Compatible oriented charts for all open balls in a manifold

Question

Let $M$ be an orientable smooth manifold. Can I find for every set $U \subset M$ that is diffeomorphic to $\mathbb{R}^n$ a chart $\phi_U : U \to \mathbb{R}^n$ so that the transitions functions $(\phi_U|_{U \cap V} \circ \phi_V^{-1})$ for two such charts are orientation-preserving?

Answer 1

Pick an orientation of $M$. This automatically orients all open subsets $U\subseteq M$. Now if $U\subseteq M$ is diffeomorphic to $\mathbb{R}^n$, pick a diffeomorphism $\phi_U\colon U\rightarrow\mathbb{R}^n$. This is a diffeomorphism between oriented, connected smooth manifolds, so it is either orientation-preserving or -reversing. In the latter case, post-compose with an orientation-reversing automorphism of $\mathbb{R}^n$, so we may WLOG assume that $\phi_U$ is orientation-preserving. Since the $\phi_U$ for all smooth balls $U$ are orientation-preserving, so are their transition functions.