Definition:Inner Product on Cotangent Space

Definition
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:


 * $\innerprod \omega \eta = g^{ij} \omega_i \eta_j$