About the definition of $\Pi^\sharp:\Omega^1(M)\longrightarrow \mathfrak{X}^1(M)$?

Question

Let $M$ be a manifold. A $p$-field on $M$ is a section of the bundle $\Lambda^p (TM)$. Let us denote the space of $p$-fields on $M$ by $\mathfrak{X}^p(M)$. Notice, $\mathfrak{X}^1(M)$ is the space of vector fields on $M$.

Furtheremore, let us denote by $\Omega^p(M)$ the space of $p$-forms on $M$, that is, the sections of the bundle $\Lambda^p T^*M$.

I came across the following construction:

Every $\Pi\in \mathfrak{X}^2(M)$ induces a map $$\Pi^{\sharp}:\Omega^1(M)\longrightarrow \mathfrak{X}^1(M),$$ defined by:

$$(\Pi^\sharp \alpha)(\beta):=\Pi(\alpha, \beta), \alpha, \beta\in \Omega^1(M).$$

But I don't understand this definition. Why is $\Pi^{\sharp} \alpha$ being applied on $\beta\in \Omega^1(M)$?

Can anyone explain it?

Thanks.

Answer 1

For any finite-dimensional vector space we have a canonical isomorphism $V \cong (V^*)^*$. This gives an isomorphism $\mathfrak{X}^1(M) \cong (\Omega^1(M))^*$.

So the data of a map $f :\Omega^1(M)\longrightarrow \mathfrak{X}^1(M) $ is really the same as a map $f:\Omega^1(M)\longrightarrow (\Omega^1(M))^*$ which is exactly what is given by $\Pi^{\sharp}$.

For more informations, you can looks for "musical isomorphism".