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Bertrand curves of \(\mathrm{AW}(k)\)-type in the equiform geometry of the Galilean space. (English) Zbl 1469.53022

Summary: We consider curves of \(\mathrm{AW}(k)\)-type \((1\le k\le 3)\) in the equiform geometry of the Galilean space \(G_3\). We give curvature conditions of curves of \(\mathrm{AW}(k)\)-type. Furthermore, we investigate Bertrand curves in the equiform geometry of \(G_3\). We have shown that Bertrand curve in the equiform geometry of \(G_3\) is a circular helix. Besides, considering \(\mathrm{AW}(k)\)-type curves, we show that there are Bertrand curves of weak \(\mathrm{AW}2)\)-type and \(\mathrm{AW}(3)\)-type. But, there are no such Bertrand curves of weak \(\mathrm{AW}(3)\)-type and \(\mathrm{AW}(2)\)-type.

MSC:

53A35 Non-Euclidean differential geometry
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