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Let $\mathcal{X}^m, \mathcal{Y}^n,\mathcal{Z}^p$ be manifolds. Let $f : \mathcal{X} \to \mathcal{Z}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be maps such that $f \pitchfork g$. Prove that $$\mathcal{M}=\{(x, y) \in X \times Y : f(x) = g(y)\}$$ is a smooth submanifold of $\mathcal{X} \times \mathcal{Y}$.

It seems what I have attempted is completely wrong...

1LiterTears
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  • Is this a final exam question? I would suggest you do some serious proofreading of what you've typed. Much of it makes no sense. – Ted Shifrin Aug 18 '13 at 02:32
  • Hi @TedShifrin, it is a practice question. – 1LiterTears Aug 18 '13 at 02:34
  • I see the transversality part has much problem, but I can't find where the transversal of two functions are defined....@TedShifrin... :_( – 1LiterTears Aug 18 '13 at 02:37
  • $f,g$ transverse means that, for all $x \in X, y \in Y$ such that $f(x) = g(y) = z \in Z$, it occurs that $T_z(Z) = \operatorname{ran}(Df_x) + \operatorname{ran}(Dg_y)$. Maybe give the problem a try on your own first? A vague hint: study the proof that preimages of regular values are manifolds. – Mike F Aug 18 '13 at 03:37

1 Answers1

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Here's a start. $M = (f\times g)^{-1}(\Delta)$, where $\Delta$ is the diagonal in $Z\times Z$. You will need a linear algebra lemma: If $U$ and $W$ are subspaces of a vector space $V$, then $U+W=V\implies U\times W + \Delta = V\times V$, where here $\Delta$ is the diagonal of $V\times V$.

Ted Shifrin
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