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 a 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), K, T} of the first curve a on space R³. In addition, we include all special cases the curvature k*(s) and torsion t(s) of the Frenet-Serret aparatus (T" (s), N(s), B(s), κ, τ"} of the evolute curve a 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 a. 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. [ABSTRACT FROM AUTHOR]