A weak pseudo-Riemannian metric on a smooth manifold M is a (0, 2) tensor field g which is symmetric and for which, at each m element M, g_m(v_m, w_m) = 0 for all w_m element T_m M implies that v_m = 0. This latter condition is most commonly referred to as non-degeneracy though, in the presence of so-called strong non-degeneracy, is more accurately described as weak non-degeneracy.

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