----- > [!proposition] Proposition. ([[the space of differential 1-forms is dual to that of vector fields]]) > Let $M$ be a [[smooth manifold]], $TM$ its [[tangent bundle]], and $T^{*}M$ its [[cotangent bundle]]. Then there is [[dual vector space|an]] (I think natural) [[isomorphism]] of $C^{\infty}(M)$-[[module|modules]] $\Omega^{1}(M) \xrightarrow{\sim} \big(\Gamma(TM)\big)^{\vee} $ identifying [[differential form|differential 1-forms]] $\theta:M \to T^{*}M$ with $C^{\infty}(M)$-[[linear map|linear]] maps $\hat{\theta}: \Gamma(TM) \to C^{\infty}(M)$. > Explicitly, the map takes $\theta \in \Omega^{1}(M)$ to $\hat{\theta}:\Gamma(TM) \to C^{\infty}(M)$ defined by $\hat{\theta}(X)(p):= \theta(p)(X_{p}).$ > > [!note] Note. > Note that these spaces are, in general, infinite-dimensional. So *this result does not identify vector fields with 1-forms* — for that, one needs additional structure like a [[metric tensor|metric]]; see e.g. [[musical isomorphism induced by a nondegenerate bilinear form|musical isomorphism between tangent and cotangent bundle]]. ^note (Is there sense in trying to generalize this to [[differential form with values in a vector bundle|bundle-valued 1-forms]]?) ^proposition > [!proof]- Proof. ([[the space of differential 1-forms is dual to that of vector fields]]) > Going from $\omega \in \Omega^{1}(M)$ to a map $\Gamma(TM) \to C^{\infty}(M)$ is done in the usual way. to go the other direction, see [here](https://math.stackexchange.com/questions/2622390/differential-1-forms-as-linear-maps). ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```