# 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 $$N_2$$:

• $$N_1$$ 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?

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

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

• Oh! Thanks a lot, I'll have a look! – Klaus Niederkrüger May 13 '20 at 23:48

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.

• If you spot any errors or if you know how to simplify the proof, I would be very happy to know. – Klaus Niederkrüger Oct 18 '19 at 12:11