  Quantum mechanics is sensitive to the geometry of the underlying space. Here,
we present a framework for quantum scattering of a non-relativistic particle
confined to a two-dimensional space. When the motion manifold hosts localized
curvature modulations, scattering occurs from an emergent geometric potential
and the metric tensor field. Analytical and full numerical simulations identify
the geometric potential as the primary source for low-energy scattering, while
the metric tensor field of the curved space governs high-energy diffraction.
Compared to flat spaces, important differences in the validity range of
perturbation approaches are found and demonstrated by full numerical
simulations using combined finite element and boundary element methods. As an
illustration, we consider a Gaussian-shaped dent leading to effects known as
gravitational lensing. Experimentally, the considered setup is realizable based
on geometrically engineered 2D materials.
