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:

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

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds