# Let $M$ be a smooth manifold. Does there exist a canonical isomorphism between $\Gamma(TM)^*$ and $\Gamma(T^*M)$?

Let $$M$$ be a smooth manifold and for each point $$p \in M$$, let $$T_pM$$ denote the tangent space at $$p \in M$$. We define the set $$TM = \bigsqcup_{p \in M}T_pM$$ and equip it with the initial topology and canonical projection $$\pi$$ to construct the tangent bundle $$TM \overset{\pi}{\to} M$$ (the standard way to construct the tangent bundle, just including for completeness). Similarly, let $$T_p^*M$$ denote the cotangent space at $$p \in M$$ and define the set $$T^*M = \bigsqcup_{p \in M}T_p^*M$$. We construct the cotangent bundle $$T^*M \overset{\pi'}{\to}M$$ analogous to how we did for the tangent bundle.

Let $$\Gamma(TM):=$$ { $$\sigma: M \to TM| \pi \circ \sigma = \text{id}_M, \ \sigma \ \text{smooth}$$} be the $$C^{\infty}(M)$$-module of smooth vector fields on $$M$$.

Let $$\Gamma(T^*M):=$$ { $$\omega: M \to T^*M| \pi' \circ \omega = \text{id}_M, \ \omega \ \text{smooth}$$} be the $$C^{\infty}(M)$$-module of smooth covector fields on $$M$$.

Let $$\Gamma(TM)^*:=$$ { $$\psi: \Gamma(TM) \overset{\sim}{\to}C^{\infty}(M)$$} denote the dual of $$\Gamma(TM)$$.

Does there exist a canonical isomorphism $$\Phi: \Gamma(TM)^* \to \Gamma(T^*M)$$?

• Do they even have the same dimension? Aug 4, 2023 at 19:41
• @pancini They're not even guaranteed a basis in ZFC because $C^{\infty}(M)$ isn't a division ring, so a notion of dimension is in general undefined for $\Gamma(TM)$, unless it admits a basis (someone please correct me if there is an alternative definition of dimension that can work here). However, locally once you choose a chart at a point, you can express the vector / covector fields in terms of the induced chart basis. Aug 4, 2023 at 19:44

Given $$\alpha\in \Gamma(T^*M)$$ there is a natural map $$\alpha': \Gamma(TM)\to C^\infty(M)$$. This is given by applying the natural pairing $$\langle \cdot,\cdot\rangle:T_p^*M\times T_pM\to \mathbb{R}$$ at each point, $$\alpha': X\mapsto (p\mapsto \langle \langle \alpha_p, X_p\rangle)$$. It is clear that such a map is injective and the fact that it is an isomorphism is a consequence of the Tensor Characterization Lemma. This lemma can be found in John Lee's book Introduction to Smooth manifolds, specifically chapter 12.