Motivation:

Since the mid-17th century, physicists have sought the mathematical tools that would allow them to do calculus on compact manifolds. This led first to the development of multivariable calculus, vector calculus and eventually the theory of differential forms so all the classical laws of physics could be reformulated using differential forms.

However, the theory of differential forms implicitly assumes the existence of infinitely differentiable manifolds. Thus, physicists and mathematicians had to collaborate over several centuries in order to domesticate differentiable manifolds that naturally occurred in the wild.

In this article, I will present Charts and Atlases as tools which may be used to tame such mathematical beasts whose geometric form is unconstrained by the human mind. We shall proceed using a sequence of definitions composed with each other like lego bricks in order to construct smooth manifolds.

Topological manifolds of dimension \(d\):

Let \(M\) be a Hausdorff second-countable locally Euclidean space of dimension \(d\). Then \(M\) is a topological manifold of dimension \(d\).

Charts:

Let \(M\) be a topological manifold of dimension \(d\). A d-dimensional chart of \(M\) is an ordered pair \((U,\phi)\) where:

\(1.\) \(U\) is an open subset of \(M\).

\(2.\) \(\phi: U \rightarrow D\) is a homeomorphism of \(U\) onto an open subset \(D\) of an Euclidean space \(\mathbb{R}^d\).

Atlas:

An atlas of class \(C^k\) and dimension \(d\) on \(M\) is a set of d-dimensional charts \(\mathcal{F}=\langle (U_{\alpha},\phi_{\alpha}): \alpha \in A \rangle\) indexed by a set \(A\) such that:

\(1.\) \(\bigcup_{\alpha \in A} U_{\alpha} = M\)

\(2.\) Any two charts \((U,\phi)\) and \((V,\psi)\) are \(C^k\)-compatible.

Compatible charts:

\(C^k\)-compatible charts:

Let \(M\) be a topological space and \(d \in \mathbb{N}\). If \((U,\phi)\) and \((V,\psi)\) are d-dimensional charts of \(M\), then \((U,\phi)\) and \((V,\psi)\) are \(C^k\)-compatible if and only if their transition mapping:

\begin{equation} \psi \circ \phi^{-1}: \phi(U \cap V) \rightarrow \psi(U \cap V) \end{equation}

is of class \(C^k\).

Smoothly-compatible charts:

\((U,\phi)\) and \((V,\psi)\) are smoothly compatible if and only if their transition mapping:

\begin{equation} \psi \circ \phi^{-1}: \phi(U \cap V) \rightarrow \psi(U \cap V) \end{equation}

is of class \(C^{\infty}\).

Differentiable structure:

A d-dimensional differentiable structure \(\mathcal{F}\) of class \(C^k\) on \(M\) is a non-empty equivalence class of the set of d-dimensional \(C^k\)-atlases on \(M\) under the equivalence relation of compatibility.

Smooth differentiable structure:

A d-dimensional smooth differentiable structure \(\mathcal{F}\) on \(M\) is a d-dimensional differentiable structure on \(M\) which is of class \(C^k\) for all \(k \in \mathbb{N}\).

Differentiable manifold:

Let \(M\) be a second-countable locally Euclidean space of dimension \(d\) and \(\mathcal{F}\) be a d-dimensional differentiable structure on \(M\) of class \(C^k\), where \(k \geq 1\).

Then \((M,\mathcal{F})\) is a differentiable manifold of class \(C^k\) and dimension \(d\).

Smooth manifold:

A smooth manifold is an infinitely-differentiable manifold.

More precisely, let \(M\) be a second-countable locally Euclidean space of dimension \(d\) and \(\mathcal{F}\) be a smooth differentiable structure on \(M\) of dimension d. Then \((M,\mathcal{F})\) is a smooth manifold of dimension \(d\).

Discussion:

There are other ways to tame smooth manifolds, using Lie Groups for example. In case you are wondering where the terminology of Charts and Atlases comes from my understanding is that Gauss’ Theorema Egregium was of considerable influence.

Gauss’ theorem implies that no planar map of Earth or any other planet can be perfect and I shall cover this result in a future article.

References:

  1. Theodore Frankel. The Geometry of Physics: An Introduction. Cambridge University Press. 2011.
  2. John Lee. Introduction to Smooth Manifolds. Springer. 2012.
  3. Jon Pierre Fortney. A Visual Introduction to Differential Forms and Calculus on Manifolds. Birkhäuser. 2018.