How do I prove the following:
Let $M$ be a Riemannian manifold. Let $\omega: M\rightarrow T^{*}M$ be a smooth 1 form on $M$. Then, there exists a unique smooth vector field $Y$ on $M$, such that
$\omega(X)=\langle X,Y\rangle$ for every smooth vector field $X$ on $M$.
I'm not sure how to construct such a smooth vector field $Y$. I initially thought of using Riesz representation theorem, but I wasn't sure how to proceed.
May I have hints? (without using Christoffel symbols)