A strong Riemannian metric on a smooth manifold is a tensor field which is both a strong pseudo-Riemannian metric and positive definite.
In a very precise way, the condition of being a strong Riemannian metric is considerably more stringent than the condition of being a weak Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, strong Riemannian metrics provide an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to .