# conormal bundle

Let $X$ be an immersed submanifold of $M$, with immersion $i:X\to M$. Then as with the normal bundle, we can pull the cotangent bundle back to $X$, forming a bundle $i^{*}T^{*}M$. This has a canonical pairing with $i^{*}TM$, essentially by definition. Since $TX$ is a natural subbundle of $i^{*}TM$, we can consider its annihilator: the subbundle of $i^{*}T^{*}M$ given by

 $\{(x,\lambda)|x\in X,\lambda\in T^{*}_{i(x)}M,\lambda(v)=0\forall v\in T_{x}X\}.$

This subbundle is denoted $N^{*}X$, and called the conormal bundle of $X$.

The conormal bundle to any submanifold is a natural Lagrangian submanifold of $T^{*}M$.

Title conormal bundle ConormalBundle 2013-03-22 13:59:09 2013-03-22 13:59:09 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 58A32