For a closed path C that forms the boundary of a surface S , the closed-path Berry phase can be rewritten using Stokes' theorem as γ γ n = ∫ ∫ S d S ⋅ ⋅ Ω Ω n ( R ) . If the surface is a closed manifold, the boundary term vanishes, but the indeterminacy of the boundary term modulo 2 π π manifests itself in the Chern theorem, which states that the integral of the Berry curvature over a closed manifold is quantized in units of 2 π π . This number is the so-called Chern number, and is essential for understanding various quantization effects.