#### Abstract

Let be a surface and let be the unit tangent bundle of endowed with the Sasaki metric. We know that any curve in consist of a curve in and as unit vector field along . In this paper we study the geometric properties and satisfying when is a slant geodesic.

#### 1. Introduction

Let be a 3-dimensional contact metric manifold. The slant curves in are generalization of Legendrian curves which form a constant angle with the Reeb vector field . Cho et al. [1] studied Lancret type problem for curves in Sasakian 3-manifold. They showed that a curve is slant if and only if is constant where and are torsion and curvature of , 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 -Kenmotsu manifolds. In [3], Călin and Crasmareanu studied slant curves in normal almost contact manifolds.

Let be a Riemannian manifold. Sasaki [4, 5] studied the geometries of endowed with the Sasaki metric and introduced the almost complex structure in which is compatible with . Tashiro [6] constructed an almost contact metric structure in the unit tangent bundle of which is induced from the almost complex structure in . Klingenberg and Sasaki [7] studied geodesics in the unit tangent bundle of -sphere endowed with Sasaki metric and showed that is isometric to . 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 of some surface . For any curve in , let be the tangent vector field of and let be the sectional curvature of at , we have the following theorems.

Theorem 1. *Let be a Legendrian geodesic parameterized by arc length in with domain . If the set consisting of points such that is discrete, then is a geodesic of velocity and is the normal direction of in .*

Theorem 2. *Let be a slant geodesic parameterized by arc length in which is not Legendrian. Under the assumptions of as in Theorem 1, we have the following.*(1)*If , then is a geodesic of velocity 2 and is a parallel vector field along .*(2)*If , then is a curve of velocity with constant curvature .*

#### 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 be a real -dimensional manifold and the Lie algebra of vector fields on . An *almost cocomplex structure* on is defined by a (1,1)-tensor , a vector field and a 1-form on such that for any point we have
where denotes the identity transformation of the tangent space at . Manifolds equipped with an almost cocomplex structure are called *almost contact manifolds*. A Riemannian manifold with a metric and an almost contact structure is called *almost contact metric* or *almost co-Hermitian manifold* if and satisfies
for any vector fields .

As in Kahler geometry, we can define the fundamental 2-form on the almost contact metric manifold as
for any vector fields . Obviously, this form satisfies which means every almost contact metric manifold is orientable and that defines an *almost cosymplectic structure* on . If the fundamental 2-form , then we call a contact metric manifold.

Let be the locally coordinate systems on the tangent bundle of . Sasaki [4, 5] defined a metric on which is the natural lifts of the metric on as follows: where and and are the horizontal and vertical lifts of at with respect to Levi-Civita connection of , respectively, and are the Christoffel symbols of . The Levi-Civita connection of is defined as where and is the curvature tensor on . The almost complex structure on which is compatible with is given by for any vector field .

We know that the normal vector field of the unit tangent bundle of at in is . Let be the Lie algebra of vector fields on where and are the horizontal and tangential lifts of at , respectively. Let be the including map from to and this map induces a metric in as follows: for any vector fields .

From Tashiro [6], we know that there is an almost contact metric structure in which is induced from the almost complex structure in such that This implies that From Blair et al. [10], at any point we have where and . Hence, we know that there is a contact metric structure in such that

By (5), we know that the Levi-Civita connection of is determined by the following: for any and .

#### 3. Geodesic Slant Curves in

Let be a surface and let be a curve in . Let be a curve in , where the contact metric structure is given by (11).

*Definition 3. *We say that is a slant curve in if the angle between the tangent vector field of and is constant.

Assume that is parameterized by the arc length. We know that where . Since and we have Let be the angle between and . We have Taking the derivative on both sides of (15) with respect to , we derive that where , , and and are the curvature and the direction of the acceleration of , respectively.

If is a slant curve, that is, , from (16) we have Let be the Frenet frame on . It follows from (15) that we have where and is a function. It follows that Substituting (19) into (14), we have Since , from (15) and (19) we have It follows that It follows from (19) and (22) that (17) turns into

For the Legendrian curves, we have the following theorem.

Theorem 1. *Let and let be a Legendrian geodesic parameterized by arc length in with domain . Suppose that the set consisting of points such that the sectional curvature of at satisfying is discrete. Then is a geodesic of velocity and is the normal direction of in .*

*Proof. *Since is a Legendrian geodesic, we have
Substituting these into (20) and (23), we have
By the second equation of (25) and the assumption of we have which means is a geodesic in . Substituting into the first equation of (25) we have . This completes the proof of Theorem 1.

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

Theorem 2. *Let be a slant geodesic in which is not Legendrian. Under the assumptions of as in Theorem 1, we have the following.*(1)*If , then is a geodesic of velocity and is a parallel vector field along .*(2)*If , then is a curve of velocity with constant curvature .*

*Proof. *By the assumption we know that , and form which and (18) we have . Since , from (18) we have
from which we derive that
Suppose that , then we have
Taking derivative on both side of this equation with respect to , we have
It follows that
Substituting (30) into (23), we have
*Remark 4.* (31) also holds for .

For proving Theorem 2, we need the following lemma.

Lemma 5. *If there is some ** such that **, then ** is vanishing everywhere. **Proof.* Substituting (30) into (20), we have
and from which we derive that
Since and under the assumption of , from (31) and (33) we have
Let be a subset of such that . Suppose that is not empty. From (34) we have
for any . It is a contradiction to the continuity of unless ; that is, or . Substituting or into (34) we have which means . This completes the proof of the Lemma.

Firstly, we consider which implies . It follows from which and (18) we have .

From (34) we have
It follows from which and (27) we have . Substituting into (33) we have .

Substituting , , and into (19), we have . This proves Case in Theorem 2.

Secondly, we consider which implies . It follows from which and (18) we have and .

Substituting and into (33) we have
which proves Case 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.