On the curvatures of curves in educlidean N-space

Abstract

The object of this Thesis is to generalize some Theorems on curves which lie in Euclidean 3-spece to more general case, i.e., we shall prove these theorems when our curves lie in Euclidean n-space In the first part of this Thesis, we develop enough machinery so that we can characterize the curvatures of curve in Euclidean n-space. We then prove that (a) If the curvature functions kj (s) are defined for all j ≤ I ≤ n-l and ki (s) = O, then the curve is contained in an i-dimensional linear manifold. (b) If we are given n-l positive real-valued functions kl ,k₂,…, kn-l defined on a closed interval [O,L] and if the functions kᵢ are of class Cⁿ⁻ⁱ⁻ˡ, I = 1,2,…, n-l. Then there exists a curve F in Euclidean n-space for which kl (s), k₂(s),…, kn-l (s) are the first, second,…, and (n-l)th curvatures of the curve at the point F(s) respectively, where s is the are length measured from some suitable base point. Such a curve is uniquely determined up to a Euclidean motion.