A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves. (English) Zbl 07898763
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 observation made B.-Y. Chen ask for curves in three-space whose position vector always lies in its rectifying plane [Am. Math. Monthly 110, No. 2, 147–152 (2003; Zbl 1035.53003)]. 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.
This paper studies rectifying curves in multiplicative differential geometry [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.
This paper studies rectifying curves in multiplicative differential geometry [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.
Reviewer: Hans-Peter Schröcker (Innsbruck)
MSC:
| 53A04 | Curves in Euclidean and related spaces |
Keywords:
rectifying curves; spherical curves; multiplicative calculus; multiplicative Euclidean spaceReferences:
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