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.