Fibered manifold

A smooth manifold E equipped with a surjective submersion onto a base manifold M.

Fibered manifold

Let $E$ and $M$ be smooth manifolds and let $\pi:E\to M$ be a smooth map . The triple $(E,\pi,M)$ is called a fibered manifold if $\pi$ is a surjective submersion, i.e. $\pi$ is surjective and for every $e\in E$ the differential

d\pi_e:T_eE\longrightarrow T_{\pi(e)}M

is surjective.

For each $x\in M$ , the fiber over $x$ is $E_x:=\pi^{-1}(x)$ . By the submersion theorem, each fiber $E_x$ is an embedded submanifold of $E$ of dimension $\dim E-\dim M$ , and the inclusion $E_x\hookrightarrow E$ identifies

T_eE_x=\ker(d\pi_e).

This kernel is the vertical tangent space at $e$ .

Equivalently, $\pi$ induces a fiberwise surjective bundle map $d\pi:TE\to TM$ between the tangent bundles . A smooth fiber bundle is a fibered manifold equipped with local product charts; vector bundles and principal G-bundles are standard examples.

Examples

Product projection: for any manifolds $M$ and $F$ , the map $\mathrm{pr}_1:M\times F\to M$ is a surjective submersion, hence a fibered manifold.
Tangent bundle: the projection $\tau:TM\to M$ is a surjective submersion.
Any smooth fiber bundle: if $\pi:E\to M$ is a smooth fiber bundle, then $(E,\pi,M)$ is automatically a fibered manifold.