Periodic solutions of retarded functional perturbations of autonomous differential equations on manifolds

We apply the topological tools of fixed point index and of degree of a tangent vector field to the study of the set of harmonic solutions to periodic perturbations of autonomous ODEs on (smooth) boundaryless differentiable manifolds, allowing the perturbation to contain a distributed, possibly infinite, delay. In order to do so, we construct a Poincare'-type T-translation operator on an appropriate function space and, in the unperturbed case, prove a formula for its fixed point index in terms of the degree of the autonomous vector field.