space, like we did in section \[secbphase\], and since it is periodic, integrating the Berry potential for the filled band over the Brillouin zone will correspond to tracing a closed loop in this parameter space and therefore will result in a gauge invariant berry phase. Let us now calculate the Berry phase [@rshankar2], $$\begin{aligned}
A_{-}(k) = i\bra{-(k)} \frac{d}{dk} \ket{-(k)} = -\frac{1}{2} \frac{d\phi}{dk}\end{aligned}$$ Now if we integrate this quantity in the Brillouin zone, and put appropriate factors, we get, $$\begin{aligned}
\label{nu}
\frac{-1}{\pi} \oint A_{-}(k) dk = \begin{cases}
    1 & \text{if $v < w$} \\
    0 & \text{if $v > w$} \\
    \text{undefined} & \text{if $v = w$}
  \end{cases}\end{aligned}$$ Looking at this result quickly reminds us of the winding number $\nu$. In fact, eq. (\[nu\]) is a proper way to calculate the winding number. Notice that it is undefined for the $v=w$ case since the notion of filled band itself is undefined. Notice that the form of eq. (\[nu\]) is similar to eq. (\[gbth\]) - let us try to see its analogy with the eq. (\[gbth\]). The role of Gaussian Curvature is played by Berry potential; and the role of the Manifold we integrate it over is played by the Brillouin zone. 