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Bangyen Chen - One of the best experts on this subject based on the ideXlab platform.
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representation of flat lagrangian h umbilical submanifolds in complex euclidean spaces
Tohoku Mathematical Journal, 1999Co-Authors: Bangyen ChenAbstract:The author proved earlier that, a Lagrangian //-umbilical submanifold in complex Euclidean «-space with n>2 is either a complex extensor, a Lagrangian pseudo-sphere, or a flat Lagrangian //-umbilical submanifold. Explicit descriptions of complex extensors and of Lagrangian pseudo-spheres are given earlier. The purpose of this article is to complete the investigation of Lagrangian //-umbilical submanifolds in complex Euclidean spaces by establishing the explicit description of flat Lagrangian //-umbilical submanifolds in complex Euclidean spaces. 1. Statements of theorems. We follow the notation and definitions given in (2). In order to establish the complete classification of Lagrangian //-umbilical submanifolds in Cn we need to introduce the notion of special Legendre Curves as follows. Let z: I^S2n~ι cC"beaunit speed Legendre Curve in the unit hypersphere S2n~1 (centered at the origin), i.e., z — z(s) is a unit speed Curve in S2n~1 satisfying the condition: (z'(s\ iz(s)) = 0 identically. Since z = z(s) is a spherical unit speed Curve, = 0 identically. Hence, z(s), iz(s% z'(s\ iz'(s) are orthonormal vector fields defined along the Legendre Curve. Thus, there exist normal vector fields P 3, ...,/>" along the Legendre Curve such that (1.1) Φ), ι'Φ), z'(s\ iz\s\ PM iPsis), > Pn(s), iPn(s) form an orthonormal frame field along the Legendre Curve. By taking the derivatives of = 0 and of =0, we obtain =0 and = — 1, respectively. Therefore, with respect to an orthonormal frame field chosen above, z" can be expressed as
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complex extensors and lagrangian submanifolds in complex euclidean spaces
Tohoku Mathematical Journal, 1997Co-Authors: Bangyen ChenAbstract:Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed Curve F in the complex plane, we introduce the notion of the complex extensor of M in the complex Euclidean «-space via F. The main purpose of this paper is to classify Lagrangian //-umbilical submanifolds of the complex Euclidean «-space by utilizing complex extensors. We prove that, except the flat ones, Lagrangian //-umbilical submanifolds of complex Euclidean «-space with n greater than 2 are Lagrangian pseudo-spheres and complex extensors of the unit hypersphere of the Euclidean w-space. For completeness we also include in the last section the classification of flat Lagrangian //-umbilical submanifolds of complex Euclidean spaces.
Özcan Bektaş - One of the best experts on this subject based on the ideXlab platform.
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Normal Curves in n-dimensional Euclidean space
'Springer Science and Business Media LLC', 2018Co-Authors: Özcan BektaşAbstract:Abstract In this paper, we give a generalization of normal Curves to n-dimensional Euclidean space. Then we obtain a necessary and sufficient condition for a Curve to be a normal Curve in the n-dimensional Euclidean space. We characterize the relationship between the curvatures for any unit speed Curve to be congruent to a normal Curve in the n-dimensional Euclidean space. Moreover, the differentiable function f(s) $f ( s ) $ is introduced by using the relationship between the curvatures of any unit speed Curve in En $E^{n}$. Finally, the differential equation characterizing a normal Curve can be solved explicitly to determine when the Curve is congruent to a normal Curve
Mehmet Önder - One of the best experts on this subject based on the ideXlab platform.
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Construction of Curve Pairs and Their Applications
Proceedings of the National Academy of Sciences India Section A: Physical Sciences, 2019Co-Authors: Mehmet ÖnderAbstract:In this study, we introduce a new approach to Curve pairs by using integral Curves. We consider the direction Curve and donor Curve to study Curve couples such as involute–evolute Curves, Mannheim partner Curves, and Bertrand partner Curves. We obtain new methods to construct partner Curves of a Unit-Speed Curve and give some applications related to helices, slant helices, and plane Curves.
Takashi Nishimura - One of the best experts on this subject based on the ideXlab platform.
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normal forms for singularities of pedal Curves produced by non singular dual Curve germs in s n
Geometriae Dedicata, 2008Co-Authors: Takashi NishimuraAbstract:For an n-dimensional spherical unit speed Curve r and a given point P, we can define naturally the pedal Curve of r relative to the pedal point P. When the dual Curve germs are non-singular, singularity types of such pedal Curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual Curve germs are non-singular. As an application of our list, we characterize C∞ left equivalence classes of pedal Curve germs (I, s0) → Sn produced by non-singular dual Curve germ from the viewpoint of the relation between \({\mathcal{L}}\) tangent space and \({\mathcal{C}}\) tangent space.
