conormal bundle

Definition

Let be a n-dimensional smooth manifold, and a k-dimensional submanifold. The conormal space at is

or equivalently

where is a linear subspace of .

The conormal bundle of is the bundle of the conormal spaces for . That is

Dimension

The conormal bundle is an -dimensional submanifold of .

Proof

Let be a coordinate chart on adapted to . Thus, on . Then we may consider the basis for for

In which case, for , whereas the other tangent vectors are non-zero. Thus, in , the annihilator of has the first co-basis vectors to be 0, so the dimension will be .

Thus, is .

Importance

The conormal bundle is a good example of Lagrangian submanifolds of the cotangent bundle.