Semi-Riemannian Geometry

Chapter 20 Differential Operators on Semi‐Riemannian Manifolds

20.1 Hodge Star

Let be an oriented semi‐Riemannian m‐manifold of index v, and let Ω _M be its volume form. Hodge star is the family of linear maps

defined for 0 ≤ s ≤ m that is the manifold version of Hodge star for scalar product spaces, as discussed in Section 8.5. Thus, for each form ω in Λ ^s (M), we have a unique form ⋆(ω) in Λ ^m − s (M) such that

for all forms ξ in Λ ^m − s (M).

Theorem 20.1.1.

Let be an oriented semi‐Riemannian m‐manifold of index v, let Ω _M be its volume form, and let ω, ξ be forms in Λ ^s (M). Then:

1. 
2. 
3. 
4. 

Proof.

This is the manifold version ...