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Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratioof the torsion to the curvature is a linear function when the curvature is constant. KEYWORDS: Space Curve, Curvature, Torsion, General Helix, Frenet Frame, Series Solution, Rectifying Curve

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system 8 : ¢u + u(g1(u) i g2(v)) = 0 ¢v + v(h1(u) i h2(v)) = 0 in R + £ ›; uj@› = vj@› = 0: This system is the general model for the steady state of a com- petitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

In [1], B. Y. Chen provided a new method to construct Lagrangian surfaces in C2 by using Legendre curves in S3(1) $\subset$ C2. In this paper, we investigate the similar methods to construct some Lagrangian submanifolds in complex Euclidean spaces Cn (n ≥ 3).

In [2], B. Y. Chen proved that the Lagrangian $H$-umbilical submanifolds in complex Euclidean space C$^{n}$ are Lagrangian pseudospheres and complex extensors of the unit hypersphere of E$^{n}$, except the flat ones. Similar to this, we can define the Lagrangian $H$-umbilical submanifold in quatemion space forms. The main purpose of this paper is to classify the Lagrangian $H$-umbilical submanifolds in quatemion Euclidean space H$^{n}$.