Index raising converts a covariant tensor index into a contravariant tensor index by tensor
contraction with the inverse of a metric tensor.
If 
are the components of the matrix
inverse of 
,
then a covariant tensor of rank 1 with components
has raised components
 |
where the repeated index 
is summed. The same operation applies to any chosen covariant
tensor index and is invertible when the metric
is nondegenerate; its inverse is index lowering.
Geometrically, index raising is the sharp isomorphism
from covectors to tangent
vectors.
