Problem
Problem 6.40 Let $(M, g)$ be a Riemannian $n$-manifold. Prove:
(i) Given $\alpha, \beta \in T_{p}^{*} M$ and an orthonormal basis $\left\{e_{i}\right\}, i=1, \ldots, n$, of $T_{p} M$, and denoting by $g^{-1}$ the contravariant metric associated to $g$, one has
$$
g^{-1}(\alpha, \beta)=\sum_{i} \alpha\left(e_{i}\right) \beta\left(e_{i}\right)
$$
(ii) For $X \in T_{p} M$, one has
$$
g^{-1}\left(\alpha, X^{b}\right)=\alpha(X)=g\left(\alpha^{\sharp}, X\right)
$$
where
$$
\begin{aligned}
& \text { b: } T_{p} M \rightarrow T_{p}^{*} M, \quad X^{b}=g(X, \cdot), \quad \sharp: T_{p}^{*} M \rightarrow T_{p} M, \\
& \alpha^{\sharp}=g^{-1}(\alpha, \cdot),
\end{aligned}
$$
are the musical isomorphisms (named "flat" and "sharp", respectively) associated to $g$.