> [!definition] > > Let $X$ be a [[Manifold|manifold]] and $T^*X$ be its [[Cotangent Bundle|cotangent bundle]]. A **covector field**/**differential**/**1-form** is a [[Section|section]] of $T^*X$. The set $\vf^*(X)$ is the space of all covector fields. > > If $\omega$ is a right inverse of the projection but not a [[Manifold Morphism|morphism]], then $\omega$ is a *rough* covector field. > [!definition] > > Let $X$ be a [[n-Manifold|n-manifold]] and $(U, \varphi = (x^i))$ be a [[Atlas|chart]] in $X$, then the maps > $ > dx^i: U \to T^*X \quad p \mapsto dx^i > $ > form the **coordinate covector fields** with respect to the chart. If $\omega \in \vf^*(X)$ is a rough vector field, then > $ > \omega = \omega_idx^i \quad \text{where} \quad \omega_i(p) = \angles{\omega(p), \ppip} > $ > where $\seqf{\omega_i}$ are the **component functions** of $\omega$. > [!definition] > > Let $X$ be a $n$-manifold, $\omega$ be a rough covector field, and $\xi \in \vf(X)$ be a [[Vector Field|vector field]], then there exists a function > $ > \angles{\omega, \xi}: X \to \real \quad p \mapsto \angles{\omega_p, \xi_p} > $ > If the fields have coordinate representations $\omega = \omega_i dx^i$ and $\xi = \xi^i \ppi$, then > $ > \angles{\omega, \xi}(p) = \omega_i(p) \xi^i(p) > $ > [!theorem] > > Let $U$ be a smooth $n$-manifold, $(U, \varphi = (x^i))$ be a global coordinate chart for $U$, and $\omega$ be a rough covector field. Then the following are equivalent: > 1. $\omega$ is smooth. > 2. $\omega$ has smooth component functions. > 3. For every smooth vector field $\xi \in \vf(U)$, the [[Function on Manifold|function]] $\angles{\omega, \xi} \in C^\infty(X)$. > > *Proof*. The first equivalence comes from [[Sections of Vector Bundles]]. > > Let $\xi \in \vf(U)$ be a vector field, then > $ > \angles{\omega, \xi}(p) = \omega_i(p)\xi^i(p) > $ > Hence if $\omega$ has smooth component functions, then $\angles{\omega, \xi}$ is smooth for every $\xi \in \vf(U)$. Now let $x^j$ be the $j$-th coordinate vector field, then > $ > \angles{\omega, \xi}(p) = \omega_i(p) (x^j)^i(p) = \omega_i(p) > $ > so if $\angles{\omega, \xi} \in C^\infty(U)$ for all $\xi \in \vf(U)$, then each one of its component is smooth. > [!theorem] > > Let $X$ be a smooth $n$-manifold, and $\omega$ be a rough covector field. Then the following are equivalent: > 1. $\omega$ is smooth. > 2. For every smooth coordinate chart, the component functions of $\omega$ are smooth. > 3. For every $p \in X$, there exists $(U, \varphi)$ at $p$ in which $\omega$ has smooth component functions. > 4. For every $\xi \in \vf(X)$, $\angles{\omega, \xi} \in C^\infty(X)$. > 5. For every open subset $U \subset X$ and $\xi \in \vf(U)$, $\angles{\omega, \xi} \in C^\infty(U)$. > > ### The First Three[^1] > > $(1) \Rightarrow (2)$: Suppose that $\omega$ is smooth, then for any chart $(U, \varphi)$, $\omega$ has smooth local representation with respect to $(U, \varphi)$ and its corresponding bundle chart. Therefore $\omega$ has smooth component functions by the previous theorem. > > $(2) \Rightarrow (3)$: For any $p \in X$, choosing any $(U, \varphi)$ at $p$ yields a chart in which $\omega$ has smooth component functions. > > $(3) \Rightarrow (1)$: By choosing a chart at each $p$, we get an open cover $\seq{U_i}$ of $X$. Since for each $U_i$, $\omega|_{U_i}$ has smooth component functions, and is therefore smooth, by the [[Gluing Lemma]], $\omega$ is smooth. > > ### The Last Two > > $(3) \Rightarrow (4)$: Let $\xi \in \vf(X)$ and $p \in X$, then there exists a chart $(U, \varphi)$ at $p$ in which $\omega$ has smooth components. By the previous theorem, $\angles{\omega, \xi}|_{U}$ is smooth. As such a chart exists for each $p \in X$, by the gluing lemma, $\angles{\omega, \xi}$ is smooth. > > $(4) \Rightarrow (5)$: Let $U \subset X$ and $\xi \in \vf(U)$. Let $p \in U$, then there exists $V \in \cn^o(p)$ with $\ol{V} \subset U$ and $\mu \in \vf(X)$ such that $\mu|_{V} = \xi|_V$ (see [[Sections of Vector Bundles]]). In which case, $\angles{\omega, \xi}|_V = \angles{\omega, \mu}|_V$ is smooth. Since such a neighbourhood can be found for any $p \in U$, by the gluing lemma, $\angles{\omega, \xi}$ is smooth. > > $(5) \Rightarrow (2)$: Let $(U, \varphi)$ be a chart at $p$ and $x^i$ be the $i$-th coordinate function. Then $\angles{\omega, x^i} = \omega_i \in C^\infty(U)$. [^1]: Also see [[Sections of Vector Bundles]].