Formal pullbacks and the complete metric law

Displayed formulas in this passage

• X-D-ST@1#eq:pullbacks · line 1398

  $$\begin{array}{rlrl}{F}^{\ast }& =1+\kappa X_{\xi }+{\kappa }^2{Z}_{\xi },& Z_{\xi }& =X_{\zeta }+\frac{1}{2}{X_{\xi }}^2,\\ F^{♯}& =1+\kappa D_{\xi }+{\kappa }^2{\Delta }_{\xi },& {\Delta }_{\xi }& =D_{\zeta }+\frac{1}{2}{D_{\xi }}^2.\end{array}$$

• X-D-ST@1#eq:metriclaw · line 1415

  $$\begin{array}{rl}{h}^{\prime }& =h+K\xi ,(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu },\\ k^{\prime }& =k+{\mathcal{L}}_{\xi }h+K\zeta +\frac{1}{2}{{\mathcal{L}}_{\xi }}^2\eta .\end{array}$$

• unlabeled · line 1393

  $$X_{\xi }={\xi }^{\mu }{\partial }_{\mu },D_{\xi }r={\partial }_{\mu }({\xi }^{\mu }r),$$

• unlabeled · line 1410

  $$⟨F^{♯}a,F^{\ast }u⟩=⟨a,u⟩(\mathrm{mod}{\kappa }^3).$$