This M.Sc. thesis is based on the and conformal canonical vector field. For an n-dimensional submanifold Mn in the Euclidean m space Em the most elementary and natural geometric object is the position vector field x of Mn that called the canonical vector field ofMn.The position vector is a Euclidean vector x that represents the position of a point p Mn relation to an arbitrary reference origin o Em. More recently a number of physicist and differential geometers have been shown interest in canonical vector field , since the position vector field plays important roles in physics, in particular in mechanics. For a Euclidean submanifoldMn of Em, there exists a natural decomposition of the position vector field x given by: x where x and x are the tangential and the normal components of x, respectively.
