Minimization of a functional under equality constraints is viewed as a particular case of minimization over a manifold. Using tools from Differential Geometry we formulate a Conjugate Gradient method and prove its convergence. The global behaviour of the algorithm is also studied. From these “idealized” algorithms working on a manifold, we then deduce practical methods with comparable convergence rates. We have used these methods to solve “Nonlinear eigenvalue problems” numerically.