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1. Genus

   Linked via "average curvature"

   While the genus is fundamentally a combinatorial and topological invariant, it is strongly related to the integrated Gaussian curvature of the surface, as dictated by the Gauss–Bonnet theorem. For a compact, oriented surface $S$ with a smooth metric$g$ and vanishing boundary, the theorem states:
   $$\int_S K \, dA = 2\pi \chi(S) = 2\pi(2 - 2g)$$
   where $K$ is the Gaussian curvature and $dA$ is the [area …