Proposition

Suppose $\mathcal{C}$ is a collection of coordinate systems for $M$ such that

1. For each $x \in M$, there is $f \in \mathcal{C}$ which is a coordinate system around $x$;
2. if $f, g \in \mathcal{C}$, then $\det(f^{-1} \circ g)’ > 0$.

Show that there is a unique orientation of $M$ such that $f$ is orientation-preserving if $f \in \mathcal{C}$.

Solution

Let $x \in M$. Let $f \in \mathcal{C}$ be a coordinate system around $x$ such that $f:W \rightarrow \mathbb{R}^n$. Let $a = f^{-1}(x) \in \mathbb{R}^k$. Let \(\mu_x = [f_*((e_1)_a), \cdots, f_*((e_k)_a)]\).

Using the procedure above, we can define $\mu_x$ for each $x \in M$.

• Is this independent of the choice of $f$?

Let $x \in M$ and $f, g \in \mathcal{C}$ be coordinate systems around $x$. Let $a, b$ be chosen such that $f(a) = g(b) = x$. Then $f^{-1} \circ g$ maps $\mathbb{R}^k$ into itself and $\det(f^{-1} \circ g)’ > 0$, so

Therefore, the choice of orientation $\mu_x$ is independent of the choice of a coordinate system.

• Is the orientation consistent?

Let $f: W \rightarrow \mathbb{R}^n$ be a coordinate system in $\mathcal{C}$. Let $a, b \in W$ be given. Then $f$ is a coordinate system around $a$ and $b$. Thus by the definition of $\mu$ and the above argument regarding independence of a choice, \([f_*((e_1)_a), \cdots, f_*((e_1)_a)] = \mu_{f(a)}\) and \([f_*((e_1)_b), \cdots, f_*((e_1)_b)] = \mu_{f(b)}\). Therefore, the orientation is consistent.

• Is $f$ orientation preserving for each $f \in \mathcal{C}$?

By the definition of $\mu$ and the above argument regarding independence of a choice, \([f_*((e_1)_a), \cdots, f_*((e_1)_a)] = \mu_{f(a)}\) for all $a \in W$ if $f \in \mathcal{C}$.