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ON GENERALIZED ROTATIONAL SURFACES IN EUCLIDEAN SPACES
Arslan, Kadri,Bulca, Betul,Kosova, Didem Korean Mathematical Society 2017 대한수학회지 Vol.54 No.3
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean (n + 1)-space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces $\mathbb{E}^3$ and $\mathbb{E}^4$, respectively. We have also obtained some basic properties of generalized rotational surfaces in $\mathbb{E}^4$ and some results of their curvatures. Finally, we give some examples of generalized Beltrami surfaces in $\mathbb{E}^3$ and $\mathbb{E}^4$, respectively.
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.
MERIDIAN SURFACES IN <sup>4</sup> WITH POINTWISE 1-TYPE GAUSS MAP
Arslan, Kadri,Bulca, Betul,Milousheva, Velichka Korean Mathematical Society 2014 대한수학회보 Vol.51 No.3
In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.
ON GENERALIZED SPHERICAL SURFACES IN EUCLIDEAN SPACES
Bayram, Bengu,Arslan, Kadri,Bulca, Betul The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.3
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean (n + 1)-space ${\mathbb{E}}^{n+1}$. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$ respectively. We have shown that the generalized spherical surfaces of first kind in ${\mathbb{E}}^4$ are known as rotational surfaces, and the second kind generalized spherical surfaces are known as meridian surfaces in ${\mathbb{E}}^4$. We have also calculated the Gaussian, normal and mean curvatures of these kind of surfaces. Finally, we give some examples.
MERIDIAN SURFACES IN E4 WITH POINTWISE 1-TYPE GAUSS MAP
Kadri Arslan,Betul Bulca,Velichka Milousheva 대한수학회 2014 대한수학회보 Vol.51 No.3
In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.
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.
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.
ON GENERALIZED ROTATIONAL SURFACES IN EUCLIDEAN SPACES
Kadri ARSLAN,Betul Bulca,Didem Kosova 대한수학회 2017 대한수학회지 Vol.54 No.3
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean $ (n+1)$-space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces $\mathbb{E}^{3}$ and $ \mathbb{E}^{4}$, respectively. We have also obtained some basic properties of generalized rotational surfaces in $\mathbb{E}^{4}$ and some results of their curvatures. Finally, we give some examples of generalized Beltrami surfaces in $\mathbb{E}^{3}$ and $\mathbb{E}^{4}$, respectively.
ON GENERALIZED SPHERICAL SURFACES IN EUCLIDEAN SPACES
( Bengu Bayram ),( Kadri Arslan ),( Betul Bulca ) 호남수학회 2017 호남수학학술지 Vol.39 No.3
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean (n + 1)-space E<sup>n+1</sup>. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces E<sup>3</sup> and E<sup>4</sup> respectively. We have shown that the generalized spherical surfaces of first kind in E<sup>4</sup> are known as rotational surfaces, and the second kind generalized spherical surfaces are known as meridian surfaces in E<sup>4</sup>. We have also calculated the Gaussian, normal and mean curvatures of these kind of surfaces. Finally, we give some examples.