Cotangent space explained

In differential geometry, the cotangent space is a vector space associated with a point

on a smooth (or differentiable) manifold

; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space,

	*
T
	xlM

is defined as the dual space of the tangent space at x

,

T_xlM

, 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 at points 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

be a smooth manifold and let

be a point in

. Let

T_xlM

be the tangent space at

. Then the cotangent space at x is defined as the dual space of

	*
T
	xlM

=(T_xlM)^*

Concretely, elements of the cotangent space are linear functionals on

T_xlM

. That is, every element

\alpha\in

	*
T
	xlM

is a linear map

\alpha:T_xlM\toF

where

is the underlying field of the vector space being considered, for example, the field of real numbers. The elements of

	*
T
	xlM

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

. Informally, we will say that two smooth functions f and g are equivalent at a point

if they have the same first-order behavior near

, analogous to their linear Taylor polynomials; two functions f and g have the same first order behavior near

if and only if the derivative of the function f − g vanishes at

. The cotangent space will then consist of all the possible first-order behaviors of a function near

Let

be a smooth manifold and let x be a point in

. Let

I_x

be the ideal of all functions in

C^infty(lM)

vanishing at

, and let

	2
I
	x

be the set of functions of the form

\sum_i f_i g_i

, where

f_i,g_i\inI_x

. Then

I_x

and

	2
I
	x

are both real vector spaces and the cotangent space can be defined as the quotient space

	*
T
	xlM

	2
I
	x

by showing that the two spaces are isomorphic to each other.

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

be a smooth manifold and let

f\inC^infty(M)

be a smooth function. The differential of

at a point

is the map

df_x(X_x)=X_x(f)

where

X_x

is a tangent vector at

, thought of as a derivation. That is

X(f)=l{L}_Xf

is the Lie derivative of

in the direction

, and one has

df(X)=X(f)

. Equivalently, we can think of tangent vectors as tangents to curves, and write

df_{x(\gamma'(0))=(f\circ\gamma)'(0)}

In either case,

df_x

is a linear map on

T_xM

and hence it is a tangent covector at

We can then define the differential map

d:C^infty(M)\to

	*(M)
T
	x

at a point

as the map which sends

df_x

. Properties of the differential map include:

is a linear map:

d(af+bg)=adf+bdg

for constants

and

d(fg)_x=f(x)dg_x+g(x)df_x

The differential map provides the link between the two alternate definitions of the cotangent space given above. Since for all

f\in

	2
I
	x

there exist

g_i,h_i\inI_x

such that

f=\sum_i g_i h_i

, we have,

\mathrm d f_x

=\sum_i \mathrm d (g_i h_i)_x =

\sum_i (g_i(x)\mathrm d(h_i)_x+\mathrm d(g_i)_x h_(x))=

\sum_i (0\mathrm d(h_i)_x+\mathrm d(g_i)_x 0)=0

i.e. All function in

	2
I
	x

have differential zero, it follows that for every two functions

f\in

	2
I
	x

g\inI_x

, we have

d(f+g)=d(g)

. We can now construct an isomorphism between

	*
T
	xlM

and

	2
I
	x

by sending linear maps

\alpha

to the corresponding cosets

\alpha+

	2
I
	x

. Since there is a unique linear map for a given kernel and slope, this is an isomorphism, establishing the equivalence of the two definitions.

The pullback of a smooth map

Just as every differentiable map

f:M\toN

between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces


f
	*

\colonT_xM\toT_f(x)N

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

f^*\colon

	*
T
	f(x)

N\to

	*
T
	x

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

(f^*\theta)(X_x)=


\theta(f
	*

X_x),

where

\theta\in

	*N
T
	f(x)

and

X_x\inT_xM

. 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

be a smooth function on

vanishing at

f(x)

. Then the pullback of the covector determined by

(denoted

) is given by

f^*dg=d(g\circf).

That is, it is the equivalence class of functions on

vanishing at

determined by

g\circf

Exterior powers

The

-th exterior power of the cotangent space, denoted

	*l{M})
Λ
	x

, is another important object in differential and algebraic geometry. Vectors in the

-th exterior power, or more precisely sections of the

-th exterior power of the cotangent bundle, are called differential

-forms. They can be thought of as alternating, multilinear maps on

tangent vectors. For this reason, tangent covectors are frequently called one-forms