• Journal of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.695-704
• /
• 2018

In this paper we derive some basic inequalities connecting the Minkowski measure of symmetry, the Minkowski symmetral and the Banach-Mazur distance. We then explore the geometric contents of these inequalities and shed light on the structure of the quotient 𝔅/Aff of the space of convex bodies modulo the affine transformations.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.2
• /
• pp.377-394
• /
• 2015

In this paper, we generalize some results about Bertrand curves and Razzaboni surfaces in Euclidean 3-space to the case that the ambient space is Minkowski 3-space. Our discussion is divided into three different cases, i.e., the parent Bertrand curve being timelike, spacelike with timelike principal normal, and spacelike with spacelike principal normal. For each case, first we show that Razzaboni surfaces and their mates are related by a reciprocal transformation; then we give B$\ddot{a}$cklund transformations for Bertrand curves and for Razzaboni surfaces; finally we prove that the reciprocal and B$\ddot{a}$cklund transformations on Razzaboni surfaces commute.

• Journal of the Korean Mathematical Society
• /
• v.41 no.6
• /
• pp.1007-1021
• /
• 2004

In this paper, we study rotation surfaces in the Minkowski 3-dimensional space with pointwise 1-type Gauss map and obtain by the use of the concept of pointwise finite type Gauss map, a characterization theorem concerning rotation surfaces and constancy of the mean curvature of certain open subsets on these surfaces.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.3
• /
• pp.429-438
• /
• 2007

In this paper, we study the position vectors of a spacelike W-curve (or a helix), i.e., curve with constant curvatures, with spacelike, timelike and null principal normal in the Minkowski 3-space $\mathbb{E}_1^3$. We give some characterizations for spacelike W - curves whose image lies on the pseudohyperbolical space $\mathbb{H}_0^2$ and Lorentzian sphere $\mathbb{S}_1^2$ by using the positions vectors of the curve.

• Kyungpook Mathematical Journal
• /
• v.54 no.4
• /
• pp.685-697
• /
• 2014

In this paper, we prove that there are no pseudo null Bertrand curve with curvature functions $k_1(s)=1$, $k_2(s){\neq}0$ and $k_3(s)$ other than itself in Minkowski spacetime ${\mathbb{E}}_1^4$ and by using the similar idea of Matsuda and Yorozu [13], we define a new kind of Bertrand curve and called it pseudo null (1,3)-Bertrand curve. Also we give some characterizations and an example of pseudo null (1,3)-Bertrand curves in Minkowski spacetime.

• Journal of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.849-866
• /
• 2011

In this paper, we give the differential equations characterizing the timelike and spacelike curves of constant breadth in Minkowski 3-space $E^3_1$. Furthermore, we give a criterion for a timelike or spacelike curve to be the curve of constant breadth in $E^3_1$. As an example, the obtained results are applied to the case $\rho$ = const. and $k_2$ = const., and are discussed.

• Journal of the Korean Mathematical Society
• /
• v.48 no.1
• /
• pp.159-167
• /
• 2011

We consider a curve $\alpha$= $\alpha$(s) in Minkowski 3-space $E_1^3$ and denote by {T, N, B} the Frenet frame of $\alpha$. We say that $\alpha$ is a slant helix if there exists a fixed direction U of $E_1^3$ such that the function is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of $\alpha$. Finally, we discuss the tangent and binormal indicatrices of slant curves, proving that they are helices in $E_1^3$.

• Honam Mathematical Journal
• /
• v.40 no.3
• /
• pp.561-576
• /
• 2018

In this paper, we define null Cartan and pseudo null Bertrand curves in Minkowski space ${\mathbb{E}}^3_1$ according to their Bishop frames. We obtain the necessary and sufficient conditions for pseudo null curves to be Bertand B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Bertrand B-curves, by considering the cases when their Bertrand B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Bertrand B-curve pairs.

• Honam Mathematical Journal
• /
• v.36 no.2
• /
• pp.233-251
• /
• 2014

In this paper, position vector of a spacelike slant helix with respect to standard frame are deduced in Minkowski space $E^3_1$. Some new characterizations of a spacelike slant helices are presented. Also, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike curve. In terms of solution, we determine the parametric representation of the spacelike slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of some special spacelike slant helices such as: Salkowski and anti-Salkowski curves.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.4
• /
• pp.1339-1351
• /
• 2015

In this paper, using pseudo-spherical Frenet frame of pseudo-spherical curves in hyperbolic space, we define the notion of the structure functions on the non-developable ruled surfaces with timelike ruling. Then we obtain the properties of the structure functions and a complete classification of the non-developable ruled surfaces with timelike ruling in Minkowski 3-space by the theories of the structure functions.