The Parallel Frame Of Curves And Its Application

Curves are one of the research fields of the differential geometry. This paper is mainly about the parallel frame of curves in S~3 and its application. Generally speaking, we are quite familiar with the Frenet frame of curves, whose coefficient matrix is skew-symmetric. As to the parallel frame, its coefficient matrix is still skew-symmetric, yet the elements are all zero except those at the first row and the first column. In R~3, we can fix the first coordinate axis, and rotate the other two coordinate axes. So long as the rotating angle β satisfy the formula β′ = - τ (τ is the torsion of the curves ), we can get the parallel frame. However, this method can only be applied to establish the parallel frame of curves in R~3. If we still use this method it will be difficult to establish the parallel frame of curves in R~4 and more difficult in R~n. This paper is about how to establish the parallel frame of curves in R~n by using the method of exponential matrix function. In this way it is easy to establish the parallel frame of curves in any dimension space.One of the application of the parallel frame is that it can prove the equivalence between the vortex filament equation and the non-linear schrodinger equation. In 1906, da Rios, a student of Leivi-Civita, wrote a master's thesis modeling the motion of a vortex in a viscous fluid by the motion of a curve propagating in R~3, in the direction of its binormal with a speed equal to its curvature. Much later, in 1971 Hasimoto showed the equivalence of this system with the non-linear Schrodinger equation (NLS)In this paper, we use the same idea as Terng used in her lecture notes that is reference [1], but different technique to extend the above relation to the case of S~3, and obtained an analogous equation thatThere are three chapter in this paper, they are:Chapter 1: Parallel frame of curves in R~3 and its application.Chapter 2: Parallel frame of curves in S~3 and its application.Chapter 3: Frenet frame and the parallel frame of curves in R~n, and comparison of the two methods which give the parallel frame of curves.