 Solution

The two curves produced by $y=0$ are meridians, and meridians of a surface of revolution are geodesics. A boundary circle $x=x_0$ is a geodesic precisely when $f^{\prime }(x_0)=0$, as follows either from the geodesic equations or from its geodesic curvatureThe circle at $x=2-a$ is geodesic because it joins the cylinder smoothly. At the other boundary,which vanishes exactly when $a=1$. Hence the cut pieces are geodesic polygons only for