In a very precise way, the condition of being a weak Riemannian metric is considerably less stringent than the condition of being a strong
Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy
but not vice versa. More precisely, any strong Riemannian metric provides an isomorphism
between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injectivelinear maps from to (Marsden et al. 2002).