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Suppose we have two manifolds $A,B$ in $\mathbb{R}^n$. I heard that the intersection $A\cap B$ is again a manifold in $\mathbb{R}^n$ if $A$ and $B$ intersect transversally in any point of the intersection (i.e the the tangent spaces fulfill $T_A(x)+T_B(x)=\mathbb{R}^n$ for any $x\in A\cap B$)

Moreover, for the tangent space, we have $T_{A\cap B}(x)=T_A(x)\cap T_B(x)$.

Do you know any references for these results or do you know how to prove them?

Kluyvert
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1 Answers1

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This result is a generalization of the pre-image theorem (which you can find on page 21 of Guillemin & Pollack) which states

If $y$ is a regular value of $f: X \to Y$, then the preimage $f^{-1}(y)$ is a submanifold of $X$, with $\dim f^{-1}(y) = \dim X - \dim Y$.

Given two manifolds that intersect transversely, we can generalize this to (from page 28 of Guillemin & Pollack)

If $f:X \to Y$ is transverse to a submanifold $Z \subset Y$, then the preimage $f^{-1}(Z)$ is a submanifold in $X$. Moreover the codimensions of $Z$ and $f^{-1}(Z)$ are equal

Lastly, suppose we have submanifolds $X$ and $Y$ of $\mathbb{R}^n$. $X$ and $Y$ intersecting transversely is the same as saying the inclusion $i:X \to \mathbb{R}^n$ is transverse to $Y$. So, we get that $i^{-1}(Y)$ is a submanifold of $X$. But $i^{-1}(Y) = X \cap Y$. So we get that $X \cap Y$ is indeed a manifold.

Osama Ghani
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  • Thank you very much for your instant reply. I Just didn't get what so you mean by the variable Z in your last Part... – Kluyvert May 31 '18 at 18:02
  • My bad, I meant to write $Y$ instead of $Z$! – Osama Ghani May 31 '18 at 18:06
  • Now I see, thank you! Thus, my question ist more or less a corollary. Is it also named somehow? – Kluyvert May 31 '18 at 18:09
  • I don't know of a standard name. It's a standard result in differential topology and is used a lot. The mantra usually goes "The codimension of the intersection of two manifolds is the sum of their codimensions" or "codimensions add under intersection", but I don't know a name for the result. We use it frequently because every smooth map can be homotoped to be made transverse to a certain subspace, and so for example, two manifolds whose dimensions add to the dimension of the ambient space intersect in only finitely many points. This marks the beginning of degree theory. – Osama Ghani May 31 '18 at 18:14
  • ok I see. But the fact that the intersection of the tangent spaces is the tangent space of the intersection is not implied by your solution, or is it just me who can't see the implication? – Kluyvert May 31 '18 at 18:18
  • You’re right! It doesn’t immediately show that. There’s a proof here https://math.stackexchange.com/questions/439071/the-tangent-space-to-the-intersection-is-the-intersection-of-the-tangent-spaces hope that helps! If you found my answer helpful, feel free to indicate that by voting or accepting my answer :) – Osama Ghani May 31 '18 at 18:43