Surface pencil with a common line of curvature in Minkowski 3-space

In this paper, we analyze the problem of constructing a surface pencil from a given spacelie （timelike） line of curvature. Using the Frenet frame of the given line of curvature in Minkowski 3-space, we express the surface pencil as a linear combination of this frame and derive the necessary and sufficient conditions for the coefficients to satisfy the line of curvature requirement. We illustrate this method by presenting some examples.     

Space-like loxodromes on rotational surfaces in Minkowski 3-space

We find all space-like loxodromes on rotational surfaces which have space-like meridians or time-like meridians, respectively by using a relevant Lorentzian angle in Minkowski 3-space. To understand loxodromes better, we draw some pictures of them via Mathematica computer program.     

二维Minkowski空间R~(2.1)中混合型时空曲面的等温参数

对于三维Minkowski空间中的混合型时空曲面，证明了其在类空部分和类时部分上分别存在光滑的，并且连续到交界线上的等温参数．进一步，给出了混合型时空曲面存在在其类空部分和类时部分上是光滑的，而在交界线上是连续的等温多数的一个必要条件．     

A new version of Bishop frame and an application to spherical images

In this work, we introduce a new version of Bishop frame using a common vector field as binormal vector field of a regular curve and call this frame as “Type-2 Bishop Frame”. Thereafter, by translating type-2 Bishop frame vectors to the center of unit sphere of three-dimensional Euclidean space, we introduce new spherical images and call them as type-2 Bishop spherical images. Frenet-Serret apparatus of these new spherical images are obtained in terms of base curve's type-2 Bishop invariants. Additionally, we express some interesting relations and illustrate two examples of our main results.     

Helicoidal surfaces under the cubic screw motion in Minkowski 3-space

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H²=K and prove that these surfaces have equal constant principal curvatures.     

Position vectors of slant helices in Euclidean 3-space

In this paper, position vector of a slant helix with respect to standard frame in Euclidean space E³ is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine a position vector of an arbitrary slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of a Salkowski curve, anti-Salkowski curve and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.     

An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space

We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike principal normal in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at points where the curve experiences high torsion. In our construction the tubular surface’s twist is bounded by the spacelike curve’s curvature and is independent of the spacelike curve’s torsion.      

三维Minkowski空间中类空轴旋转曲面的分类

研究了三维Minkowski空间中满足ΔG=φ(G+C)条件的类空轴旋转曲面,并给出了该类曲面的分类.主要结论为上述条件中当C为零向量时该曲面拥有常值平均曲率;当C为非零向量时,该曲面或是一类圆锥面、或是二类圆锥面.     

Position vectors of spacelike general helices in Minkowski 3-space

In this paper, the position vectors of a spacelike general helix with respect to the standard frame in Minkowski space are studied in terms of the Frenet equations. First, a vector differential equation of third order is constructed to determine the position vectors of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of spacelike general helices with a spacelike and timelike principal normal vector.     

Contributions to differential geometry of isotropic curves in the complex space

This work deals with classical differential geometry of isotropic curves in the complex space C⁴. First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.     

A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Let be a Minkowski 3-space of Randers type with , where is the Euclidean metric and . We consider minimal surfaces in and prove that if a connected surface M in is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of , M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x ³-axis.      

Normals in a Minkowski Plane

In this paper we will establish bounds on the average number of normals through a point in a convex body in a Minkowski plane for certain classes of convex bodies. Also, a related Euler relation is discussed.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Complete flat surfaces with two isolated singularities in hyperbolic 3-space

We construct examples of flat surfaces in H³ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H³ with only one end and at most two isolated singularities.     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

The singularities of null surfaces in Anti de Sitter 3-space

We study the geometric properties of degenerate surfaces, which are called AdS null surfaces, in Anti de Sitter 3-space from a contact viewpoint. These surfaces are associated to spacelike curves in Anti de Sitter 3-space. We define a map which is called the torus Gauss image. We also define two families of functions and use them to investigate the singularities of AdS null surfaces and torus Gauss images as applications of singularity theory of functions.     

On the generalized Weyl problem for flat metrics in the hyperbolic 3-space

The paper deals with the study of complete embedded flat surfaces in H³${\mathbb{H}}^3$ with a finite number of isolated singularities. We give a detailed information about its topology, conformal type and metric properties. We show how to solve the generalized Weyl?s problem of realizing isometrically any complete flat metric with Euclidean singularities in H³${\mathbb{H}}^3$ which gives the existence of complete embedded flat surfaces with a finite arbitrary number of isolated singularities.     

Ruled surfaces of Weingarten type in Minkowski 3-space, II

In our previous paper of the same title, we did not study the ruled surfaces of Weingarten type M : x(s, t)=α(s)+t β (s) in Minkowski 3-space with vector fields β and β′ along the base curve β such that β is nowhere null but β′ is null everywhere. We here fulfill our project by investigating this remaining case.