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 $T^{*}M$.