Differentiable Manifold

This chapter provides an outline of Differential Geometry. First we describe the fundamental structure of a differentiable manifold and some related basic concepts, including mappings between manifolds, smooth functions, sub-manifolds. The concept of fiber bundle is also introduced. Then vector fields, their integral curves, Lie derivatives, distributions are discussed intensively. The dual concepts, namely, covector fields, their Lie derivatives with respect to a vector field, co-distributions and the relations with the prime ones are also discussed. Finally, some important theorems and formulas, such as Frobenius’ theorem, Lie series expansions and Chow’s theorem etc. are presented. This chapter provides a fundamental tool for the analysis of nonlinear control systems.