In Euclidean spaces, we define the Euclidean norm of a vector $\vec{x} = (x\_1,x\_2,...x\_n)$ as $\|\vec{x}\|:=\sqrt{x\_1^2+x\_2^2+ \cdots +x\_n^2 }$

Does the metric tensor field of a Riemannian manifold allow us to establish something similar to a norm in Euclidean space? If so, why is this notion of length preserved from point to point on the manifold. 

REMARK:
By the definition of a differentiable manifold, I understand that a manifold is locally homeomorphic to $\Bbb{R}^{n}$. But if I take the Euclidean norm of a tangent vector at a point $p$ on the manifold, shouldn't its norm only be valid in the tangent space $T\_{p}M$?