Pushforward (differential)

We wish to generalize this to the case that φ is a smooth function between any smooth manifolds M and N.

Let φ : MN be a smooth map of smooth manifolds. Given some xM, the differential of φ at x is a linear map

from the tangent space of M at x to the tangent space of N at φ(x). The application of x to a tangent vector X is sometimes called the pushforward of X by φ. The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).

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

After choosing two charts around x and around φ(x), φ is locally determined by a smooth map

in the Einstein summation notation, where the partial derivatives are evaluated at the point in U corresponding to x in the given chart.

The differential is frequently expressed using a variety of other notations such as

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps.

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

The differential of a smooth map φ induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of M to the tangent bundle of N, denoted by or φ, which fits into the following commutative diagram:

where πM and πN denote the bundle projections of the tangent bundles of M and N respectively.

Given a smooth map φ : MN and a vector field X on M, it is not usually possible to identify a pushforward of X by φ with some vector field Y 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 φTN 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.

Any vector field Y on N defines a pullback section φY of φTN with (φY)x = Yφ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φX = φY as vector fields along φ. In other words, for all x in M, x(X) = Yφ(x).

In some situations, given a X vector field on M, there is a unique vector field Y on N 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