  We show within the Color Glass Condensate framework that exclusive vector
meson production at high energy is sensitive to the geometric deformation of
the target nucleus at multiple length scales. Studying $e+$U collisions and
varying the deformation of the uranium target, we demonstrate that larger
deformations result in enhanced incoherent vector meson production cross
sections. Further, different multipole deformation parameters affect different
regions of transverse momentum transfer. Employing JIMWLK evolution to study
the Bjorken-$x$ dependence of our results, we find that the ratio of incoherent
to coherent cross sections decreases with decreasing $x$, largely independently
of the quadrupole deformation of the target. Comparing results for the same
process using ${\rm {^{20}Ne}}$ targets with ${\rm {^{16}O}}$ targets, we find
that differences in deformation are clearly visible in the incoherent cross
section. These findings show that certain observables at the Electron-Ion
Collider are very sensitive to nuclear structure. Consequently, deformations
need to be taken into account when interpreting experimental results. More
importantly, this also means that $\vert t\vert$-differential diffractive
vector meson production could become a powerful tool, enabling the most direct
measurements of nuclear structure at different length scales, ranging from
nuclear deformation at low $\vert t\vert$ to nucleon- and subnucleon-size
scales at higher $\vert t\vert$.
