---- > [!definition] Definition. ([[normal bundle]]) > Let $M$ be a [[smooth manifold]], $N \subset_{i} M$ a smooth [[embedded submanifold|submanifold]], so that $TN$ is a [[vector subbundle]] of the [[pullback of a vector bundle|pullback]] $i^{*}TM$.[^1] > > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQAVAWRAF9T0mXPkIoyARmp0mrdj36DseAkXGlJNBizaIQAOT4CQGJSNUUpW2bqwA9AFTdDi4SpTkLmmTs4HeUmCgAc3giUAAzACcIAFskABYaHAgkAGYaAAsYeih2SDA2GkYsAvYoCBwcQJAvbXYAHXq4ZgAjOBgcGpBGehaYRgAFIWVREBLsWGcQKNi0pJTEYgVp6LjF+aQ1EBx6LEZ2DIgIAGsuqx8sLp6+weGzXUisIIzO5Zm1j22F8X9eIA > \begin{tikzcd} > TN \arrow[r, "\subset" description, no head, dotted] & i^*TM \arrow[d] & TM \arrow[d] \\ > & N \arrow[r, "i"', hook] & M > \end{tikzcd} > \end{document} > ``` The **normal bundle** of $N$ in $M$ is the [[quotient vector bundle|quotient bundle]] $\nu_{N \subset M}:= \frac{i^{*}TM}{TN}.$ When there is an [[inner product on a vector bundle|inner product]], the normal bundle can be understood as an [[orthogonal vector subbundle]] $(TN)^{\perp}$, which underlies the name. [[vector field|Sections]] of $\nu_{N \subset M}$ are called **normal vector fields**. > The [[tubular neighborhood theorem]] says that the $\varepsilon$-neighborhood about a smooth submanifold $N \subset M$ always looks like the [[vector bundle]] $\nu_{N \subset M}$. ---- #### [^1]: With old notation $M=X$, $N=Y$: $\begin{align} i^{*}TX &=\{ \big(y , (x, v_{x}) \big) \in Y \times TX : i(y)=\pi(x) \} \\ &=\{ \big( y, (x, v_{x}) \big) \in Y \times T X : y=x\}\\ & \cong \{ (y, v_{y}) : y \in Y, v_{y} \in T_{y}X \} \\ & '\supset'\{ y, v_{y}: y \in Y, v_{y} \in T_{y} Y \} \\ &= TY. \end{align}$ (That's way too explicit, I don't know why I wrote it out.) ---- #### 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 > ```