User:Lethe/covariance

This page does not deal with the statistical concept covariance of random variables, nor with the computer science concepts of covariance and contravariance.

In mathematics and theoretical physics, covariance and contravariance are terms used to describe how objects defined on some spaces behave under their induced transformations with respect to the transformations of their underlying spaces. A covariant object is one which transforms in the same direction as the transformation of its underlying space, while a contravariant object is one which transforms in the opposite direction. Depending on how one describes the objects and which transformations of the underlying space one considers, different conventions can be reached, which is the source of some confusion.

Change of coordinates

The basic observation is that under the change of coordinates

$x^{a}\mapsto y^{b}=F^{b}(x^{1},\cdots ,x^{m})$

expressions like dx^a behave like the coordinates x^a

$dx^{a}\mapsto dy^{b}=d(F^{b}(x))={\frac {\partial F^{b}}{\partial x^{a}}}dx^{a}$

so that differentials of the new coordinates are obtained from differentials of the old coordinates using (the derivatives of) the change of coordinates function. On the other hand, derivatives behave in the opposite manner

${\frac {\partial }{\partial y^{b}}}\mapsto {\frac {\partial }{\partial x^{a}}}={\frac {\partial F^{b}}{\partial x^{a}}}{\frac {\partial }{\partial y^{b}}}$

so that derivatives with respect to the old coordinates are obtained from the derivatives with respect to the new coordinates by means of the change of coordinates function. In general the map F need not not be invertible, so the directions of the change of coordinates of differentials necessarily goes forward, while the change of coordinates of derivatives necessarily goes backwards with respect to the direction of the function F. Quantities like the former are called covariant while quantities like the latter are called contravariant.

The terms are most often encountered describing constructions on manifolds. Manifolds are spaces which, by definition, come equipped with a set of local coordinate charts, maps from Rⁿ. Any construction on the underlying space can be expressed as a construction on Rⁿ by means of these maps. Thus one may choose to think of objects as living in an abstract space or as living in Rⁿ. The latter case, while more concrete, has the disadvantage that the constructions will be dependent on exactly which map from Rⁿ was chosen; the construction which lives in Rⁿ will change form under this transition funciton, while the construction in the abstract space will not. Both the abstract object and the object in Rⁿ will change under a map between manifolds. Exactly how the object changes under these maps determines whether it is covariant or contravariant.

Active transformations

Coordinate free description

Let φ: M → N be a smooth map of manifolds. For any real function g on N, one constructes the pullback of g, a real function on M, denoted φ^*g, and given by

$\phi ^{*}g(p)=g(\alpha (p)).$

Thus the map φ, which takes points in M to points in N, induces the pullback, which takes functions on N to functions on M. Thus functions are contravariant objects.

The pushforward of φ, φ_*: TM → TN takes tangent vectors on M to tangent vectors on N. This map is given in terms of the directional derivative. If v is a vector on M, and f is a real function on M, denote the directional derivative of f along v as v(f). Given a real function g on N, the map φ induces a directional derivative of g given by

$\phi _{*}(\mathbf {v} )(g)=\mathbf {v} (\phi ^{*}g).$

Thus φ induces an action on tangent vectors which takes tangent vectors on M to tangent vectors on N.

The map φ also induces an action on cotangent vectors defined by

$\phi ^{*}\sigma (\mathbf {v} )=\sigma (\phi _{*}\mathbf {v} ).$

This induced map of cotangent vectors is also referred to as the pullback.

To summarize, functions and cotagent vectors transform contravariantly under active transformations while tangent vectors transform covariantly. The tensor product of covariant objects is again covariant and so get "pushed forward" and the tensor product of a contravariant objects is again contravariant and so get "pushed back" (this includes differential forms and the metric tensor). Tensors of mixed type, that is the tensor product of a covariant object with a contravariant object, cannot get pushed forward or pulled back along φ in general, although if φ is a diffeomorphism (and so invertible), then one can employ the inverse to move any tensors forward or back along φ.

Descending to local coordinates

Every point p has a neighborhood U diffeomorphic to R^m. Let ψ: R → U be such a diffeomorphism. Given the smooth map φ: M → N, let V be a neighborhood of q = φ(p) and χ: Rⁿ → V a diffeomorphism. Then the composition

$F=\chi ^{-1}\circ \phi \circ \psi :\mathbb {R} ^{m}\to \mathbb {R} ^{n}$

is the local coordinate version of the active transformation φ.

Labelling the coordinates of R^m x^a with a an index ranging from 1 to m and the coordinates of Rⁿ y^b, one has

$y^{b}=F^{b}(x^{1},\cdots ,x^{m})$

Then the directional derivative along a vector v in R^m is given by

$\mathbf {v} =v^{a}{\frac {\partial }{\partial x^{a}}}$

and its pushforward to Rⁿ is

$F_{*}(\mathbf {v} )=v^{a}{\frac {\partial F^{b}}{\partial x^{a}}}{\frac {\partial }{\partial y^{b}}}.$