Reference request: clean intersection between submanifolds Let $N_1,N_2$ be two smooth submanifolds in an ambient manifold $M$.
There are two definitions of clean intersection between $N_1$ and $N_2$:


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*$N_1$ and $N_2$ intersect cleanly if $N_1\cap N_2$ is a smooth submanifold such that $T_x(N_1\cap N_2) = T_xN_1\cap T_xN_2$ for every $x\in N_1\cap N_2$.

*There is  around every $x\in N_1\cap N_2$ a chart $(U,\phi)$ such that $\phi(N_1\cap U)$ and $\phi(N_2\cap U)$ are open subsets of affine subspaces $V_1, V_2 \subset \mathbb{K}^d$.


No doubt, using the implicit function theorem you are able to produce an elegant (or not so elegant) proof that both definitions are equivalent.  But if possible, I would like to have a reference (to a standard textbook?). Do you know any?
 A: Guillemin and Pollack Differential Topology covers transversality.  Perhaps have a look in there for some ideas.
A: I didn't find a reference so for completeness I'll give instead a proof.  This is of course all quite elementary, but I plan to include it in the appendix of a preprint, I'll hopefully will soon upload.
Proof:  If $N_1$ and $N_2$ look in a chart like affine subspaces then the other statements follow directly.
Assume now that $N_1$ and $N_2$ are of dimension $k_1$ and $k_2$ respectively and that $L = N_1 \cap N_2$ is in a neighborhood of a point $x$ a submanifold (of $M$) of dimension $k$.  It is easy to see that $L$ is close
to $x$ also a submanifold of $N_1$, thus we can find a chart $(U_x , \varphi_x)$ with $\varphi_x(x) = \mathbf{0}_d$ in which $N_1 \cap U_x$ is represented by $\bigl(\mathbb{R}^{k_1}\times \{\mathbf{0}_{d−k_1}\}\bigr) \cap  \varphi_x(U_x)$, and $L\cap U_x$ corresponds to $\bigl(\mathbb{R}^k \times \{\mathbf{0}_{d−k}\}\bigr) \cap \varphi_x (U_x)$.
Furthermore we can assume that $\varphi_x(N_2\cap U_x)$ is along $\varphi_x (L\cap U_x)$ tangent to $\mathbb{R}^k  \times \{\mathbf{0}_{k_1 −k}\} \times \mathbb{R}^{k_2 −k} \times \{\mathbf{0}_{d−k_1−k_2 +k}\}$, for example, we may choose a frame of $\Bigl.T N_2\Bigr|_L$ along $L$, extract with Gram-Schmidt a subframe that is orthogonal to $TL$.
By our assumptions, this subframe will be transverse to $\Bigl.TN_1\Bigr|_L$, thus we can map it with $D\varphi_x$ to $\mathbb{R}^d$, smoothly extend it along $\bigl(\mathbb{R}^{k_1} \times \{\mathbf{0}_{d−k_1}\}\bigr)\cap \varphi_x(U_x)$,
giving vector fields $X_1, \dotsc, X_{k_2−k}$ which we complete by $X_{k_2−k+1}, \dotsc, X_{d−k_1}$ to obtain a frame along $\bigl(\mathbb{R}^{k_1} \times \{\mathbf{0}_{d−k_1}\}\bigr)\cap \varphi_x (U_x)$ spanning a transverse bundle to $\mathbb{R}^{k_1}\times \{\mathbf{0}_{d−k_1}\}$.
Denote $(y_1,\dotsc, y_d)$ by $\mathbf{y}$, $(y_1, \dotsc , y_k)$ by $\mathbf{y}_L$, $(y_{k+1}, \dotsc, y_{k_1 −k})$ by $\mathbf{y}_{N_1}$, and $(y_{k_1 +1}, \dotsc, y_{k_1 + k_2 −k})$ by $\mathbf{y}_{N_2}$ and define the map
$$ \Psi(\mathbf{y}) = \bigl(\mathbf{y}_L ; \mathbf{y}_{N_1} ; y_{k_1 +1} \cdot X_1 (\mathbf{y}_L ; \mathbf{y}_{N_1}), \dotsc , y_d \cdot X_{d−k_1} (\mathbf{y}_L ; \mathbf{y}_{N_1}) \bigr)$$
which is a local diffeomorphism on a neighborhood of the origin in $\mathbb{R}^d$.
Composing $\varphi_x$ with $\Psi^{-1}$ and possibly shrinking the size of $U_x$ gives a new chart with the desired properties.   For simplicity,  we denote this new chart also with  $(U_x , \varphi_x)$.
We can parametrize $\varphi_x (N_2 \cap U_x)$ as a graph over $\mathbb{R}^k \times \{\mathbf{0}_{k_1 −k}\} \times \mathbb{R}^{k_1+k_2 −k} \times \{\mathbf{0}_{d−k_1−k_2 +k} \}$.  For this restrict the projection $\mathbf{y}\mapsto (\mathbf{y}_L ; \mathbf{y}_{N_2})$ to the submanifold $\varphi_x(N_2 \cap U_x)$.   Clearly this restriction
is (after possibly decreasing again the size of $U_x$) a submersion, and for dimension reasons thus also a local diffeomorphism.  This allows us to find a map
$$ F(\mathbf{x}) = \Bigl(\mathbf{x}_L; F_{k+1} (\mathbf{x}), \dotsc , F_{k_1} (\mathbf{x}); \mathbf{x}_{N_2} ; F_{k_1+k_2 −k+1}(\mathbf{x}), \dotsc , F_d (\mathbf{x})\Bigr)$$
with $\mathbf{x} = (x_1, \dotsc , x_{k_2})$, $\mathbf{x}_L = (x_1 , \dotsc , x_k)$, and $\mathbf{x}_{N_2} = (x_{k+1}, \dotsc , x_{k_2})$ that parametrizes $\varphi_x(N_2 \cap U_x)$.
Furthermore, all $F_j$ vanish at the points $(\mathbf{x}_L , \mathbf{0}_{k_2 −k})$ that correspond to the intersection between $N_1$ and $N_2$.
Define now a diffeomorphism $\Phi$ between two neighborhoods of the origin in $\mathbb{R}^d$ by
$$ \Phi(\mathbf{y}) = \mathbf{y} − \Bigl(\mathbf{0}_k ; F_{k+1}(\mathbf{y}_L ; \mathbf{y}_{N_2}), \dotsc , F_{k_1} (\mathbf{y}_L; \mathbf{y}_{N_2}); \mathbf{0}_{k_2 −k} ;
F_{k_1 +k_2−k+1} (\mathbf{y}_L; \mathbf{y}_{N_2}), \dotsc , F_d(\mathbf{y}_L; \mathbf{y}_{N_2})\Bigr) .$$
Composing $\varphi_x$ with $\Phi$ (and possibly shrinking the size of $U_x$), we find a new chart that we have constructed in such a way that $N_2 \cap U_x$ is flattened to the linear subspace  $\mathbb{R}^k \times \{\mathbf{0}_{k_1 −k}\} \times \mathbb{R}^{k_2 −k} \times \{\mathbf{0}_{d−k_1 −k_2 +k}\}$.  Furthermore, $N_1$ lies in the initial chart $(U_x, \varphi_x)$ in $\mathbb{R}^{k_1} \times \{\mathbf{0}_{d−k_1}\}$ along which $F_j$ vanishes. Thus $\Phi$ preserves this subspace pointwise, and it follows that $\Phi\circ \varphi_x$ is a chart with the desired properties.
A: so far I have only found the following reference:
Hörmander - The Analysis of Linear PD Operators III, appendix C.3.
Hope that will help, Jan
