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For two smooth manifolds [itex]M[/itex] and [itex]P[/itex] It is true that

[tex]T(M\times P)\simeq TM\times TP [/tex]

If I have local coordinates [itex]\lambda[/itex] on [itex]M[/itex] and [itex]q[/itex] on [itex]P[/itex] then ([itex]\lambda[/itex], [itex]q[/itex]) are local coordinates on [itex]M\times P[/itex] (right?). This means that in these local coordinates the tanget vectors are of the form [itex]a^{i}\frac{\partial}{\partial\lambda^{i}}+b^{i}\frac{\partial}{\partial q^{i}}[/itex]

Now, I can compute push forwards in local coordinates. For example, for a function

[tex]f(\lambda, q)\rightarrow(\lambda,Q(q,\lambda))[/tex]

Then

[tex]f^{*}\left(\frac{\partial}{\partial\lambda}\right)=\frac{\partial}{\partial\lambda}+\frac{\partial Q}{\partial\lambda}\frac{\partial}{\partial q}[/tex]

where I just had to do the matrix product of the Jacobian to the column vector [itex](1,0)^{T}[/itex].

Actual Question.

For a function [itex]f:\, TM\times TP\longrightarrow TM\times TP[/itex]

and without using local coordinates what can be said about the Push forward [itex]f^{*}:\, TM\times TP\longrightarrow TM\times TP[/itex] ?.

Particularly interested if the push forward can be descomposed into something in [itex]TM[/itex] product something in [itex]TP[/itex].

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# Push Forward on a Product Manifold.

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