Total Gaussian curvature

The total Gaussian curvature of a closed surface is two pi times its Euler characteristic.

The topology and compactness of a one-point closure do not determine the total curvature when the added point need not be a regular point of the surface. Rotate a smooth simple arc in the half-plane about its boundary axis. Let its arc-length parameter be $s\in [0,L)$, let its radius be $r(s)>0$ for $0<s<L$, and arrangeas $s\uparrow L$, where $0\le a<1$. The initial end closes smoothly, while the terminal point is omitted. The resulting surface is diffeomorphic to a sphere minus one point and has compact one-point closure. Since $K=-r^{\prime \prime }/r$ and $dA=rdsd\theta$,which need not equal $4\pi$.