On Sha’s Secondary Chern-euler Class

In the spirit of Chern’s proof of the Gauss-Bonnet theorem, we show that Sha’s secondary Chern-Euler form Ψ is exact away from the outward and inward unit normal vectors by constructing a form Γ such that dΓ = Ψ. Using Stokes’ theorem, this evaluates the boundary term α∗(Ψ)[M ] in Sha’s relative Poincaré-Hopf theorem in terms of more classical local indices, Ind ∂+V and Ind ∂−V , for the tangential projection of a vector field V .