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RISS 인기검색어
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- 저자 : Pascual Lucas (검색결과 6 건)
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HYPERSURFACES IN 𝕊<sup>4</sup> THAT ARE OF L<sub>k</sub>-2-TYPE
Lucas, Pascual,Ramirez-Ospina, Hector-Fabian Korean Mathematical Society 2016 대한수학회보 Vol.53 No.3
In this paper we begin the study of $L_k$-2-type hypersurfaces of a hypersphere ${\mathbb{S}}^{n+1}{\subset}{\mathbb{R}}^{n+2}$ for $k{\geq}1$ Let ${\psi}:M^3{\rightarrow}{\mathbb{S}}^4$ be an orientable $H_k$-hypersurface, which is not an open portion of a hypersphere. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is a Clifford tori ${\mathbb{S}}^1(r_1){\times}{\mathbb{S}}^2(r_2)$, $r^2_1+r^2_2=1$, for appropriate radii, or a tube $T^r(V^2)$ of appropriate constant radius r around the Veronese embedding of the real projective plane ${\mathbb{R}}P^2({\sqrt{3}})$.
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Hypersurfaces in ${\mathbb{S}}^4$ that are of $L_k$-2-type
Pascual Lucas,H\'ector-Fabi\'an Ram\'\i rez-Ospina 대한수학회 2016 대한수학회보 Vol.53 No.3
In this paper we begin the study of $L_k$-2-type hypersurfaces of a hypersphere $\S^{n+1}\subset\R^{n+2}$ for $k\geq 1$. Let $\psi:\M\rightarrow\S^{4}$ be an orientable $H_k$-hypersurface, which is not an open portion of a hypersphere. Then $\M$ is of $L_k$-2-type if and only if $\M$ is a Clifford tori $\S^1(r_1)\times\S^2(r_2)$, $r_1^2+r_2^2=1$, for appropriate radii, or a tube $T^r(V^2)$ of appropriate constant radius $r$ around the Veronese embedding of the real projective plane $\R P^2(\sqrt3)$.
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BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS
Lucas, Pascual,Ortega-Yagues, Jose Antonio Korean Mathematical Society 2013 대한수학회보 Vol.50 No.4
Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.
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Bertrand curves in non-flat 3-dimensional (Riemannian or Lorentzian) space forms
Pascual Lucas,Jose Antonio Ortega-Yagues 대한수학회 2013 대한수학회보 Vol.50 No.4
Let M3 q(c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c 6= 0. A curve α immersed in M3 q(c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: nonnull Bertrand curves in M3 q(c) correspond with curves for which there exist two constants λ 6= 0 and μ such that λκ + μ = 1, where τ and stand for the curvature and torsion of the curve. As a consequence, non-null helices in M3 q(c) are the only twisted curves in M3 q(c) having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.
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SLANT HELICES IN THE THREE-DIMENSIONAL SPHERE
Pascual Lucas,Jose Antonio Ortega-Yagues 대한수학회 2017 대한수학회지 Vol.54 No.4
A curve $\gamma$ immersed in the three-dimensional sphere $\S3$ is said to be a slant helix if there exists a Killing vector field $V(s)$ with constant length along $\gamma$ and such that the angle between $V$ and the principal normal is constant along $\gamma$. In this paper we characterize slant helices in $\S3$ by means of a differential equation in the curvature $\kappa$ and the torsion $\tau$ of the curve. We define a helix surface in $\S3$ and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in $\S3$. Finally, we obtain a geometric approach to the problem of solving natural equations for slant helices in the three-dimensional sphere. We prove that the slant helices in $\S3$ are exactly the geodesics of helix surfaces.
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SLANT HELICES IN THE THREE-DIMENSIONAL SPHERE
Lucas, Pascual,Ortega-Yagues, Jose Antonio Korean Mathematical Society 2017 대한수학회지 Vol.54 No.4
A curve ${\gamma}$ immersed in the three-dimensional sphere ${\mathbb{S}}^3$ is said to be a slant helix if there exists a Killing vector field V(s) with constant length along ${\gamma}$ and such that the angle between V and the principal normal is constant along ${\gamma}$. In this paper we characterize slant helices in ${\mathbb{S}}^3$ by means of a differential equation in the curvature ${\kappa}$ and the torsion ${\tau}$ of the curve. We define a helix surface in ${\mathbb{S}}^3$ and give a method to construct any helix surface. This method is based on the Kitagawa representation of flat surfaces in ${\mathbb{S}}^3$. Finally, we obtain a geometric approach to the problem of solving natural equations for slant helices in the three-dimensional sphere. We prove that the slant helices in ${\mathbb{S}}^3$ are exactly the geodesics of helix surfaces.
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