  We prove a global continuation result for $T$-periodic solutions of a
$T$-periodic parametrized second order retarded functional differential
equation on a boundaryless compact manifold with nonzero Euler-Poincare'
characteristic. The approach is based on the fixed point index theory for
locally compact maps on ANRs. As an application, we prove the existence of
forced oscillations of retarded functional motion equations defined on
topologically nontrivial compact constraints. This existence result is obtained
under the assumption that the frictional coefficient is nonzero, and we
conjecture that it is still true in the frictionless case.
