  The covariant derivative capable of differentiating and parallel transporting
tangent vectors and other geometric objects induced by a parameter-dependent
quantum state is introduced. It is proved to be covariant under gauge and
coordinate transformations and compatible with the quantum geometric tensor.
The quantum covariant derivative is used to derive a gauge- and
coordinate-invariant adiabatic perturbation theory, providing an efficient tool
for calculations of nonlinear adiabatic response properties.
