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 injectivelinear maps from to .