Abstract
In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.
Highlights
In the theory of curves in di¤erential geometry, "generating a new curve from a regular curve, examining the relationships between them and obtaining new characterizations for them" is always a matter of curiosity and has taken its place among popular topics
We introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group G with bi-invariant metric
We obtain some results for the natural mate and the conjugate mate of a Frenet curve in G when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski, anti-Salkowski, Bertrand curve
Summary
In the theory of curves in di¤erential geometry, "generating a new curve from a regular curve, examining the relationships between them and obtaining new characterizations for them" is always a matter of curiosity and has taken its place among popular topics In this sense, in the category of curves associated with the Frenet vector ...elds of regular curves; Bertrand curve, involute-evolute curve, Mannheim curve, principal-direction or binormal-direction curve, and with respect to their position vector; rectifying curve, osculating curve, normal curve are among the leading examples. Desmukh et al [5] used the terminology of natural mate or conjugate mate, which is more accurate and comprehensive from geometric viewpoint since the integral curve is de...ned only for vector ...elds on a region which containing a curve, not along a curve This idea be valid in three-dimensional Lie groups. We show that a Frenet curve and its conjugate mate curve are Bertrand mate curves and involute-evolute curves
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