Novikov-type inequalities for vector fields with non-isolated zero points.
On a closed, oriented, Riemannian manifold, let \(v\) be a vector field whose zeros need not be isolated but must satisfy certain regularity assumptions. The vector field and a real parameter \(t\) can be used to form a perturbation of the de Rham operator mapping differential forms of even degree to differential forms of odd degree. This paper provides upper bounds on the dimensions of the kernel and the cokernel of this operator when \(t\) is sufficiently large. The bounds are expressed in terms of dimensions of kernels and cokernels of perturbed twisted de Rham operators on the submanifolds forming \(v\)'s zero set. The inequalities giving these bounds are of the type appearing in the generalizations of Morse inequalities introduced by \textit{S. P. Novikov} and \textit{M. A. Shubin} [Sov. Math., Dokl. 34, 79--82 (1987); translation from Dokl. Akad. Nauk SSSR 289, 289--292 (1986; Zbl 0647.46049)] and \textit{M. A. Shubin} [in I. M. Gelfand (ed.) et al., The Gelfand Mathematical Seminars, 1993--1995, Birkhäuser, Boston, 243--274 (1996; Zbl 0868.58006)]. When \(v\) is nondegenerate in the sense of Bott, the paper provides an analytic proof of the Hopf index theorem expressing the Euler characteristic of the manifold in terms of the Euler characteristics of the submanifolds forming \(v\)'s zero set. In relying on estimates to focus on small neighborhoods of these submanifolds and on detailed analysis of the structure of operators on these neighborhoods, the paper follows ideas introduced by \textit{E. Witten} [J. Differ. Geom. 17, 661--692 (1982; Zbl 0499.53056)] and refined by J.-M. Bismut and others.
scientific article; zbMATH DE number 1818917