A strong Riemannian metric on a smooth manifold 
is a 
tensor field 
which is both a strong pseudo-Riemannian
metric and positive definite.
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 injective linear maps from 
to 
.
