Push Forward on a Product Manifold.

1. Sep 23, 2014

andresB

Some words before the question.
For two smooth manifolds $M$ and $P$ It is true that
$$T(M\times P)\simeq TM\times TP$$
If I have local coordinates $\lambda$ on $M$ and $q$ on $P$ then ($\lambda$, $q$) are local coordinates on $M\times P$ (right?). This means that in these local coordinates the tanget vectors are of the form $a^{i}\frac{\partial}{\partial\lambda^{i}}+b^{i}\frac{\partial}{\partial q^{i}}$

Now, I can compute push forwards in local coordinates. For example, for a function
$$f(\lambda, q)\rightarrow(\lambda,Q(q,\lambda))$$
Then
$$f^{*}\left(\frac{\partial}{\partial\lambda}\right)=\frac{\partial}{\partial\lambda}+\frac{\partial Q}{\partial\lambda}\frac{\partial}{\partial q}$$
where I just had to do the matrix product of the Jacobian to the column vector $(1,0)^{T}$.

Actual Question.

For a function $f:\, TM\times TP\longrightarrow TM\times TP$
and without using local coordinates what can be said about the Push forward $f^{*}:\, TM\times TP\longrightarrow TM\times TP$ ?.

Particularly interested if the push forward can be descomposed into something in $TM$ product something in $TP$.

2. Sep 23, 2014

WWGD

There is a result that for m in M , n in N, $T_{(m,n)} (M \times N) = T_m M (+) T_n N$ , where $(+)$ is the direct sum of (tangent) vectors. Where by '=' I mean isomorphic.

3. Sep 23, 2014

andresB

Thanks for replying.

Yes, I'm aware of the result, I actually implicitly used it in the example above.

4. Sep 23, 2014

WWGD

I am not sure of what you are looking for, but you can also use properties of duality, since $TM:= \cup (T_p M)^{*}$
Then $T(M\times N) = (T_{(m,n)} M \times N )^{*}=T_m^{*}M (+) T_n^{*}N = TM \times TN$.

Then a finite direct sum is a direct product .

You may want to play with these properties of duals , duals of products, etc. to look for the
result you want.

Last edited: Sep 23, 2014
5. Sep 23, 2014

andresB

I'm not sure to understand what you mean. Either way, What I'm intersted is in the push forward.

If I have local coordinates, like in the example in the OP, for a given a function of $M\times P$ to itself I can compute the push forward of that function. If I do not have coordinates then I have no idea what to do.

For example, let $G$ be a lie group (I'm interested in SU(n)). The left translation is given by $L_{a}b=ab$. The push forward of the left translation aplied to a tangent vector at the identity, $E\in T_{e}G$ , would give a tangent vector at the new group element $(L_{a})_{*}E\in T_{a}G$.

Now, for $(g,g\text{´)}\in G\times G$ I would like to define $L_{(1,h)}(g,g\text{´)}=(g,hg)$, and find the push forward of this function as $$L_{(1,h)*}(E\oplus0)= Something \oplus Something$$ .

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