Cotangent space

In differential geometry, one can attach to every point formula_1 of a smooth (or differentiable) manifold, formula_2, a vector space called the cotangent space at "formula_1". Typically, the cotangent space, formula_4 is defined as the dual space of the tangent space at "formula_1", formula_6, 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 "formula_2" be a smooth manifold and let "formula_1" be a point in "formula_2". Let "formula_6" be the tangent space at "formula_1". Then the cotangent space at "x" is defined as the dual space of "formula_6":
Concretely, elements of the cotangent space are linear functionals on "formula_6". That is, every element formula_15 is a linear map
where formula_17 is the underlying field of the vector space being considered, for example, the field of real numbers. The elements of formula_4 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 "formula_2". Informally, we will say that two smooth functions "f" and "g" are equivalent at a point "formula_1" if they have the same first-order behavior near "formula_1", analogous to their linear Taylor polynomials; two functions "f" and "g" have the same first order behavior near "formula_1" if and only if the derivative of the function "f"-"g" vanishes at "formula_1". The cotangent space will then consist of all the possible first-order behaviors of a function near "formula_1".
Let "M" be a smooth manifold and let "x" be a point in "formula_2". Let formula_26be the ideal of all functions in formula_27 vanishing at "formula_1", and let formula_29 be the set of functions of the form formula_30, where formula_31. Then "formula_26" and formula_29 are real vector spaces and the cotangent space is defined as the quotient space formula_34.
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 be a smooth function. The differential of "f" at a point "x" is the map
where "X""x" is a tangent vector at "x", thought of as a derivation. That is formula_35 is the Lie derivative of "f" in the direction "X", and one has . Equivalently, we can think of tangent vectors as tangents to curves, and write
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 at a point "x" as the map which sends "f" to d"f""x". Properties of the differential map include:
The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function (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 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 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 and . 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 .
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".