---- > [!definition] Definition. ([[transverse submanifolds]]) > Let $M$ be a [[smooth manifold]] and $N,L \subset M$ smooth [[embedded submanifold|submanifolds]]. We say $N,L$ are **transverse** if for all $x \in N \cap L$, the [[tangent space to a manifold|tangent spaces]] sum nondegenerately: $T_{x}N + T_{x}L=T_{x}M$ (note that this is not a direct sum). > In this case, $N \cap W$ is again [[smooth manifold|smooth]], and $\begin{align} (\nu_{N \cap L \subset M} )_{x} &\cong (\nu_{N \subset M})_{x} \oplus (\nu_{L \subset M})_{x} \\ \nu_{N \cap L \subset M} &= i^{*}_{N}(\nu_{N \subset M}) \oplus i^{*}_{L}(\nu_{L \subset M}) \end{align}$ > >![[Pasted image 20250523112622.png|300]] > [!basicexample] > With $\mathbb{R}^{2}$ the ambient [[manifold]]: > ![[Pasted image 20250523103424.png]] > > - Any two disjoint submanifolds are transverse > - Two $1$-manifolds in $\mathbb{R}^{3}$ are transverse iff they are disjoint > - Two planes in $\mathbb{R}^{3}$ are transverse iff they're not equal ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```