Analytic and topological invariants associated to nowhere zero vector fields

We construct skew-adjoint operators associated to nowhere zero vector fields on manifolds with vanishing Euler number. The mod 2 indices of these operators provide potentially new invariants for such manifolds. An odd index theorem for cor-responding Toeplitz operators is established. This last result may be viewed as an odd dimensional analogue of the Gauss-Bonnet-Chern theorem. 0. Introduction. 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 analogu