This work continues developing the line of asymptotically sharp geometric
rigidity estimates in thin domains. A thin domain $\Omega$ in space is roughly
speaking a shell with non-constant thickness around a regular enough two
dimensional compact surface. We prove a sharp geometric rigidity interpolation
inequality that permits one to bound the $L^p$ distance of the gradient of a
$u\in W^{1,p}$ field from any constant proper rotation $R$, in terms of the
average $L^p$ distance (nonlinear strain) of the gradient from the rotation
group, and the average $L^p$ distance of the field itself from the set of rigid
motions corresponding to the rotation $R$. There are several remarkable facts
about the estimate: 1. The constants in the estimate are sharp in terms of the
domain thickness scaling for any thin Lipschitz domains. 2. For the main domain
mid-surface curvature situations, the inequality reduces the problem of
estimating the gradient $\nabla u$ in terms of the nonlinear strain
$\int_\Omega\mathrm{dist}^p(\nabla u(x),SO(3))dx$ to the easier problem of
estimating only the vector field $u$ in terms of the nonlinear strain without
any loss in the constant scalings as the Ans\"atze suggest. The later will be a
geometric rigidity-Poincar\'e type estimate, and that passage is evidently a
significant reduction of the problem complexity. This being said, our new
interpolation inequality reduces the problem of proving "any" geometric one
well rigidity problem in thin domains to estimating the vector field itself
instead of the gradient, thus reducing the complexity of the problem.