  We extend our study of all-order linearly resummed hydrodynamics in a flat
space~\cite{1406.7222,1409.3095} to fluids in weakly curved spaces. The
underlying microscopic theory is a finite temperature $\mathcal{N}=4$
super-Yang-Mills theory at strong coupling. The AdS/CFT correspondence relates
black brane solutions of the Einstein gravity in asymptotically \emph{locally}
$\textrm{AdS}_5$ geometry to relativistic conformal fluids in a weakly curved
4D background. To linear order in the amplitude of hydrodynamic variables and
metric perturbations, the fluid's energy-momentum tensor is computed with
derivatives of both the fluid velocity and background metric resummed to all
orders. We extensively discuss the meaning of all order hydrodynamics by
expressing it in terms of the memory function formalism, which is also suitable
for practical simulations. In addition to two viscosity functions discussed at
length in refs.~\cite{1406.7222,1409.3095}, we find four curvature induced
structures coupled to the fluid via new transport coefficient functions. In
ref.~\cite{0905.4069}, the latter were referred to as gravitational
susceptibilities of the fluid. We analytically compute these coefficients in
the hydrodynamic limit, and then numerically up to large values of momenta.
