> [!definition] > > Let $X$ be a [[Manifold|n-manifold]]. A **pointwise orientation** on $X$ is a choice of [[Orientation|orientation]] on each $T_pX$. > [!definition] > > A pointwise orientation on $X$ is $C^k$ if the following equivalent conditions hold: > 1. For any $p \in X$, there exists a [[Neighbourhood|neighbourhood]] $U \in \cn^o(p)$ and a $C^k$ local [[Frame|frame]] $(U, \{\xi^j\}_1^n)$ such that $\{\xi^j_q\}_1^n$ is positively oriented for each $q \in U$. > 2. For any $p \in X$, there exists a $C^k$ [[Atlas|chart]] $(U, \varphi) \in X$ at $p$ such that the local coordinates $\{\ppi|_q\}_1^n$ are positively oriented at each $q \in U$. In this case, $(U, \varphi)$ is a **positively oriented** chart. > 3. There exists a $C^k$-[[Atlas|atlas]] $\bracs{(U_i, \varphi_i)}_{i \in I}$ for $X$, such that each $(U_i, \varphi_i)$ is positively oriented, and the transition map $D(\varphi_i \circ \varphi_j)$ has positive determinant. > 4. There exists a non-vanishing $C^k$ [[Differential Form|n-form]] $\omega \in \Omega^n$ such that $\{v_j\}_1^n \subset T_pX$ is positively oriented if and only if $\omega(v_1, \cdots, v_n) > 0$. > > *Proof*. Since each chart corresponds to a frame and each frame corresponds to a chart, the first two are equivalent. > > Suppose that there is an [[Atlas|atlas]] $\bracs{(U_i, \varphi_i)}_{i \in I}$ consisting of positively oriented charts, then since $\phi \circ T = \det{T} \cdot \phi$ for any alternating tensor $\phi$ and linear map $T$, the transition maps must have positive determinants. > > Let $\omega_i = dx_i^1 \wedge \cdots \wedge dx_i^n$ where $\varphi_i = (x_i^1, \cdots, x_i^n)$, and $\seqi{\psi}$ be a $C^k$-[[Partition of Unity on n-Manifold|partition of unity]] subordinate to $\seqi{U}$. Define > $ > \omega = \sum_{i \in I}\omega_i \psi_i > $ > then for any positively oriented basis $\bracs{e_j}_1^n \subset T_pX$, > $ > \omega_p(e_1, \cdots, e_n) = \sum_{i \in I}\omega_i(e_1, \cdots, e_n)\psi_i > 0 > $ > so $\omega$ is a non-vanishing $C^k$ $n$-form corresponding to the given orientation. > > Lastly, if $\omega \in \Omega^n$ is a non-vanishing $C^k$ $n$-form, then choosing a connected neighbourhood yields the desired chart. > [!definition] > > Let $X, Y$ be $n$-manifolds. A [[Manifold Morphism|morphism]] $F: X \to Y$ is **orientation preserving** if the [[Differential|differential]] $df: TX \to TY$ maps positively oriented bases to positively oriented bases.