# 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 |
---|---|

Canonical name | ConormalBundle |

Date of creation | 2013-03-22 13:59:09 |

Last modified on | 2013-03-22 13:59:09 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 58A32 |