: ISRN Geometry
 (2013)

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## 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.


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## Body

## 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)/kγ is constant where τ and kγ 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 Gs. 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,Gs,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 bundleU(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 KM(s) be the sectional curvature of M at γ(s), we have the following theorems.Theorem 1.
LetΓ(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 KM(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.
LetΓ(s)=(γ(s),X(s)) be a slant geodesic parameterized by arc length in U(TM) which is not Legendrian. Under the assumptions of KM(s) as in Theorem 1, we have the following.(1)
IfX(s)≠±E(s), then γ(s) is a geodesic of velocity 2 and X is a parallel vector field along γ(s).(2)
IfX(s)=±E(s), then γ(s) is a curve of velocity 2|cosθ| with constant curvature kγ=±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
(1)ϕx2=-I+ηx⨂ξx,ηx(ξx)=1,
where I denotes the identity transformation of the tangent space TxM 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
(2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),
for any vector fields X,Y∈𝔛(M).As in Kahler geometry, we can define the fundamental 2-formΦ on the almost contact metric manifold (M,g,ϕ,ξ,η) as
(3)Φ(X,Y)=g(X,ϕY),
for any vector fields X,Y∈𝔛(M). Obviously, this form satisfies η∧Φn≠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(xi,ui) be the locally coordinate systems on the tangent bundle TM of M. Sasaki [4, 5] defined a metric Gs on TM which is the natural lifts of the metric g on M as follows:
(4)Gs(Xh,Yh)=g(X,Y),Gs(Xh,Yv)=0,Gs(Xv,Yv)=g(X,Y),
where X,Y∈𝔛(M) and Xh=Xi∂xi-XiujΓijk∂uk and Xv=Xi∂ui are the horizontal and vertical lifts of X at (x,u) with respect to Levi-Civita connection ∇ of g, respectively, and {Γijk} are the Christoffel symbols of ∇. The Levi-Civita connection ∇¯ of Gs is defined as
(5)∇¯XhYh=(∇XY)h-12(RXYMu)v,∇¯XvYh=12(RuXMY)h,∇¯XhYv=(∇XY)v+12(RuYMX)h,∇¯XvYv=0,
where X,Y∈𝔛(M) and RM is the curvature tensor on (M,g). The almost complex structure J on TM which is compatible with Gs is given by
(6)JXh=Xv,JXv=-Xh,
for any vector field X∈𝔛(M).We know that the normal vector field of the unit tangent bundleU(TM)={(x,u)∈TM;g(u,u)=1} of M at (x,u)∈U(TM) in (TM,Gs) is uv. Let 𝔛(U(TM))={Xh+Yt;X,Y∈𝔛(M)} be the Lie algebra of 𝒞∞ vector fields on U(TM) where Xh=Xi∂xi-XiujΓijk and Xt=(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*Gs in U(TM) as follows:
(7)G′(Xh,Yh)=g(X,Y),G′(Xh,Yt)=0,G′(Xt,Yt)=g(X,Y)-g(X,u)g(Y,u),
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
(8)J∘i*=i*∘ϕ′+η′⨂uv,J∘i*(ξ′)=uv.
This implies that
(9)η′(ξ′)=1,ϕ′(ξ′)=0,η′∘ϕ′=0.
From Blair et al. [10], at any point (x,u)∈U(TM) we have
(10)ξ′=uh,η′(X¯)=G′(X¯,uh),Φ′(X¯,Y¯)=G′(X¯,ϕ′Y¯)=2dη′(X¯,Y¯),ϕ′Xh=Xt,ϕ′Xt=-Xh+η′(Xh)ξ′,
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
(11)η=12η′,ξ=2ξ′,ϕ=ϕ′,G=14G′.By (5), we know that the Levi-Civita connection D of (U(TM),G) is determined by the following:
(12)DXhYh=(∇¯XhYh)⊤=(∇XY)h-12(RXYu)t,DXtYh=(∇¯XtYh)⊤=12(RuXY)h,DXhYt=(∇¯XhYt)⊤=(∇XY)t+12(RuYX)h,DXtYt=(∇¯XtYt)⊤=-g(Y,u)Xt,
for any (x,u)∈U(TM) and X,Y∈𝔛(U(TM)).


## 3. Geodesic Slant Curves inU(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 thatΓ(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 thatΓ(s) is parameterized by the arc length. We know that
(13)Γ′(s)=dxids∂xi+dXids∂ui=dxids(∂xi)h(Γ(s))+(dXids+dxjdsXkΓjki)∂ui(Γ(s))=(Th+(∇TX)v)(Γ(s)),
where T(s)=γ′(s). Since G(Γ′,Γ′)=1 and G=G′/4 we have
(14)g(T,T)+g(∇TX,∇TX)=4.
Let θ be the angle between Γ′(s) and ξ. We have
(15)cosθ(s)=G(Γ′(s),ξ)=g(T(s),X(s))2.
Taking the derivative on both sides of (15) with respect to s, we derive that
(16)-sinθ·θ′=ddsG(Γ′(s),ξ)=G(DΓ′(s)Γ′(s),ξ)+G(Γ′(s),DΓ′(s)ξ)=kΓη(NΓ)+G((Ru∇TXu)hΓ′(s),2(∇TX)h=kΓη(NΓ)mGGG+(Ru∇TXu)h-(RTuu)t)=kΓη(NΓ)+8g(∇TX,T)-8R(T,X,X,∇TX),
where θ′=dθ/ds, ξ(s)=2Xh(s), and kΓ and NΓ are the curvature and the direction of the acceleration of Γ(s), respectively.IfΓ(s) is a slant curve, that is, θ≡Const, from (16) we have
(17)kΓη(NΓ)+8g(∇TX,T)-8R(T,X,X,∇TX)=0.
Let {E,N} be the Frenet frame on γ(s). It follows from (15) that we have
(18)X=2cosθαE+2βN,
where α=|T| and β is a 𝒞∞ function. It follows that
(19)∇TX=(2cosθα)′E+2cosθ·kγN+2β′N-2αβkγE=-2(cosθα2α′+αβkγ)E+2(kγcosθ+β′)N.
Substituting (19) into (14), we have
(20)α2+4(cosθα2α′+αβkγ)2+4(kγcosθ+β′)2=4.
Since g(X,∇TX)=0, from (15) and (19) we have
(21)T=2cosθX-2g(∇TX,∇TX)(cosθαα′+α2βkγ)∇TX.
It follows that
(22)R(T,X,X,∇TX)=-2((cosθ/α)α′+α2βkγ)R(∇TX,X,X,∇TX)g(∇TX,∇TX)=-2(cosθαα′+α2βkγ)KM(s).
It follows from (19) and (22) that (17) turns into
(23)kΓη(NΓ)+2(cosθαα′+α2βkγ)(KM-1)=0.For the Legendrian curves, we have the following theorem.Theorem 1.
LetΓ(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 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.Proof.
SinceΓ(s) is a Legendrian geodesic, we have
(24)cosθ=0,β=±12,X=±N,kΓ=0.
Substituting these into (20) and (23), we have
(25)α2(1+kγ2)=4,α2kγ(KM-1)=0.
By the second equation of (25) and the assumption of K(s) we have kγ=0 which means γ(s) is a geodesic in M. Substituting kγ=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.
LetΓ(s)=(γ(s),X(s)) be a slant geodesic in U(TM) which is not Legendrian. Under the assumptions of KM as in Theorem 1, we have the following.(1)
IfX(s)≠±E(s), then γ(s) is a geodesic of velocity 2 and X is a parallel vector field along γ(s).(2)
IfX(s)=±E(s), then γ(s) is a curve of velocity 2|cosθ| with constant curvature kγ=±tanθ.Proof.
By the assumption we know thatθ≠π/2, and form which and (18) we have β≠±1/2. Since g(X,X)=1, from (18) we have
(26)4(cosθα)2+4β2=1,
from which we derive that
(27)cosθα={1-4β22,θ∈[0,π2),-1-4β22,θ∈(π2,π].
Suppose that θ∈[0,π/2), then we have
(28)α=2cosθ1-4β2.
Taking derivative on both side of this equation with respect to s, we have
(29)cosθα2α′=2ββ′1-4β2.
It follows that
(30)cosθαα′+α2βkγ=4βcosθ1-4β2(β′+kγcosθ).
Substituting (30) into (23), we have
(31)kΓη(NΓ)+8βcosθ1-4β2(β′+kγcosθ)(KM-1)=0.
Remark  4. (31) also holds for θ∈(π/2,π].
For proving Theorem2, we need the following lemma.
Lemma  5.If there is somes0∈[a,b]such thatβ(s0)=0, thenβis vanishing everywhere.
Proof. Substituting (30) into (20), we have
(32)11-4β2(cos2θ+(kγcosθ+β′)2)=1,
and from which we derive that
(33)kγcosθ+β′=±1-4β2-cos2θ.
Since kΓ=0 and under the assumption of K(s), from (31) and (33) we have
(34)β(1-4β2-cos2θ)=0.
Let I be a subset of [a,b] such that β|I=0. Suppose that I is not empty. From (34) we have
(35)β(s)=±1-cos2θ2,
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 considerI=∅ which implies β≠0. It follows from which and (18) we have X≠±E.
From (34) we have
(36)β≡±sinθ2.
It follows from which and (27) we have α=2. Substituting β into (33) we have kγ=0.
Substitutingα, β, and kγ into (19), we have ∇TX=0. This proves Case (1) in Theorem 2.
Secondly, we considerI=[a,b] which implies β≡0. It follows from which and (18) we have X=±E and α=2|cosθ|.
Substitutingα and β into (33) we have
(37)kγ=±1cos2θ-1=±tanθ,
which proves Case (2) in Theorem 2. This completes the proof of Theorem 2.



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