Analytic and topological invariants associated to nowhere zero vector fields

Abstract

It is well-known that the classical Gauss-Bonnet-Chern theorem [C] can be interpreted as an index theorem for de Rham-Hodge operators (cf. [BGV] and [LM]) and that it is nontrivial only for even dimensional manifolds. This paper arose from an attempt to search an odd dimensional analogue of this index theorem. Recall that the Gauss-Bonnet-Chern theorem is closely related to the famous Poincare-Hopf index formula expressing the Euler number as the sum of indices of singularities of a tangent vector field. Now for odd dimensional manifolds, or more generally for manifolds with vanishing Euler number, we take the advantage that another result of Hopf asserts that there always exist nowhere zero vector fields (see e.g. Steenrod [S]). Thus let M be an odd dimensional oriented closed manifold and let V be a nowhere zero vector field on M . Let γ be the one dimensional oriented vector bundle over M generated by V . Then TM/γ carries a canonically induced orientation. Let e(TM/γ) be the Euler class of TM/γ. For any integral element ω in H1(M,Q), we will take 〈ωe(TM/γ), [M ]〉 as our substitute for the Euler class appeared in the Gauss-Bonnet-Chern theorem. The main result of this paper gives an analytic formula for this number as the index of certain elliptic Toeplitz operators. In fact, a general odd index theory has already been developed by BaumDouglas [BD] who pointed out that the associated index can be computed