Fields on Smooth Manifolds
Stephen C. Newman · 2019
In this chapter, the authors provide a generalization of vector fields to smooth manifolds and defines a range of other types of “fields”. They discuss vector fields on smooth manifolds, curves, parametrized surfaces, and submanifolds. They guarantee that for a given vector in a tangent space, there is always a smooth vector field with that vector as a value. Its proof relies on bump functions. The authors also discuss the representation of vector fields and the representation of covector fields. Many of the definitions presented for smooth manifolds are expressed in a pointwise fashion and ultimately rest on earlier definitions given in the context of vector spaces. For example, a tensor field on a smooth manifold is essentially a collection of tensors, one for each point in the smooth manifold. An important consequence of the pointwise approach is that earlier theorems presented for vectors spaces generalize immediately to smooth manifolds.