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I have just proved that: Given $M, M', N, N' $ smooth manifolds and $f : M \longrightarrow M'$ and $g : N\longrightarrow N'$ are differentiable k class maps then:

$f \times g : M \times N \longrightarrow M' \times N'$ is a differentiable k class map .

But I would like to know if the other implication is also true, or is it possible to find a counterexample.

J. Salieri
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1 Answers1

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The other implication is simple: If $f \times g$ is $C^k$, then picking $n_0 \in N$, since $i_M : M \to M \times N : m \mapsto (m, n_0)$ is $C^\infty$ and so is $\pi: M' \times N' \to M' : (u, v) \mapsto u$, we have that $$ \pi \circ (f \times g) \circ i_M $$ is $C^k$ as well. But that map is just $f$.

John Hughes
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