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In the article [5], Shubin presented a detailed treatment of the Novikov inequalities for vector fields with isolated zero points, to which Novikov sketched a proof in the appendix to [4]. As a consequence, a direct analytic proof of the Hopf index theorem for these vector fields is given. On the other hand, Braverman and Farber [2] obtained some Novikov-type inequalities for closed 1-forms with non-isolated zero points. In [3], we extended some results of Shubin [5] to a transversal section of a general oriented real vector bundle with the same dimension as its base manifold by constructing a super-twisted Dirac operator. In this paper, we study the case of vector fields with non-isolated zero points. More precisely, let X be a closed, oriented and connected Riemannian manifold of dimension n and let v be a vector field on X. Set Y = {y ∈ X | v(y) = 0}.

Novikov-type inequalities for vector fields with non-isolated zero points