Reference request: clean intersection between submanifolds

Question

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 $N_2$:

• $N_1$ 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?

Answer 1

Guillemin and Pollack Differential Topology covers transversality. Perhaps have a look in there for some ideas.

Answer 2

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.

Answer 3

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