Knowlpedia

Fiber-preserving map

A smooth map between total spaces that sends fibers to fibers over a base map.

Let E,M,E′,M′E,M,E',M' be and let π:E→M\pi:E\to M and π′:E′→M′\pi':E'\to M' be . A smooth map F:E→E′F:E\to E' is fiber-preserving if there exists a smooth map f:M→M′f:M\to M' such that

π′∘F  =  f∘π.\pi'\circ F \;=\; f\circ \pi.

In this situation one says that FF covers ff, or that FF is a map over ff. If π\pi is surjective, then the base map ff is uniquely determined by FF.

When π\pi and π′\pi' are surjective submersions (i.e. when they define ), a fiber-preserving smooth map is the same structure as a . A basic example is the differential dφ:TM→TNd\varphi:TM\to TN associated to a smooth map φ:M→N\varphi:M\to N, which is fiber-preserving between the and covers φ\varphi.

Examples
  1. Maps between products: given f:M→M′f:M\to M' and a smooth map g:M×F→F′g:M\times F\to F', the map F:M×F→M′×F′F:M\times F\to M'\times F', F(x,u)=(f(x),g(x,u))F(x,u)=(f(x),g(x,u)), is fiber-preserving over ff.
  2. Differential of a map: for any φ:M→N\varphi:M\to N, the differential dφ:TM→TNd\varphi:TM\to TN satisfies τN∘dφ=φ∘τM\tau_N\circ d\varphi=\varphi\circ\tau_M.
  3. Restriction to an open set: for an open inclusion i:U↪Mi:U\hookrightarrow M, the inclusion π−1(U)↪E\pi^{-1}(U)\hookrightarrow E is fiber-preserving over ii.