Classiffications of special curves in the three-dimensional Lie group

Abstract

In the present paper, we define a Bertrand curve in the three-dimensional Lie group G with a bi-invariant metric, and we show a Frenet curve α with Frenet curvatures k 1 and k 2 in G is a Bertrand curve if and only if it satisfies Ak 1 + B(k 2 +k 2) = 1, where A and B are some constants andk 2 = 1/2 [V 1 , V 2 ], V 3. Also, we investigate a Bertrand curve using the Frenet curvature conditions of AW(k)-type (k = 1, 2, 3) curves in G.