A strong pseudo-Riemannian metric on a smooth manifold is a tensor field which is symmetric and for which, at each , the map
is an isomorphism of onto . This latter condition is called strong non-degeneracy.
Strong pseudo-Riemannian metrics which are also positive definite are called strong 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. Every strong pseudo-Riemannian (respectively, strong Riemannian) metric is weak pseudo-Riemannian (respectively, weak Riemannian) due to the fact that strong non-degeneracy implies weak non-degeneracy.