tautological form

Definition

The tautological 1-form is a canonical differential 1-form that always exists for the cotangent bundle of a smooth manifold .

Coordinate version

Let be a cotangent coordinate chart for . The tautological 1-form is the one form

This definition does not depend on a choice of coordinates. Consider coordinates and . In order to change coordinates we have

This gives

Coordinate free version

For a more coordinate free definition, we can take the canonical projection map

Then we can define the one form as

In picture form we have

so using this we have

Why are these the same

Though it should be pretty clear, here is an explicit reason the two definitions are the same. Given coordinates then we have that the projection induces the map

So following through, using the canonical definition we have

So clearly, .

Relation to sections

Let be a de Rham differential 1-form, then let

be the image of the section. Consider the map, as the 1-form only as a map,

Let be the tautological 1-form on . Then

Proof

Note that the image of is , therefore, for , then . By definition, for . Thus, we can regard which gives . Then we have (using pullback notation throughout)

since is the identity map by definition of a section.

Or actually computing the pullbacks directly gives

This can be summed up with the following commutative diagram