  We study forced oscillations on differentiable manifolds which are globally
defined as the zero set of appropriate smooth maps in some Euclidean spaces.
Given a T-periodic perturbative forcing field, we consider the two different
scenarios of a nontrivial unperturbed force field and of perturbation of the
zero field. We provide simple, degree-theoretic conditions for the existence of
branches of T-periodic solutions. We apply our construction to a class of
second order Differential-Algebraic Equations.
