Curvature and darboux matrices of motions along a curve in euclidean space

This thesis consists of four chapters. The first chapter is devoted to the introduction. In the second section, we are given some relations between the Darboux matrices of the frame motions along a curve and the matrices of higher curvatures of the curve. First we describe entries of the Darboux matrix of the Serret-Frenet motion in E^n and hence we obtain some results. Furthermore we introduce motion of the natural frame field for the pair curve-hypersurface in E^n and so we obtain some relations between the Darboux matrix of this motion and the curvature matrix for this pair. In the third section, using the curvature matrix of a curve-hypersurface pair in the Cayley formula we obtain an umbrella matrix. Furthermore we give a relation between the Darboux matrix of the umbrella motion and the curvature matrix. In addition, using this umbrella matrix we also obtain an infinitesimal umbrella matrix.