Cotangent space

In differential geometry, a branch of mathematics, the cotangent space is a vector space, which is zugeordert a point of a differentiable manifold. It is the dual space of the corresponding tangent space.

Definition

Let be a differentiable manifold and its tangent space at the point. Then, the cotangent is defined as the dual space of. That is, the cotangent consists of all linear shapes on the tangent space.

Alternative definition

In the following, a different approach is shown in which the dual space is defined directly, without reference to the tangent.

It should be a -dimensional differentiable manifold. Next are the set of all smooth curves through

And the set of the smooth functions, which are defined in an area of ​​:

If we denote by the following equivalence relation on

Then the factor space of the vector space of germs is over. about

Is then defined a formal combination that is linear in the first component. Now

A linear subspace of, more precisely with respect to the null space and

Is the -dimensional cotangent space at the point. For the Kotangentialvektor to write well.

Related to the tangent space

With the above definition, one can define an equivalence relation as follows:

The factor space just describes the -dimensional tangent space.

Now form a basis of, so you can select for each basis vector a representative. is a differentiable map and for each can get a curve

Define, where the -th unit vector in. because of

And are dual to each other and to write for it.

Justification of the spellings

Let, , be an arbitrary function and for the curves, the canonical basis vectors. Then, in the above notation:

Thus, the notation is justified.

Next is the linear map just the total differential. It is therefore the notation is justified.