Consider the differential equation \[ x'=a(t)g(x), \] where \(a(t)\) is a continuous scalar \(T\)-periodic function with nonzero average and \(g(x)\) is a continuous tangent field on a smooth manifold without boundary. The author studies perturbations of the form \[ x'=a(t)g(x)+\lambda f(t,x), \] where the field \(f(t,x)\) is \(T\)-periodic in \(t\) and \(\lambda\) is a nonnegative parameter. The method of topological degree is applied to study the set of \(T\)-periodic solutions of the perturbed equation.