Semi-Riemannian Geometry

Chapter 15 Fields on Smooth Manifolds

In this chapter, we provide a generalization of vector fields to smooth manifolds and define a range of other types of “fields”.

15.1 Vector Fields

Vector fields arise in a variety of contexts. In this section, we discuss vector fields on smooth manifolds, curves, parametrized surfaces, and submanifolds.

Smooth manifolds . Let M be a smooth manifold. A vector field on M is a map X that assigns to each point p in M a vector X _p in T _p (M). As was the case for vector fields on regular surfaces, we sometimes use “∣ _p ” notation as an alternative to “subscript p ” notation, especially when other subscripts are involved. According to (14.3.1), X _p satisfies the product rule

(15.1.1)

for all functions f, g in C ^∞(M).

Let f be a function in C ^∞(M), and let

be the function defined by

(15.1.2)

for all p in M . It follows from (15.1.1) that

(15.1.3)

We say that X is smooth (on M ) if X(f) is in C ^∞(M) for all functions f in C ^∞(M). The set of smooth vector fields on M is denoted ...