In a (pseudo-) Riemannian manifold, we have a metric tensor defined on each point in the manifold. I.e. a tensor field.
With this metric tensor we can define also a metric on the manifold by integrating over the metric tensor along a curve, making the manifold a metric space.
However, generally the manifold is not necessarily a normed space, since a norm induces a metric but a metric not a norm.
However, when we look at the tangent space at each point in the manifold, is it correct to say that the metric tensor $g$ is a norm of a vector $v$ on that tangent space, if we take the metric tensor of $v$ with itself ($g(v,v)$)?