The flat isomorphism on a Riemannian manifold with Riemannian metric 
is the map from the tangent
bundle to the cotangent bundle defined by
 |
Thus a tangent vector 
is sent to the covector whose
value on a tangent vector 
is 
. In coordinates, this sends components 
to 
, so it is also called index
lowering. The nondegeneracy of the metric makes the flat isomorphism invertible,
with inverse the sharp isomorphism.
