A weak Riemannian metric on a smooth manifold 
is a 
tensor field 
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 
and 
, respectively, for all 
; conversely, weak Riemannian metrics are merely injective linear maps from 
to 
(Marsden et al. 2002).
