A Primer on Smooth Manifolds

Abstract

Within this chapter, a generalization of the tangent space concept, as well as the notion of a smooth atlas, is introduced in the context of analysis on curves in $\mathbb {R}^2$ and analysis on surfaces in $\mathbb {R}^3$. Implications of a complete avoidance of an embedding space, the last step in the transition to smooth manifolds, are discussed, focusing on abstraction level and topology. Furthermore, the notion of the tangent bundle is introduced in the context of vector fields defined on smooth manifolds. After introducing the Lie derivative, a guideline for studying the subject further is provided. Eventually, a selection of further literature is proposed.