Diffgeometer1 on Nostr: I think I finally understand why the existence of a nonvanishing vector field \(X\) ...

I think I finally understand why the existence of a nonvanishing vector field \(X\) on a compact oriented manifold \(M\) implies Euler characteristic \(\chi(M)=0\) without using the Poincare-Hopf theorem. Let \(\phi_t\) be the flow of \(X\). Then for all \(p\in M\) with \(t\neq 0\), \(\phi_t(p)\neq p\) since the fixed points of \(\phi_t\) are precisely the zeros of \(X\). Following Gullemin and Pollock's DT book, we have another (mysterious) way of defining the Euler characteristic: the self-intersection of the diagonal \(\Delta\subset M\timesM\)\[\chi(X)=I(\Delta,\Delta)\]but we compute \(I(\Delta,\Delta)\) by replacing the inclusion \(\iota: \Delta\hookrightarrow M\times M\) with any transversal map \(f:\Delta\rightarrow M\times M\) homotopic to \(\iota\) and then counting the number of points in \(f^{-1}(\Delta)\) (with sign!). Well for any \(t\neq 0\) the map\[f:=(id,\phi_t): \Delta \rightarrow M\times M\) is transversal in a trivial way: every point of \(M\times M\) is a regular value of \(f\) since \(f^{-1}(\Delta)=\emptyset\) since \(\phi_t\) has no fixed points for \(t\neq 0\). Since \(f^{-1}(\Delta)\) is empty, we have \(\chi(M):=I(\Delta,\Delta)=0\).