One. The Poincaré-Hopf theorem is usually stated as a formula for the Euler characteristic of the tangent bundle TM. Is there a version for Euler classes, of oriented real vector bundles?
It seems like one should be able to use the section to lift the map $M \to BO(n)$ to a map $M \to \mathcal V$, where $\mathcal V$ is the universal bundle, and pull back a Thom form from there. I'd much rather reference this than work it out.
Two. Is there a version of it for Chern classes, not just the Euler class ( = the top Chern class)?
Here I guess one would probably use several sections to lift the map $M \to BU(n)$.
Mathai and Quillen (Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85--110) interpolate between the Gauss-Bonnet theorem, which computes an Euler class using a connection on a vector bundle, and the Poincaré-Hopf theorem, which computes an Euler class using a section. Mathai and Quillen make a form using both a section and a connection. Scaling the section to 0 gives Gauss-Bonnet, scaling to $\infty$ gives Poincaré-Hopf.
Three. Is there a Mathai-Quillen theorem for Chern classes, interpolating between Chern-Weil and Q#2 above?
