A weak pseudo-Riemannian metric on a smooth manifold is a tensor field which is symmetric and for which, at each ,
for all implies that . This latter condition is most commonly referred to as non-degeneracy though, in the presence of so-called strong non-degeneracy, is more accurately described as weak non-degeneracy.
Weak pseudo-Riemannian metrics which are also positive definite are called weak Riemannian metrics. This use of terminology is in stark contrast to the case of pseudo-Riemannian (and hence, semi-Riemannian) metrics which fundamentally cannot be Riemannian due to the existence of negative-squared terms in their metric signatures. In most literature, weak Riemannian metrics are simply called Riemannian.