2019
Том 71
№ 1

Yayli Y.

Articles: 2
Brief Communications (English)

Special Space Curves Characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 571-576

By using the facts that the condition$\det(α^{(1)}, α^{(2)}, α^{(3)}) = 0$ characterizes a plane curve and the condition $\det(α^{(2)}, α^{(3)}, α^{(4)}) = 0$ characterizes a curve of constant slope, we present special space curves characterized by the condition $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$, in different approaches. It is shown that the space curve is Salkowski if and only if $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$. The approach used in our investigation can be useful in understanding the role of the curves characterized by $\det(α^{(3)}, α^{(4)}, α^{(5)}) = 0$ in differential geometry.

Article (English)

Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 307-317

We show that a hyperquadric $M$ in $\mathbb{R}^4_2$ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric $M$. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Morever, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in $\mathbb{R}^4_2$.