In the same way that the metric maps a vector 
into a one- form, it maps a M/N tensor into a M-1/N+1 tensor, i.e.
it lowers an index. Similarly the inverse metric maps
a M/N tensor into a M+1/N-1 tensor, i.e. it raises an index.
So for example
In Special Relativity, raising or lowering a 0 component changes the sign of the component; raising or lowering 1, 2, or 3 components has no effect.
We can operate the inverse metric on the metric to get the Kroneker delta [ Assignment 3 ]:
So far we have confined our attention to Lorentz frames
[ i.e. inertial frames ]. We can also allow more general coordinate transformations in a more general space i.e. 
. We then define
Tensors will then transform as before, for example
Old fashioned texts regard the above as the definition of a tensor. Raised indices are called contravariant because they transform ``contrary'' to basis vectors:
Lowered indices are called covariant :
In particular one- forms are sometimes called covariant
vectors , while
ordinary vectors are called contravariant vectors .