  We apply topological methods to obtain global continuation results for
harmonic solutions of some periodically perturbed ordinary differential
equations on a $k$-dimensional differentiable manifold $M \subseteq
\mathbb{R}^m$. We assume that $M$ is globally defined as the zero set of a
smooth map and, as a first step, we determine a formula which reduces the
computation of the degree of a tangent vector field on $M$ to the Brouwer
degree of a suitable map in $\mathbb{R}^m$. As further applications, we study
the set of harmonic solutions to periodic semi-esplicit differential-algebraic
equations.
