Is a Riemannian metric positive definite or positive semidefinite? From Wikipedia

The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space,

But a Riemannian metric is defined to be a family of (positive definite) inner products.
So I was wondering whether a Riemannian metric is positive definite or positive semidefinite?
Thanks and regards!

A Riemannian metric is positive definite and produces a Riemannian manifold. If you take the metric tensor to be positive semidefinite then you get what is called a semi-Riemannian manifold or pseudo-Riemannian manifold. With regards to the Fisher information metric on a statistical manifold it is usually assumed that the metric is positive definite. I would not trust Wikipedia. The standard reference on information geometry is Amari's Methods of Information Geometry.
