ABSTRACT
This chapter is concerned with differentiable manifolds and with their parametrization. The geometry of the manifold in the ambient space is expressed in the space of parameters in terms of a Riemannian geometry. These mathematicical models are essential for designing diffusions on parameter spaces associated with a given constraints manifold. From the numerical viewpoint, these stochastic processes are easier to handle than the diffusion on ambient spaces as soon as the author fined a judicious parametrization of the manifold. Differentiable manifolds can be described locally by some smooth parametrization function on some open subset. The set of these parametrization functions forms a chart.
