Slant Curves in the Unit Tangent Bundles of Surfaces

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, C？lin et al. [2] studied the slant curves in -Kenmotsu manifolds. In [3], C？lin 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