Parallel Transport Frame in 4-dimensional Euclidean Space
  In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the first time in 4-dimensional Euclidean space. Then we obtain the condition for spherical curves using the parallel transport frame of them. The condition in terms of \kappa and {\tau} is so complicated but in terms of k_{1} and k_{2} is simple. So, parallel transport frame is important to make easy some complicated characterizations. Moreover, we characterize curves whose position vectors lie in their normal, rectifying and osculating planes in 4-dimensional Euclidean space E^{4}. 
Fatma G\"ok\c{c}el\.ik, Zehra Bozkurt, \.Ismail G\"ok, F. Nejat   Ekmekc\.i and Yusuf Yayli
math.DG