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.
