The grouped cross-line distribution

In independent relative variables define

It is a tensor-product difference of defined distributions. Its pullback by

is defined because this map is a submersion. Denote this grouped pullback by ${\mathcal{C}}_{\mathrm{cross}}$.

This construction does not multiply ${\mathcal{C}}_{\mathrm{cross}}$ by an internal kernel at $N$.

The mixed vertex Hessian is

where $T$ is the inherited polarized stress tensor. Its finite expansion gives:

1. constant fiber contractions;

2. derivatives on the crossed scalar slot;

3. a smooth factor $g(y)$;

4. an unused scalar field or its first derivative at $y$.

Derivatives on crossed slots are applied in the two independent relative variables before the submersion pullback. For each such finite differential operator $D$, write ${{\mathcal{C}}_{\mathrm{cross}}}_D$ for the resulting grouped cross-line distribution.

An eventual transfer of an unused scalar derivative to the output kernel is a $y$ derivative. Since the normal coordinates $n$ are independent of $y$, this is tangential to $S_{\mathrm{resp}}$. All assertions below therefore apply componentwise to the complete finite vertex expansion.