1-form on the Riemannian manifold

Question

Let $\omega$ a 1-form on a riemannian manifold $(M,g)$, and for a point $x\in M$, there is a notation: $|\omega_x|_g$, what does $|\omega_x|_g$ mean?

Answer 1

It's the norm of the linear functional $\omega_x$ with respect to the scalar product induced on the cotangent bundle by the Riemannian metric. E.g. in coordinates, $$|\omega_x|^2= \sum_{i,j} g^{ij}(x)\omega_i(x) \omega_j(x)$$ where, in commonly used notation, $(g^{ij})$ is the inverse of $(g_{ij})$