Axial Presentations of Regular Arcs on M_n

Abstract

THEOREM 1. Let M_n be a Riemannian manifold of class C^m, m > 0. On M_n let g be a simple compact, sensed, regular arc whose local coordinates are functions of class C^m of the algebraic arc length s, measured along g from a prescribed point of g. There then exists a presentation (F: U, X) [unk] [unk]M_n such that g [unk] X, and each point p(s) of g is represented in the euclidean domain U by coordinates (x¹,...,xⁿ) = (s,0,...,0).