Liu Haiming

• Professor
• Professor at Mudanjiang Normal University

The aim of this paper is to obtain magnetic trajectories corresponding to contact magnetic fields in Lorentzian α-Sasakian 3-manifolds. First, we provide the definitions of the magnetic field and the Lorentzian equation used in this paper and derive the expression for the Bott connection by using the Levi-Civita connection. Then we derive the expre...

In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface Σ \Sigma in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on Σ \Sigma as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types...

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away...

In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classi...

In this paper, we focus on geometry of evolutoids and pedaloids for timelike curves and spacelike curves on timelike surfaces. First, by studying an example of a pseudo-circle, we get some properties of timelike curve on timelike surface. Then, we introduce a moving frame along curves and describe the corresponding evolutoids of curves as ruled sur...

In this paper, we focus on the research and analysis of the geometric properties and symmetry of slant curves and contact magnetic curves in Lorentzian α-Sasakian 3-manifolds. To do this, we define the notion of Lorentzian cross product. From the perspectives of the Legendre and non-geodesic curves, we found the ratio relationship between the curva...

In this paper, we define the notion of Bott connection in the Heisenberg group [Formula: see text] and derive the expression of the Bott connection by using the Levi-Civita connection. Moreover, we derive the expressions of killing vector fields by using the killing equation and obtain some explicit formulas for killing magnetic curves associated t...

Canonical connections play important roles in studying the differential geometry properties of submanifolds in Lie groups. We define the first kind of canonical connection and the second canonical connection on Lorentzian approximations of the Heisenberg group. Moreover, we give the definitions of intrinsic curvature of a regular curve as well of i...

A surface with nonempty timelike, lightlike, and spacelike points in Minkowski 3‐space is a mixed type surface. The mixed type surface has a signature‐changing metric, and its lightlike points can be seen as singularities of such metric. In this paper, we study singular properties of pseudo‐spherical evolutes of lightlike loci on mixed type surface...

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane EL21,1. By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian s...

We consider the sub-Lorentzian geometry of curves and surfaces in the Lie group E 1 , 1 . Firstly, as an application of Riemannian approximants scheme, we give the definition of Lorentzian approximants scheme for E 1 , 1 which is a sequence of Lorentzian manifolds denoted by E λ 1 , λ 2 L . By using the Koszul formula, we calculate the expressions...

In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information abou...

In this paper, we consider two kinds of developable surfaces along a timelike frontal curve lying in a timelike surface in Minkowski 3-space, the Lorentz–Darboux rectifying surfaces and the Lorentz–Darboux osculating surfaces. Meanwhile, we also consider two curves generated by such a timelike frontal curve. We give two new invariants of the fronta...

The universal covering group of Euclidean motion group E (2) with the general left-invariant metric is denoted by $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2)),$$ ( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) , where $$\lambda _1\ge \lambda _2>0.$$ λ 1 ≥ λ 2 > 0 . It is one of three-dimensional unimodular Lie groups which are classified by Milnor. In this pa...

The motivation of this paper is that we wish to add some new information on Legendrian dualities for continuous families of pseudo-spheres in Minkowski space. We construct all contact diffeomorphic mappings between those contact manifolds and display them in a table which contains all information about Legendrian dualities, by the way, we extend th...

In this paper, we define the Lorentzian approximations of a $ 3 $-dimensional Lorentzian $ \alpha $-Sasakian manifold. Moreover, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and spacelike surfaces and the intrinsic Gaussian curvature of Lorentzian surf...

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of s...

In this paper, we consider the local topological structures of a class of new worldsheets, call it the rectifying worldsheets, which are generated by a class of singular worldlines. Using the classification approaches of the finite type on the tangent developables and defining the extended striction curve, this paper gives the detailed classificati...

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute t...

The rototranslation group ℛT is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in the rototranslation group away from characteristic points...

In this paper, we use a Lorentzian approximation scheme to compute the sub-Lorentzian limit of curvatures for curves and Lorentzian surfaces in the Lorentzian Bianci-Cartan-Vranceanu model of 3-dimensional Lorentzian Sasakian space forms. Based on these results, we get a Gauss-Bonnet theorem in the Lorentzian Sasakian space forms.

The motivation of this paper is that we wish to add some new information on Legendrian dualities for continuous families of pseudo-spheres in Minkowski space. We construct all contact diffeomorphic mappings between those contact manifolds and display them in a table which contains all information about Legendrian dualities, by the way, we extend th...

The roto-translation group RT is the group comprising rotations and translations of the Euclidean plane which has profound physical backgrounds. In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $C^2$-smooth surface in the roto-translation away from characteristic points and signed geodesic curvature for Euclidea...

The focal surface of a generic space curve in Euclidean 3-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory...

In this paper we study helix surfaces whose unit normals make a constant angle with a fixed direction. As an application of the singularity theory, we classify the generic singularities of helix surfaces, which are cuspidaledges and swallowtails. These singularities are deeply related to the order of contact between the generating curves of helix s...

We extend and generalize the best proximity results for Suzuki type α⁺-ψ -proximal single valued map- pings given by Hussain et al. Some novel best proximity results and coupled best proximity results are presented for Suzuki type α⁺-ψ -proximal multivalued mappings satisfying generalized conditions of existence.

In this paper, we consider evolutes of spacelike curves in de Sitter 2-space. Applying the theory of singularity theory, we find that these evolutes can be seen as one dimensional caustics which are locally diffeomorphic to lines or ordinary cusps. We establish the relationships between singularities of caustics and geometric invariants of curves u...

Legendrian dualities between spherical indicatrixes of curves in Euclidean 3-space are investigated by using the theory of Legendrian duality. Moreover, the singularities of the ruled surfaces according to Bishop frame which are deeply related to space curves are classified from the viewpoints of wave fronts. We also give some more detail descripti...

In this paper, we consider the spacelike curves in de Sitter space and we investigate the singularities of lightcone dual surfaces and hyperbolic dual surfaces of these spacelike curves in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space. We classify the singularities of these subjects and reveal the rel...

The relatively parallel adapted frame or Bishop frame is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of biology, engineering, and computer graphics. The main result of this paper is using the relatively parallel adapted frame for c...

The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspon...