On the diffetential of a mapping 2

In the case ${\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m}$ case, there is a linear map, which is “linear approximation” of ${\psi}$ . In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If ${M}$ and ${N}$ are smooth manifolds and ${\psi \colon M \rightarrow N}$ is a smooth map then for each ${m \in M}$ , the map

$\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N$

is defined by

$\displaystyle d\psi(v)(f) = v(f \circ \psi)$

is called the pushforward. Actually,

$\displaystyle d\psi \colon TM \rightarrow TN.$

Suppose that ${\dim{M} \ge \dim{N}}$ and ${f \colon M \rightarrow N}$ is a differentiable mapping. We have

Definition 1 The mapping ${f}$ is called a trivial fibration (differentiable) on ${N}$ if there exists a differential manifold ${F}$ , is called fibre of ${f}$ , and a diffeomorphism

$\displaystyle \phi \colon M \rightarrow N \times F$

such that the following diagram is commutative

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