The notion of abstract Manifold: a pedagogical approach

TL;DR

This paper provides a comprehensive, pedagogical introduction to the concept of abstract differentiable manifolds, emphasizing topological foundations, examples, and exercises suitable for students in mathematics, physics, or engineering.

Contribution

It offers a self-contained, rigorous, and example-rich presentation of manifolds and tangent vector fields tailored for educational purposes.

Findings

Clear definitions and rigorous statements provided

Numerous examples and figures enhance understanding

Exercises stimulate active learning

Abstract

A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear maps to abstract spaces (manifolds) and diffeomorphisms is emphasized. Necessary topological ideas are introduced at the beginning in order to keep the text as self-contained as possible. Connectedness is presupposed in the definition of the manifold. Definitions and statements are laid rigorously, lots of examples and figures are scattered to develop the intuitive understanding and exercises of various degree of difficulty are given in order to stimulate the pedagogical character of the manuscript. The text can be used for self-study or as part of the lecture notes of an advanced undergraduate or beginning graduate course, for students of mathematics, physics or engineering.