Push forward

This article deals with the push forward of a differentiable map as map between tangent spaces. For the push forward of a fiber bundle in the cohomology see Gysin sequence .

A mapping between tangential spaces of smooth manifolds that generalizes the directional derivative defined in Euclidean space is called pushforward .

The dual concept is usually called return transport ( pullback ).

definition

If and are smooth manifolds and is a smooth map , then the pushforward is defined ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle F\colon M\rightarrow N}$

${\displaystyle F_{\ast }\colon T_{p}M\rightarrow T_{F(p)}N}$

from at the point through ${\displaystyle F}$ ${\displaystyle p\in M}$

${\displaystyle (F_{*}v)(f)=v(f\circ F)}$

for and every smooth function on the manifold . Here tangential vectors are understood as directional derivatives ( cf. Tangent space . ${\displaystyle v\in T_{p}M}$ ${\displaystyle f\in {\mathcal {C}}^{\infty }(N)}$ ${\displaystyle N}$

In this way an image is defined. ${\displaystyle F_{\ast }\colon TM\to TN}$

Designations and spellings

Other names for push forward are derivative , differential and tangential mapping of . Other spellings are , , , , and . Often the brackets around the argument are also left out. ${\displaystyle F}$ ${\displaystyle F\,'(p)v}$ ${\displaystyle DF_{p}(v)}$ ${\displaystyle D_{p}F(v)}$ ${\displaystyle dF_{p}(v)}$ ${\displaystyle d_{p}F(v)}$ ${\displaystyle T_{p}F(v)}$ ${\displaystyle v}$

Meaning for tangential vectors of curves

If the tangential vector of a differentiable curve (here an interval in ) is in the point , then the tangential vector of the image curve is in the image point , i.e. ${\displaystyle v={\dot {c}}(t)\in T_{p}M}$ ${\displaystyle c\colon I\to M}$ ${\displaystyle I}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle p=c(t)}$ ${\displaystyle F_{*}v\,}$ ${\displaystyle {\tilde {c}}=F\circ c\colon I\to N}$ ${\displaystyle F(p)={\tilde {c}}(t)}$

${\displaystyle F_{*}v={\dot {\tilde {c}}}(t)=(F\circ c)^{\cdot }(t)}$.

Representation in coordinates

If local coordinates are around and local coordinates are around the image point , then the vectors and the representations have ${\displaystyle (x_{1},\dots ,x_{m})}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle (y_{1},\dots ,y_{n})}$ ${\displaystyle N}$ ${\displaystyle F(p)}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle w=F_{*}v\in T_{F(p)}N}$

${\displaystyle v=\sum _{j}v^{j}\,{\frac {\partial }{\partial x^{j}}}}$or . ${\displaystyle w=\sum _{i}w^{i}\,{\frac {\partial }{\partial y^{i}}}}$

If the figure is also represented by the functions , then the following applies ${\displaystyle F\colon M\to N}$ ${\displaystyle f^{1}(x_{1},\dots ,x_{m}),\dots ,f^{n}(x_{1},\dots ,x_{m})}$

${\displaystyle w^{i}=\sum _{j}{\frac {\partial f^{i}}{\partial x^{j}}}\,v^{j}}$.

Pushforward in Euclidean space

If the special case is present, it represents nothing other than the total derivative , whereby the Euclidean space is identified naturally with its tangential space (the distinction between directional derivative and total derivative does not matter here, since the function is already assumed to be sufficiently smooth) . ${\displaystyle F\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle F_{*}\,}$ ${\displaystyle DF(p)\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}$

Often the tangent space of Euclidean space is identified in the point with, i.e. the tangential bundle with . In this case, the push forward is the figure . ${\displaystyle T_{p}\mathbb {R} ^{m}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle p\in \mathbb {R} ^{m}}$ ${\displaystyle \{p\}\times \mathbb {R} ^{m}}$ ${\displaystyle T\mathbb {R} ^{m}}$ ${\displaystyle \mathbb {R} ^{m}\times \mathbb {R} ^{m}}$ ${\displaystyle F_{\ast }\colon (p,v)\mapsto (F(p),DF(p)(v))}$

properties

For Push Forward a concatenation of two figures and the true chain rule : ${\displaystyle G\circ F\colon M\to P}$ ${\displaystyle F\colon M\to N}$ ${\displaystyle G\colon N\to P}$

${\displaystyle (G\circ F)_{\ast }=G_{\ast }\circ F_{\ast }}$

or point by point

${\displaystyle (G\circ F)_{\ast p}=G_{\ast F(p)}\circ F_{\ast p}.}$

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .