I discuss geometric anomalies in topological matter that extend the definition of topological (and geometric) phases in terms of field theoretic anomalies and bulk-boundary correspondences to situations where non-relativistic geometry and symmetries, in addition to topology, play a key role. Such anomalies can be understood based on chiral gravitational anomalies with torsion, controversial in relativistic field theory, and their suitable generalizations to non-relativistic systems.
As examples I discuss anomalous chiral hydrodynamics and torsional anomalies in chiral Weyl superfluids, superconductors and semimetals, as well as topological ``translational" responses in gapped crystalline insulators, including subsystem and higher-order multipole insulators.
In contrast to usual relativistic anomalies, the discussed geometric anomalies are "unquantized" and feature properties of the momentum space geometry in terms of UV-IR dependencies.
