---- > [!theorem] Theorem. ([[tubular neighborhood theorem]]) > Let $M$ be a [[smooth manifold]] and $N$ a smooth submanifold. There is an open neighborhood $U \subset M$ of $N$ in $M$ and a [[homeomorphism]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12wmB9YATQAEnOEwBGcGDkEANAL4g5pdJlz5CKAIzkqtRizYBVRcpAZseAkW2bd9Zq0Qh+JlRfVEyt6vYNO4PMSKujBQAObwRKAAZgBOEAC2SADM1DgQSABM1AAWMHRQSNwMDNQMWGCOIHAQ5YVpdFgMbJCVID76VQC87SAMdGIwDAAKqpYaILFYYTk4riBxiUhkIOkpufmFTq2sHQ5snADGTGi9-YMjYx5OFdiwZxVVUBA4OKHzi0mI2qsZiNkgPIFFoEXZ6fZOI4nM4DIajdxWG5gO5g8ptJzPV7vJQxeJfFZrb57PwcdiHAhhYJyIA > \begin{tikzcd} > \nu_{N \subset M} \arrow[d, "\cup" description, no head, dotted] \arrow[r, "\cong"] & U \arrow[d, "\cup" description, no head, dotted] \\ > s_0 \arrow[r, "="'] & N > \end{tikzcd} > \end{document} > ``` > taking the zero section to $Y$ via the [[identity map|identity]]. Here, $\nu_{N \subset M}$ denotes the [[normal bundle]]. > > ![[Pasted image 20250422124815.png]] > [!proof]- Proof. ([[tubular neighborhood theorem]]) > We won't prove this in our course. ---- #### ----- #### 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 > ```