contravariant tensor

Much like the curvature, it can be shown that Θ behaves as a contravariant tensor under a change in frame:

Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates.

It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight 1.

A completely antisymmetric covariant tensor of order p may be referred to as a p-form, and a completely antisymmetric contravariant tensor may be referred to as a p-vector.

This tensor may be converted to the contravariant tensor by raising the indices with the metric as usual, but a minus sign is needed if the metric signature contains an odd number of negatives.

On a pseudo-Riemannian manifold, one may define coordinate-invariant covariant and contravariant tensor fields whose coordinate representations agree with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation.