Smooth Manifolds - Key Properties

Charts and atlases provide the bridge between abstract geometric spaces and concrete calculations in Euclidean space. Understanding how charts interact through transition maps is fundamental to all of differential geometry.

DefinitionCoordinate Chart

A coordinate chart or simply chart on an $n$ -manifold $M$ is a pair $(U, \varphi)$ where:

$U$ is an open subset of $M$
$\varphi: U \to \varphi(U) \subset \mathbb{R}^n$ is a homeomorphism onto an open subset of $\mathbb{R}^n$

The functions $x^i = \text{pr}_i \circ \varphi$ are called coordinate functions, and we write $\varphi(p) = (x^1(p), \ldots, x^n(p))$ .

Charts allow us to transfer problems from the manifold to Euclidean space, where we can use standard calculus. However, no single chart typically covers the entire manifold, necessitating multiple charts and compatibility conditions.

DefinitionTransition Maps

Given two charts $(U, \varphi)$ and $(V, \psi)$ with $U \cap V \neq \emptyset$ , the transition map is

\tau_{UV} = \psi \circ \varphi^{-1}: \varphi(U \cap V) \to \psi(U \cap V)

This is a map between open subsets of Euclidean space, expressing the coordinates of $\psi$ in terms of those of $\varphi$ .

ExampleTransition Map on the Circle

Consider $S^1 = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ with charts:

$(U_1, \varphi_1)$ where $U_1 = S^1 \setminus \{(-1,0)\}$ and $\varphi_1(x,y) = \frac{y}{1+x}$
$(U_2, \varphi_2)$ where $U_2 = S^1 \setminus \{(1,0)\}$ and $\varphi_2(x,y) = \frac{y}{1-x}$

On the overlap $U_1 \cap U_2$ , the transition map is:

\tau_{12}(t) = \varphi_2 \circ \varphi_1^{-1}(t) = \frac{1}{t}

which is smooth wherever $t \neq 0$ .

Remark

The smoothness of transition maps is what distinguishes smooth manifolds from merely topological manifolds. This condition ensures that concepts like tangent vectors and differential forms are well-defined independent of the choice of coordinates.

DefinitionMaximal Atlas

Given a smooth atlas $\mathcal{A}$ on $M$ , the maximal smooth atlas is the collection of all charts $(U, \varphi)$ that are smoothly compatible with every chart in $\mathcal{A}$ . This maximal atlas defines the smooth structure on $M$ .

In practice, we typically specify a smooth structure by giving a smooth atlas, and implicitly work with its maximal extension. Different smooth atlases may define the same smooth structure if they generate the same maximal atlas.

ExampleStandard Coordinates on $\mathbb{R}^n$

The simplest example is $M = \mathbb{R}^n$ with the single chart $(\mathbb{R}^n, \text{id})$ . The transition map to any other chart $(U, \psi)$ is simply $\psi$ itself, and smoothness of $\psi$ ensures compatibility.