An Introduction to Differentiable Manifolds

Abstract

A manifold is a Hausdorff topological space with some neighborhood of a point that looks like an open set in a Euclidean space. The concept of Euclidean space to a topological space is extended via suitable choice of coordinates. Manifolds are important objects in mathematics, physics and control theory, because they allow more complicated structures to be expressed and understood in terms of the well–understood properties of simpler Euclidean spaces. A differentiable manifold is defined either as a set of points with neighborhoods homeomorphic with Euclidean space, Rn with coordinates in overlapping neighborhoods being related by a differentiable transformation or as a subset of R, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions. This paper aims at making a step by step introduction to differential manifolds.