On a Riemannian manifold with Riemannian metric 
,
the musical isomorphisms identify tangent vectors
and cotangent vectors. The flat
isomorphism sends a tangent vector 
to the covector
 |
while its inverse, the sharp isomorphism, sends a covector 
to the unique tangent vector 
satisfying 
for every 
. The names refer to the musical symbols 
and 
. The maps extend by applying them to tensor indices
and are also called index lowering and index
raising.
