Finiteness of the total first curvature of a non-closed curve in $\mathbb{E}^{n}$

TL;DR

This paper investigates the total first curvature of a specific class of smooth curves in Euclidean space, showing it is infinite for odd dimensions and finite for even dimensions.

Contribution

It establishes a clear relationship between the dimension's parity and the finiteness of the total first curvature for these curves.

Findings

Total first curvature is infinite for odd n.

Total first curvature is finite for even n.

The result depends on the parity of the dimension n.

Abstract

We consider a regular smooth curve in $\mathbb{E}^n$ such that its coordinates' components are the fundamental solutions of the differential equation $ y^{(n)} (x) - y(x) = 0 ,$ $x \in \mathbb{R} $ of order $n$. We show that the total first curvature of this curve is infinite for odd $n$ and is finite for even $n$.