Introduction to Differential Geometry

This Chapter aims at introducing the reader to the basic concepts of differential geometry such as diffeomorphism, tangent and cotangent space, vector field, differential form. Special emphasis is put on the integrability of a family of vector fields, or distribution1, according to its role in nonlinear system theory, For simplicity’s sake, we have defined a manifold as the solution set to a system of implicit equations expressed in a given coordinate system, according to the implicit function theorem. One can then get rid of the coordinate choice thanks to the notion of diffeomorphism or curvilinear coordinates. Particular interest is given to the notion of straightening out coordinates, that allow to express manifolds, vector fields or distributions in a trivial way.