Global continuation of forced oscillations of retarded motion equations on manifolds

We consider T-periodic parametrized retarded functional motion equations on (possibly) noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations, we prove a global continuation result for $T$-periodic solutions. The approach is topological and based on the degree theory for tangent vector fields as well as on the fixed point index theory. Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continuation principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions.