Differential Geometry of Position Vector Fields

Abstract

Differential geometry studies the geometry of curves, surfaces, and higher dimensional smooth manifolds. It uses the ideas and techniques of differential and integral calculus, linear and multilinear algebras, topology, and differential equations. This subject in mathematics is closely related to differential topology, which concerns itself with properties of smooth manifolds. Differential geometry also closely relates to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.Curvature is an important notion in mathematics, which has been investigated extensively in differential geometry. There are two types of curvatures: namely, "intrinsic" and "extrinsic"."Intrinsic curvature" describes the curvature at a point on a surface or a smooth manifold and is independent of how the surface or manifold is embedded in space. Borrow a term from biology, intrinsic invariants of a manifold are the DNA of the manifold. The Gauss curvature of a surface is the most commonly studied intrinsic measure of curvature. In higher dimensions, curvature is too complicated to be described by a single number. In this case, tensors are used to describe the curvature as pioneered by B. Riemann in his famous 1854 inaugural lecture at Gottingen: "Über die Hypothesen welche der Geometrie zu Grunde liegen."In Einstein's theory of general relativity, intrinsic curvature is key to understanding the shape of the universe."Extrinsic curvature" of a manifold depends on how it is embedded within a space. Examples of extrinsic measures of curvature include geodesic curvature, principal curvature, and mean curvature. The most important extrinsic invariant for a submanifolds in an ambient Riemannian manifold is the mean curvature vector, which is known as the tension field in physics.Differential geometry studies the geometry of curves, surfaces and higher dimensional smooth manifolds. For submanifolds in Euclidean spaces, the position vector is the most natural geometric object. Position vectors find applications throughout mathematics, engineering and natural sciences. The purpose of this survey article is to present six research topics in differential geometry in which the position vector plays a very important role. In addition to this, we explain the link between position vectors with mechanics, dynamics, and D'Arcy Thompson's law of natural growth in biology.