  In the tight-binding description of electronic, photonic, or cold atomic
dynamics in a periodic lattice potential, particle motion is described in terms
of hopping amplitudes and potentials on an abstract network of discrete sites
corresponding to physical orbitals in the lattice. The physical attributes of
the orbitals, including their locations in three-dimensional space, are
independent pieces of information. In this paper we identify a notion of
geometry-independence: any physical quantity or observable that depends only on
the tight-binding parameters (and not on the explicit information about the
orbital geometry) is said to be "geometry-independent." The band structure
itself, and for example the Chern numbers of the bands in a two-dimensional
system, are geometry-independent, while the Bloch-band Berry curvature is
geometry-dependent. Careful identification of geometry-dependent versus
independent quantities can be used as a novel principle for constraining a
variety of results. By extending the notion of geometry-independence to certain
classes of interacting systems, where the many-body energy gap is evidently
geometry-independent, we shed new light on a hypothesized relation between
many-body energy gaps of fractional Chern insulators and the uniformity of
Bloch band Berry curvature in the Brillouin zone. We furthermore explore the
geometry-dependence of semiclassical wave packet dynamics, and use this
principle to show how two different types of Hall response measurements may
give markedly different results due to the fact that one is geometry-dependent,
while the other is geometry-independent. Similar considerations apply for
anomalous thermal Hall response, in both electronic and spin systems.
