Pushforward (differential)

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.

It can be viewed as a generalization of the total derivative of ordinary calculus.

Explicitly, the differential is a linear map from the tangent space of

(with respect to the standard coordinates) is the matrix representation of the total derivative of

If tangent vectors are defined as equivalence classes of the curves

In other words, the pushforward of the tangent vector to the curve

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by for an arbitrary function

that satisfies the Leibniz rule, see: definition of tangent space via derivations).

, and in the Einstein summation notation, where the partial derivatives are evaluated at the point in

Extending by linearity gives the following matrix Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map

Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from

Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces.

The latter map may in turn be viewed as a section of the vector bundle Hom(TM, φ∗TN) over M. The bundle map

Also, if φ is not injective there may be more than one choice of pushforward at a given point.

Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.

A section of φ∗TN over M is called a vector field along φ.

For example, if M is a submanifold of N and φ is the inclusion, then a vector field along φ is just a section of the tangent bundle of N along M; in particular, a vector field on M defines such a section via the inclusion of TM inside TN.

This idea generalizes to arbitrary smooth maps.

Suppose that X is a vector field on M, i.e., a section of TM.

yields, in the above sense, the pushforward φ∗X, which is a vector field along φ, i.e., a section of φ∗TN over M. Any vector field Y on N defines a pullback section φ∗Y of φ∗TN with (φ∗Y)x = Yφ(x).

In this case, the pushforward defines a vector field Y on N, given by A more general situation arises when φ is surjective (for example the bundle projection of a fiber bundle).

Then a vector field X on M is said to be projectable if for all y in N, dφx(Xx) is independent of the choice of x in φ−1({y}).

This is precisely the condition that guarantees that a pushforward of X, as a vector field on N, is well defined.

These maps can be used to construct left or right invariant vector fields on

This can be readily computed using the curves definition of pushforward maps.

it has Lie algebra given by the set of matrices

giving any real number in one of the upper matrix entries with

(i-th row and j-th column).

which is equal to the original set of matrices.

we have its Lie algebra as the set of matrices