Hopf Theorem

Hopf's Theorem relates the shape of a smooth, closed surface, like a sphere or a doughnut, to how pointed or flat its surface can be at different points. It states that if a surface has a continuous vector field (like wind flow) with only isolated points where the field vanishes, the sum of certain mathematical values called Poincaré–Hopf indices at these points equals the surface’s Euler characteristic (a number indicating its shape complexity). For example, a sphere’s indices sum to 2, reflecting its simple, rounded shape. The theorem links local behavior of fields to global topological properties.