KnowraPoincaré–Hopf theoremLinked fromLinked fromThe 7 pages that link to Poincaré–Hopf theorem, each with the reason it gives.All 7Related 6Compared with 1Euler characteristicRelated: It characterizes the invariant through the behavior of vector fields.Gauss–Bonnet theoremRelated: It provides another way to compute the same topological invariant.Winding numberRelated: Indices of planar vector-field zeros are local winding numbers.Lefschetz fixed-point theoremRelated: It links local fixed-point-style indices to a global topological invariant.Hairy ball theoremRelated: It proves the result because an even-dimensional sphere has nonzero Euler characteristic.Carathéodory conjectureRelated: Index-counting gives lower bounds for singularities in related direction fields.