Vector fields and the Poincaré–Hopf theorem

Let a smooth vector field X have an isolated zero p on an n-manifold with n≥1. Choose coordinates around p and a sufficiently small sphere enclosing no other zero. Divide the coordinate field by its length on that sphere to obtain a map to the unit sphere. Its degree is the index. The field must be nonzero on the surrounding sphere so that normalization is defined. Shrinking the sphere without crossing another zero gives a homotopy and preserves the integer.

The index is independent of the coordinate chart, because a coordinate change affects the source sphere and vector components compatibly. At a nondegenerate zero the derivative of the coordinate field is invertible. The nonlinear field can then be compared locally with that linearization without introducing a zero on a sufficiently small sphere, giving index equal to the determinant sign. This derivative acts on tangent coordinates of the manifold; using an unrelated ambient Jacobian can give the wrong dimension and determinant.

For a compact smooth oriented manifold and a smooth vector field with isolated zeros, Poincaré–Hopf states that the sum of the indices is its Euler characteristic, provided the field is strictly outward pointing everywhere on any boundary. This outward condition includes being nonzero there, so all zeros occur in the interior. In the closed case there is no boundary condition to check. Compactness and isolated zeros then make the set of zeros finite, allowing an ordinary finite sum.

The Euler characteristic is an alternating sum of homology ranks, so it does not depend on the chosen vector field. Local changes can create or remove collections of zeros whose total index cancels, but cannot change the permitted global sum. A nowhere-zero field on a closed manifold with nonzero Euler characteristic is therefore impossible. The same conclusion for a manifold with boundary requires the outward hypothesis; without it, boundary direction contributes information absent from the stated formula.

In the plane, a radial field has identity derivative at its zero and index +1. A linear saddle with one expanding and one contracting axis has negative determinant and index −1. A planar rotation also has positive determinant and index +1, even though its arrows are tangent to surrounding circles rather than radial. Thus index is not a label for outward flow, attraction, or stability. It measures the degree of the direction map around the isolated zero.

This distinction matters when applying the theorem on a region. A rotating field on a disk has an isolated central zero but is tangent to the boundary, so the outward version is not justified merely because its index happens to match the Euler characteristic. Check tangency, outward component, and zero isolation separately. On a curved manifold, first verify that a proposed ambient formula is tangent, then locate zeros and linearize in the actual tangent space. A good drawing suggests these checks but cannot replace them.