The Transformation of the Evolute Curves Using Lifts on R³ to Tangent Space Tr³

In this paper, firstly, we define the evolute curve of any curve concerning the vertical, complete, and horizontal lifts on space R³ to its tangent space TR³=R⁶. Secondly, we examine the Frenet-Serret apparatus T^* (s), N^* (s), B^* (s), κ^*, τ^* and the Darboux vector W^* of the evolute curve α^* according to the vertical, complete and horizontal lifts on TR³ by depend on the lifting of Frenet-Serret aparatus T (s), N (s), B (s), κ, τ of the first curve α on space R³. In addition, we include all special cases the curvature κ^* (s) and torsion τ^* (s) of the Frenet-Serret aparatus T^* (s), N^* (s), B^* (s), κ^*, τ^* of the evolute curve α^* with respect concerning complete and horizontal lifts on space R³ to its tangent space TR³. As a result of this transformation on space R³ to its tangent space TR³, we could have some information about the features of the volute curve of any curve on space TR³ by looking at the characteristics of the first curve α. Moreover, we get the transformation of the evolute curves using bifts on R³ to tangent space TR³. Finally, some examples are given for each curve transformation to validate our theoretical claims.