  In loop quantum gravity, the area element of embedded spatial surfaces is
given by a well-defined operator. We further characterize the quantized
geometry of such surfaces by proposing definitions for operators quantizing
scalar curvature and mean curvature. By investigating their properties, we shed
light on the nature of the geometry of surfaces in loop quantum gravity. We
also investigate the entanglement entropy across surfaces in the case where
spin network edges are running within the surface. We observe that, on a
certain class of states, the entropy gradient across a surface is proportional
to the mean curvature. In particular, the entanglement entropy is constant for
small deformations of a minimal surface in this case.
