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A Characterization of Involutes and Evolutes of a Given Curve in 𝔼<sup>n</sup>
Ozturk, Gunay,Arslan, Kadri,Bulca, Betul Department of Mathematics 2018 Kyungpook mathematical journal Vol.58 No.1
The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. Further, we obtain some results on these type of curves in ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$, respectively.
A NEW TYPE OF TUBULAR SURFACE HAVING POINTWISE 1-TYPE GAUSS MAP IN EUCLIDEAN 4-SPACE 𝔼<sup>4</sup>
Kisi, Ilim,Ozturk, Gunay Korean Mathematical Society 2018 대한수학회지 Vol.55 No.4
In this paper, we handle the Gauss map of a tubular surface which is constructed according to the parallel transport frame of its spine curve. We show that there is no tubular surface having harmonic Gauss map. Moreover, we give a complete classification of this kind of tubular surface having pointwise 1-type Gauss map in Euclidean 4-space ${\mathbb{E}}^4$.
Ilim Kisi,Gunay Ozturk 대한수학회 2018 대한수학회지 Vol.55 No.4
In this paper, we handle the Gauss map of a tubular surface which is constructed according to the parallel transport frame of its spine curve. We show that there is no tubular surface having harmonic Gauss map. Moreover, we give a complete classification of this kind of tubular surface having pointwise 1-type Gauss map in Euclidean $4$-space $\mathbb{E}^{4}$.
Kisi, Ilim,Ozturk, Gunay Department of Mathematics 2022 Kyungpook mathematical journal Vol.62 No.1
In this manuscript, we handle a tubular surface whose Gauss map G satisfies the equality L<sub>1</sub>G = f(G + C) for the Cheng-Yau operator L<sub>1</sub> in Galilean 3-space 𝔾<sub>3</sub>. We give an example of a tubular surface having L<sub>1</sub>-harmonic Gauss map. Moreover, we obtain a complete classification of tubular surface having L<sub>1</sub>-pointwise 1-type Gauss map of the first kind in 𝔾<sub>3</sub> and we give some visualizations of this type surface.
INVOLUTE CURVES OF ORDER k OF A GIVEN CURVE IN GALILEAN 4-SPACE G<sub>4</sub>
Kisi, Ilim,Ozturk, Gunay The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.2
In the present study, we consider the curves in Galilean 4-space ${\mathbb{G}}_4$. We find out the involute curves of order k (k = 1, 2, 3) of a given curve. We get the relationships between the Frenet apparatus of a given curve and its involute curves of order k.
INVOLUTE CURVES OF ORDER k OF A GIVEN CURVE IN GALILEAN 4-SPACE G<sub>4</sub>
( Ilim Kisi ),( Gunay Ozturk ) 호남수학회 2018 호남수학학술지 Vol.40 No.2
In the present study, we consider the curves in Galilean 4-space G<sub>4</sub>. We find out the involute curves of order k (k = 1; 2; 3) of a given curve. We get the relationships between the Frenet ap-paratus of a given curve and its involute curves of order k.
Tensor product surfaces with pointwise 1-type Gauss map
Kadri Arslan,Betul Bulca,Bengu Kilic,Young Ho Kim,Cengizhan Murathan,Gunay Ozturk 대한수학회 2011 대한수학회보 Vol.48 No.3
Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c_1 centered at origin with an Euclidean planar curve c_2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c_1 centered at origin with an Euclidean planar curve c_2 to have pointwise 1-type Gauss map.
TENSOR PRODUCT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP
Arslan, Kadri,Bulca, Betul,Kilic, Bengu,Kim, Young-Ho,Murathan, Cengizhan,Ozturk, Gunay Korean Mathematical Society 2011 대한수학회보 Vol.48 No.3
Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.