\begin{definition}[Definition:Inner Product on Cotangent Space]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $x \in M$ be a base point.
Let ${T_x}^* M$ be the cotangent space of $M$ at $x$.
Let $\omega, \eta \in {T_x}^* M$ be covector fields.
The '''inner product on the cotangent space''' is defined by:
:$\innerprod \omega \eta_g := \innerprod {\omega^\sharp} {\eta^\sharp}_g$
where $\sharp$ denotes the sharp operator.
Locally this reads:
:$\innerprod \omega \eta = g^{ij} \omega_i \eta_j$
{{Refactor|local version as separate derivative result|level = medium}}
\end{definition}