Covariance and Contravariance of Vectors - Algebra and Geometry

Algebra and Geometry

In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a contravariant functor. Some constructions of multilinear algebra are of 'mixed' variance, which prevents them from being functors.

In geometry, the same map in/map out distinction is helpful in assessing the variance of constructions. A tangent vector to a smooth manifold M is, to begin with, a curve mapping smoothly into M and passing through a given point P. It is therefore covariant, with respect to smooth mappings of M. A contravariant vector, or 1-form, is in the same way constructed from a smooth mapping from M to the real line, near P. It is in the cotangent bundle, built up from the dual spaces of the tangent spaces. Its components with respect to a local basis of one-forms dx_i will be covariant; but one-forms and differential forms in general are contravariant, in the sense that they pull back under smooth mappings. This is crucial to how they are applied; for example a differential form can be restricted to any submanifold, while this does not make the same sense for a field of tangent vectors.

Covariant and contravariant components transform in different ways under coordinate transformations. By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.