Periodic perturbations with delay of autonomous differential equations on manifolds

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincar\'e $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.