A Riemann surface is not a special case of a Riemannian manifold, which is a manifold equipped with a Riemannian metric. A Riemannian metric is not a special case of a metric. You can use a Riemannian metric to define a metric though. A Riemannian metric is an equipment of an inner product to the tangent space at each point of the manifold. With this, you can define the length of a tangent vector as the inner product with itself (sqrt-ed). So then you can compute the length of a curve in the manifold as integrating over the curve's tangent vectors. Now we can define the distance between two points in the manifold as the infimum of lengths of curves that connect them.

Theorem: The topology generated by this metric is the same as the original topology on the manifold.