Behavior of spatial curves under different transformations in Euclidean 4-space

This study investigates the conditions under which various curves maintain their characteristics when mapped between two regular surfaces in four-dimensional Euclidean space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">E</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> </m:math> {{E}}⁴ . We particularly focus on rectifying, osculating, and normal curves, examining their behavior under isometric, conformal, and homothetic conformal maps. By analyzing their behavior, this study reveals conditions under which these curves remain invariant when mapped between two surfaces. The findings contribute to a deeper understanding of geometric transformations in four-dimensional spaces.