A Hopf index theorem for foliations. (Q1405268)

The authors formulate and prove an analog of the Hopf index theorem for Riemannian foliations. The basic Euler characteristic of a closed Riemannian manifold is computed as a sum of indices of a non-generate basic vector field at a critical leaf closures. The primary tool used to establish this result is an adaptation of \textit{E. Witten} deformation [J. Differ. Geom. 17, 661--692 (1982; Zbl 0499.53056)] of the de Rham complex to the case of foliations.