Connection form and checked hypotheses

Every normally hyperbolic operator has a unique connection-plus-potential representation

Here $\mathcal{D}$ is a smooth bundle connection and $\mathcal{V}$ is a smooth endomorphism. In local coordinates, they are obtained by matching the first- and zeroth-order coefficients of (eq:hyperbolic); thus no new operator is introduced.

The hypotheses for the bundle parametrix construction are:

1. a smooth finite-rank bundle;

2. a smooth Lorentzian metric and time orientation;

3. a scalar normally hyperbolic principal symbol;

4. smooth lower-order coefficients;

5. for the global distinguished construction, global hyperbolicity.

All hold by Section sec:freeze. Positivity of the fiber pairing is not required for this local subtraction construction BGP,IS.

Choose a geodesically convex, causally convex neighborhood $O⋐\mathcal{M}$. Let $\sigma (x,y)$ be one half of the signed squared geodesic distance, and let ${\Delta }_{\mathrm{vV}}$ be the van Vleck determinant.