A weak Riemannian metric on a smooth manifold M is a (0, 2) tensor field g which is both a weak pseudo-Riemannian metric and positive definite. 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 T_m M and T_m^* M, respectively, for all m element M; conversely, weak Riemannian metrics are merely injective linear maps from T_m M to T_m^* M .

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