# Cotangent space

In differential geometry, one can attach to every point *x* of a smooth (or differentiable) manifold a vector space called the **cotangent space** at *x*. Typically, the cotangent space is defined as the dual space of the tangent space at *x*, although there are more direct definitions (see below). The elements of the cotangent space are called **cotangent vectors** or **tangent covectors**.

## Properties

All cotangent spaces on a connected manifold have the same dimension, equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold.

The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

## Formal definitions

### Definition as linear functionals

Let *M* be a smooth manifold and let *x* be a point in *M*. Let *T*_{x}*M* be the tangent space at *x*. Then the cotangent space at *x* is defined as the dual space of *T*_{x}*M*:

*T*_{x}^{*}*M*= (*T*_{x}*M*)^{*}

Concretely, elements of the cotangent space are linear functionals on *T*_{x}*M*. That is, every element α ∈ *T*_{x}^{*}*M* is a linear map

- α :
*T*_{x}*M*→**F**

where **F** is the underlying field of the vector space being considered. For example, the field of real numbers. The elements of *T*_{x}^{*}*M* are called cotangent vectors.

### Alternative definition

In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of equivalence classes of smooth functions on *M*. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x. The cotangent space will then consist of all the possible first-order behaviors of a function near x.

Let *M* be a smooth manifold and let *x* be a point in *M*. Let *I*_{x} be the ideal of all functions in C^{∞}(*M*) vanishing at *x*, and let *I*_{x}^{2} be the set of functions of the form , where *f*_{i}, *g*_{i} ∈ *I*_{x}. Then *I*_{x} and *I*_{x}^{2} are real vector spaces and the cotangent space is defined as the quotient space *T*_{x}^{*}*M* = *I*_{x} / *I*_{x}^{2}.

This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry. The construction also generalizes to locally ringed spaces.

## The differential of a function

Let *M* be a smooth manifold and let *f* ∈ C^{∞}(*M*) be a smooth function. The differential of *f* at a point *x* is the map

- d
*f*_{x}(*X*_{x}) =*X*_{x}(*f*)

where *X*_{x} is a tangent vector at *x*, thought of as a derivation. That is is the Lie derivative of *f* in the direction *X*, and one has d*f*(*X*)=*X*(*f*). Equivalently, we can think of tangent vectors as tangents to curves, and write

- d
*f*_{x}(γ′(0)) = (*f*o γ)′(0)

In either case, d*f*_{x} is a linear map on *T*_{x}*M* and hence it is a tangent covector at *x*.

We can then define the differential map d : C^{∞}(*M*) → *T*_{x}^{*}*M* at a point *x* as the map which sends *f* to d*f*_{x}. Properties of the differential map include:

- d is a linear map: d(
*af*+*bg*) =*a*d*f*+*b*d*g*for constants*a*and*b*, - d(
*fg*)_{x}=*f*(*x*)d*g*_{x}+*g*(*x*)d*f*_{x},

The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function *f* ∈ *I*_{x} (a smooth function vanishing at *x*) we can form the linear functional d*f*_{x} as above. Since the map d restricts to 0 on *I*_{x}^{2} (the reader should verify this), d descends to a map from *I*_{x} / *I*_{x}^{2} to the dual of the tangent space, (*T*_{x}*M*)^{*}. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.

## The pullback of a smooth map

Just as every differentiable map *f* : *M* → *N* between manifolds induces a linear map (called the *pushforward* or *derivative*) between the tangent spaces

every such map induces a linear map (called the *pullback*) between the cotangent spaces, only this time in the reverse direction:

The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:

where θ ∈ *T*_{f(x)}^{*}*N* and *X*_{x} ∈ *T*_{x}*M*. Note carefully where everything lives.

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let *g* be a smooth function on *N* vanishing at *f*(*x*). Then the pullback of the covector determined by *g* (denoted d*g*) is given by

That is, it is the equivalence class of functions on *M* vanishing at *x* determined by *g* o *f*.

## Exterior powers

The *k*-th exterior power of the cotangent space, denoted Λ^{k}(*T*_{x}^{*}*M*), is another important object in differential geometry. Vectors in the *k*th exterior power, or more precisely sections of the *k*-th exterior power of the cotangent bundle, are called differential *k*-forms. They can be thought of as alternating, multilinear maps on *k* tangent vectors.
For this reason, tangent covectors are frequently called *one-forms*.

## References

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