nLab pushforward of vector fields

Redirected from "pushforward of a vector field".

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a differentiable map ϕ:X 1X 2\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 xX{γC (,X)|γ(0)=x}/(γ 1γ 2dγ 1(0)=dγ 2(0)) 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 ϕ\phi induces maps of equivalence classes

T xX 1 T ϕ(x)X 2 [γ] [ϕγ] \begin{array}{ccc} T_x X_1 &\xrightarrow{\phantom{--}}& T_{\phi(x)} X_2 \\ [\gamma] &\mapsto& [\phi \circ \gamma] \end{array}

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

Literature

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

See also

Created on June 21, 2024 at 09:45:41. See the history of this page for a list of all contributions to it.