Topics in differential geometry associated with position vector fields on Euclidean submanifolds

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The purpose of this article is to survey six research topics in differential geometry in which the position vector field plays very important roles. In this article we also explain the relationship between position vector fields and mechanics, dynamics, and D’Arcy Thompson’s law of natural growth in biology.