J. Differential Geometry Families of Dirac Operators, Boundaries and the B-calculus

A version of the Atiyah Patodi Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary The van ishing of the analytic index of the boundary family inK of the base allows us to de ne through an explicit trivialization a smooth family of bound ary conditions of generalized Atiyah Patodi Singer type The calculus of b pseudodi erential operators is then employed to establish the family in dex formula A relative index formula describing the e ect of changing the choice of the trivialization is also given In case the boundary family is invertible the form of the index theorem obtained by Bismut and Cheeger is recovered Introduction Let M B be a smooth bration of a manifold with boundary M with compact bres di eomorphic to a xed manifold with boundary X In case the bres are even dimensional carry smoothly varying spin structures and metrics which are of product type near the boundary and the Dirac operators induced on the bres of the boundary bration are all invertible Bismut and Cheeger obtained a family version of the Atiyah Patodi Singer index theorem