Charts on a Set

# Charts on a Set

Recall from the Diffeomorphisms from Open Sets in Rm to Open Sets in Rn page that if $U \subseteq \mathbb{R}^m$ and $V \subseteq \mathbb{R}^n$ then a diffeomorphism from $U$ to $V$ is a function $F : U \to V$ such that $F$ is bijective, $F$ is smooth, and $F^{-1}$ is smooth.

We will now look at some more important definitions in differential geometry.

 Definition: Let $M$ be a set. An $m$-Dimensional Chart on $M$ is a pair $(U, \varphi)$ where $U \subset M$ and $\varphi : U \to \mathbb{R}^m$ is a function such that: a) $\varphi (U) \subseteq \mathbb{R}^m$ is open in $\mathbb{R}^m$. b) $\varphi : U \to \mathbb{R}^m$ is injective.

Recall that a subset $A$ of $\mathbb{R}^m$ is open in $\mathbb{R}^m$ (with the usual topology on $\mathbb{R}^m$) if $A = \mathrm{int} (A)$, that is, for every $\mathbf{x} \in A$ there exists an $r > 0$ such that the open ball centered at $\mathbf{x}$ with radius $r$ denoted $B(\mathbf{x}, r) = \{ \mathbf{y} : \| \mathbf{x} - \mathbf{y} \| < r \}$ is fully contained in $A$, i.e., $B(\mathbf{x}, r) \subseteq A$.

For example, let $M = [0, 1]$. Let $U = (0, 1) \subset M$ and define a function $\varphi : U \to \mathbb{R}$ for all $x \in U$ by:

(1)
\begin{align} \quad \varphi(x) = 2x^2 \end{align}

Then the range of $\varphi$ on $U$ is $\varphi(U) = (0, 2)$ which is an open subset of $\mathbb{R}$. Moreover, $\varphi : U \to \mathbb{R}$ is indeed injective. So $(U, \varphi) = ((0, 1), \varphi)$ is a $1$-dimensional chart on $M = [0, 1]$.

 Definition: Let $M$ be a set, and let $(U, \varphi)$ and $(V, \psi)$ be $m$-dimensional charts on $M$. Then these charts are said to be Compatible if: a) $\varphi (U \cap V ) \subseteq \mathbb{R}^m$ is open in $\mathbb{R}^m$. b) $\psi (U \cap V) \subseteq \mathbb{R}^m$ is open in $\mathbb{R}^m$. c) The function $\psi \circ \varphi^{-1}$ when restricted to $varphi (U \cap V)$, i.e., $\psi \circ \varphi^{-1} : \varphi(U \cap V) \to \psi (U \cap V)$, is a diffeomorphism from $\varphi (U \cap V)$ to $\psi (U \cap V)$.

Condition (c) for compatibility can be replaced to require that $\varphi \circ \psi^{-1} : \psi (U \cap V) \to \varphi (U \cap V)$ is a diffeomorphism (why?). Furthermore, these maps are sometimes called Transition Maps.

Note that if $(U, \varphi)$ and $(V, \psi)$ are charts on $M$ such that $U \cap V = \emptyset$ then $\varphi(U \cap V) = \varphi(\emptyset) = \emptyset$ and $\psi (U \cap V) = \psi (\emptyset) = \emptyset$ which is open in $\mathbb{R}^m$. Furthermore, $\psi \circ \varphi^{-1}$ and $\varphi \circ \psi^{-1}$ are trivially a diffeomorphism, so $(U, \varphi)$ and $(V, \psi)$ are trivially compatible if $U$ and $V$ are disjoint.