Pushforward (differential) - Pushforward of Vector Fields

Pushforward of Vector Fields

Given a smooth map φ:M→N and a vector field X on M, it is not usually possible to define a pushforward of X by φ as a vector field on N. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ. 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. Then, applying the differential pointwise to X yields 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 = Y_φ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φ_*X = φ*Y as vector fields along φ. In other words, for all x in M, dφ_x(X)=Y_φ(x).

In some situations, given a X vector field on M, there is a unique vector field Y on M which is φ-related to X. This is true in particular when φ is a diffeomorphism. 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, dφ_x(X_x) 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.