Transverse Manifolds. Intersection is a Manifold again. 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?

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.