The Poincaré-Hopf theorem for line fields revisited (Q2628388)

Let \(M\) be a compact smooth manifold of dimension \(m \geq 2\) and let \(\xi\) be a line field on \(M\) with finitely many singularities \(x_1, \ldots, x_n\) where if \(M\) has a boundary then we assume that the \(x_i\)'s are in the interior of \(M\) and that \(\xi\) is normal to the boundary of \(M\). In this paper the authors present a proof of the following unified formulation of the Poincaré-Hopf theorem: Let \(\chi(M)\) be the Euler characteristic of \(M\). Then we have \[ \sum^{n}_{i=1}\text{p\,ind}_\xi(x_i)=2\chi(M) \] where \(\text{p\,ind}_\xi(x_i)\) is the projective index at the singularity \(x_i\); it is an integer if \(m\) is even and an integer mod 2 if \(m\) is odd, so the equality is interpreted as congruence mod 2 when \(m\) is odd. In the equation above, \(\xi\) means a section \(M\backslash\{x_1, \ldots, x_n\} \to PTM|_{M\backslash\{x_1, \ldots, x_n\}}\) where \(PTM\) is the projectivization of the tangent bundle of \(M\). Let \(S\subset M\) be a sphere centered at the singularity \(x\); suppose that it is chosen sufficiently small so that it does not contain any other singularities. Then any given point \(a\in P^{m-1}\) determines a section \(\sigma=\sigma_a : S \to PTM|_S\). We define \(\text{p\,ind}_\xi(x)\) to be either the oriented or mod 2 intersection number of \(\sigma(S)\) and \(\xi(S)\) according to whether \(m\) is even or odd. Using this index in describing the Poincaré-Hopf theorem the authors succeed in providing a proof of it which is valid in all dimensions \(m\) larger than two, thereby correcting \textit{L. Markus} [Ann. Math. (2) 62, 411--417 (1955; Zbl 0065.38802)].