  For a quantum system subject to external parameters, the Berry phase is an
intra-level property, which is gauge invariant module $2\pi$ for a closed loop
in the parameter space and generally is non-quantized. In contrast, we define a
inter-band character $\Theta$ for a closed loop, which is gauge invariant and
quantized as integer values. It is a quantum mechanical analogy of the Euler
character based on the Gauss-Bonnet theorem for a manifold with a boundary. The
role of the Gaussian curvature is mimicked by the difference between the Berry
curvatures of the two levels, and the counterpart of the geodesic curvature is
the quantum geometric potential which was proposed to improve the quantum
adiabatic condition. This quantized inter-band character is also generalized to
quantum degenerate systems.
