A covariant tensor, denoted with a lowered index (e.g., 
) is a tensor having specific
transformation properties. In general, these transformation properties differ from
those of a contravariant tensor.
To examine the transformation properties of a covariant tensor, first consider the gradient
 |
(1)
|
for which
 |
(2)
|
where 
.
Now let
 |
(3)
|
then any set of quantities 
which transform according to
 |
(4)
|
or, defining
 |
(5)
|
according to
 |
(6)
|
is a covariant tensor.
Contravariant tensors are a type of tensor with differing transformation properties, denoted 
. To turn a contravariant
tensor 
into a covariant tensor 
(index lowering), use
the metric tensor 
to write
 |
(7)
|
Covariant and contravariant indices can be used simultaneously in a mixed tensor.
In Euclidean spaces, and more generally in flat Riemannian manifolds, a coordinate system can be found where the metric tensor is constant, equal to Kronecker delta
 |
(8)
|
Therefore, raising and lowering indices is trivial, hence covariant and contravariant tensors have the same coordinates, and can be identified. Such tensors are known as Cartesian tensors.
A similar result holds for flat pseudo-Riemannian manifolds, such as Minkowski space, for which covariant and contravariant tensors can be identified. However, raising and lowering indices changes the sign of the temporal components of tensors, because of the negative eigenvalue in the Minkowski metric.
