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.