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.
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.  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.  studied the slant curves in -Kenmotsu manifolds. In , 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  constructed an almost contact metric structure in the unit tangent bundle of which is induced from the almost complex structure in . Klingenberg and Sasaki  studied geodesics in the unit tangent bundle of -sphere endowed with Sasaki metric and showed that is isometric to . Sasaki  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 .
Firstly, we introduce the (almost) contact metric structure on a Riemannian manifold of odd dimension. With the same notations as in ; 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 , 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. , 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.
This work was supported in part by the fundamental research funds for the Central University.
D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 203 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 2002.View at: MathSciNet