Index lowering converts a contravariant tensor index into a covariant tensor index by tensor
contraction with a metric tensor. For a contravariant
tensor of rank 1 with components 
, the lowered components are
 |
where the repeated index 
is summed. The same operation applies to any chosen contravariant
tensor index. It depends on the metric and is invertible when the metric is nondegenerate.
Geometrically, index lowering is the flat isomorphism
from tangent vectors to covectors.
