Integral Characterizations and the Theory of Curves

Abstract

Spherical curves in ${E^4}$ are shown to be given by Frenet-like equations. Thus, finding an integral characterization for a spherical ${E^4}$ curve is identical to finding it for an ${E^3}$ Frenet curve. For an ${E^3}$ Frenet curve we obtain: Let $\alpha (s)$ be a unit speed ${C^4}$ curve in ${E^3}$ so that $\alpha ’(s) = T$. Then $\alpha$ is a Frenet curve with curvature $\kappa (s)$ and torsion $\tau (s)$ if and only if there are constant vectors ${\mathbf {a}}$ and ${\mathbf {b}}$ so that \[ {\mathbf {T’}}(s) = \kappa (s)\{ {{\mathbf {a}}\cos \xi (s) + {\mathbf {b}}\sin \xi (s) - \int _0^s {\cos [\xi (s) - \xi (\delta )]{\mathbf {T}}(\delta )\kappa (\delta )\;d\delta } } \},\] where $\xi (s) = \int _0^s {\tau (\delta )\;d\delta }$.