nLab pushforward of vector fields

Idea

Given a differentiable map $\phi \colon X_1 \xrightarrow{\;} X_2$ between differentiable manifolds (e.g. a smooth map between smooth manifolds) and thinking of vector fields as infinitesimal approximations to differentiable curves

$$T_x X \;\simeq\; \big\{ \gamma \in C^\infty\big(\mathbb{R},\, X\big) \,\big\vert\, \gamma(0) = x \big\} \Big/ \big( \gamma_1 \sim \gamma_2 \;\Leftrightarrow\; \mathrm{d}\gamma_1(0) = \mathrm{d}\gamma_2(0) \big)$$

then the postcomposition of these curves with $\phi$ induces maps of equivalence classes

$$\begin{array}{ccc} T_x X_1 &\xrightarrow{\phantom{--}}& T_{\phi(x)} X_2 \\ [\gamma] &\mapsto& [\phi \circ \gamma] \end{array}$$

alternatively denoted “$\phi_\ast$” or “$\mathrm{d}\phi$” (cf. differentiation as a functor) and called the pushforward of vector fields along $\phi$.

Related concept

• The dual notion is that of pullback of differential forms.

Literature

Most texts on differential geometry will discuss pushforward of vector fields.

See also

• Wikipedia, Pushforward_(differential)