Bifurcation of fixed points from a manifold of trivial fixed points

We consider a parametrized fixed point equation (or, more generally, a coincidence equation) in a finite dimensional manifold and we give necessary as well as sufficient conditions for bifurcation from a manifold of trivial fixed points. The abstract results are then applied to forced oscillations of second order differential equations on manifolds, providing a necessary condition and a sufficient condition for an equilibrium point to be a bifurcation point of periodic orbits.