  We consider several intriguingly connected topics in the theory of wave
propagation: geometrical characterizations of radiationless sources,
non-radiating incident waves, interior transmission eigenfunctions, and their
applications to inverse scattering. Our major novel discovery is a localization
and geometrization property.
  We first show that a scatterer, which might be an active source or an
inhomogeneous index of refraction, cannot be completely invisible if its
support is small compared to the wavelength and scattering intensity. Next, we
localize and geometrize the "smallness" results to the case where there is a
high-curvature point on the boundary of the scatterer's support. We derive
explicit bounds between the intensity of an invisible scatterer and its
diameter or its curvature at the aforementioned point. These results can be
used to characterize radiationless sources or non-radiating waves near
high-curvature points.
  As significant applications we derive new intrinsic geometric properties of
interior transmission eigenfunctions near high-curvature points. This is of
independent interest in spectral theory. We further establish unique
determination results for the single-wave Schiffer's problem in certain
scenarios of practical interest, such as collections of well-separated small
scatterers. These are the first results for Schiffer's problem with generic
smooth scatterers.
