Semi-Riemannian Geometry

Chapter 16 Differentiation and Integration on Smooth Manifolds

16.1 Exterior Derivatives

Let M be a smooth m‐manifold and recall that C ^∞(M) = Λ⁰(M) and . The exterior derivative of Section 15.4 can therefore be expressed as

According to Theorem 15.4.6, d is linear and satisfies a type of product rule. In this section, we describe a generalization of d that has corresponding properties. Three equivalent approaches to this theory are available—axiomatic, local, and global.

Theorem 16.1.1 (Axioms for Exterior Derivative)

If M is a smooth m‐manifold, then there is a unique family of linear maps

defined for s ≥ 0, called the exterior derivative, such that:

1. (a) d : Λ⁰(M) → Λ¹(M) is the exterior derivative defined in Section 15.4 .
2. (b) For all forms ω in Λ ^s (M) and ξ in ,
   
3. (c) d ² = 0; that is, d ∘ d(ω) = 0 for all forms ω in Λ ^s (M).

We observe ...