cotangent lift

Definition

Suppose that is a diffeomorphism. Then the cotangent lift of is the diffeomorphism on the cotangent spaces of and

In picture form on the relevant cotangents spaces:

Connection to tautological 1-form

The cotangent lift of a diffeomorphism pulls the tautological 1-form on back to the tautological form on . In other words,

Proof

Consider points and such that

then it remains to show that

Note that the tautological 1 form is defined such that

So this gives

from here we see that by the definition of the lift that the following diagram commutes

Thus we have

Canonical symplectic form

As a group homomorphism

Since the cotangent lift preserves the tautological 1-form, then it will also preserve the canonical symplectic form . That is, is a symplectomorphism.

Note by the definition of the lift (using the chain rule), we have that

This means in terms of groups that there is a group homomorphism

Non-surjective

The above group homomorphism is NOT surjective. There are symplectomorphisms between cotangent spaces that are not induced from a diffeomorphism of underlying manifolds.