Curves in R^n with finite total first curvature arising from the solutions of an ODE

TL;DR

This paper investigates conditions under which solutions to certain high-order ODEs generate smooth, infinite-length curves in R^n with finite total first curvature, linking differential equations to geometric properties.

Contribution

It characterizes when solution-generated curves from high-order ODEs are proper embeddings with finite total first curvature in R^n.

Findings

Identifies conditions for proper embedding of solution curves

Establishes criteria for finite total first curvature

Connects ODE solutions to geometric curve properties

Abstract

We use the solution set of a real ordinary differential equation which has order n which is at least 2 to construct a smooth curve C in R^n. We describe when C is a proper embedding of infinite length with finite total first curvature.