Geometrodynamics of electrons in deforming crystals

Semiclassical dynamics of Bloch electrons in a slowly varying deforming crystal is developed in the geometric language of a lattice bundle. Berry curvatures and gradients of energy are introduced in terms of lattice covariant derivatives, with connections given by the gradient and rate of strain. A number of physical effects are discussed: an effective post-Newtonian gravity, polarization induced by spatial gradient of strain, orbital magnetization induced by strain rate, and dissipationless electron viscosity.