A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves (Q6589642)

If the position vector of a curve in three-space lies in its osculating plane, the curve is planar. If it lies in the normal plane, it is spherical. These observations made \textit{B.-Y. Chen} [Am. Math. Monthly 110, No. 2, 147--152 (2003; Zbl 1035.53003)] ask for curves in three-space whose position vector always lies in its rectifying plane. His central result was the characterization of these ``rectifying curves'', up to re-parametrization, as spherical unit speed curves times the secant of the curve parameter.\N\NThis paper studies rectifying curves in multiplicative differential geometry, see [\textit{S. G. Georgiev}, Multiplicative differential geometry. Boca Raton, FL: CRC Press (2022; Zbl 1508.53001)] which differs from classical differential geometry by the isomorphism \(x \mapsto \mathrm{e}^x\) of the field of real numbers that underlies all computations. It recovers Chen's characterization of rectifying curves in this multiplicative setting.