In differential geometry, one can attach to every point of a smooth (or differentiable) manifold, , a vector space called the cotangent space at . Typically, the cotangent space, is defined as the dual space of the tangent space at , , although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.
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.
Definition as linear functionals
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 M be a smooth manifold and let x be a point in . Let be the ideal of all functions in vanishing at , and let be the set of functions of the form , where . Then and are real vector spaces and the cotangent space is defined as the quotient space .
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
- dfx(Xx) = Xx(f)
where Xx 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 df(X) = X(f). Equivalently, we can think of tangent vectors as tangents to curves, and write
- dfx(γ′(0)) = (f ∘ γ)′(0)
In either case, dfx is a linear map on TxM and hence it is a tangent covector at x.
We can then define the differential map d : C∞(M) → Tx*M at a point x as the map which sends f to dfx. Properties of the differential map include:
- d is a linear map: d(af + bg) = a df + b dg for constants a and b,
- d(fg)x = f(x) dgx + g(x) dfx,
The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function f ∈ Ix (a smooth function vanishing at x) we can form the linear functional dfx as above. Since the map d restricts to 0 on Ix2 (the reader should verify this), d descends to a map from Ix / Ix2 to the dual of the tangent space, (TxM)*. 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 ∈ Tf(x)*N and Xx ∈ TxM. 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 dg) is given by
That is, it is the equivalence class of functions on M vanishing at x determined by g ∘ f.
The k-th exterior power of the cotangent space, denoted Λk(Tx*M), is another important object in differential geometry. Vectors in the kth 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.
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