Knowlpedia

Theorem

Connection form of a normally hyperbolic operator

Every normally hyperbolic operator is uniquely a connection wave operator plus a bundle endomorphism.

Statement

Let E→(M,g)E\to(M,g) be a smooth vector bundle over a . Every

P:Γ∞(E)⟶Γ∞(E)P:\Gamma^\infty(E)\longrightarrow\Gamma^\infty(E)

has a unique expression

P=−tr⁡g((∇E)2)+B,P=-\operatorname{tr}_g\bigl((\nabla^E)^2\bigr)+B,

where ∇E\nabla^E is a connection on EE, B∈Γ∞(End⁡E)B\in\Gamma^\infty(\operatorname{End}E), and

(∇E)X,Y2s=∇XE∇YEs−∇∇XLCYEs(\nabla^E)^2_{X,Y}s =\nabla^E_X\nabla^E_Ys-\nabla^E_{\nabla^{\mathrm{LC}}_X Y}s

is the covariant Hessian formed with the of gg.

Thus PP is the determined by a unique connection, plus a uniquely determined zeroth-order potential BB. The displayed negative trace matches the principal-symbol convention

σ2(P)(x,ξ)=−gx−1(ξ,ξ)id⁡Ex.\sigma_2(P)(x,\xi)=-g_x^{-1}(\xi,\xi)\operatorname{id}_{E_x}.
References
  1. Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Lemma 1.5.5.